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+Subgap states and quantum phase transitions in one-dimensional
+superconductor-ferromagnetic insulator heterostructures
+Javier Feijoo,1, 2 An´ıbal Iucci,1, 2 and Alejandro M. Lobos3, 4
+1Instituto de F´ısica La Plata - CONICET, Diag 113 y 64 (1900) La Plata, Argentina
+2Departamento de F´ısica, Universidad Nacional de La Plata, cc 67, 1900 La Plata, Argentina.
+3Instituto Interdisciplinario de Ciencias B´asicas (CONICET-UNCuyo)
+4Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Cuyo, 5500 Mendoza, Argentina
+We
+theoretically
+study
+the
+spectral
+properties
+of
+a
+one
+dimensional
+semiconductor-
+superconductor-ferromagnetic insulator (SE-SU-FMI) hybrid nanostructure, motivated by recents
+experiments where such devices have been fabricated using epitaxial growing techniques. We model
+the hybrid structure as a one-dimensional single-channel semiconductor nanowire under the si-
+multaneous effect of two proximity-induced interactions: superconducting pairing and a (spatially
+inhomogeneous) Zeeman exchange field. The coexistence of these competing interactions generates
+a rich quantum phase diagram and a complex subgap Andreev bound state (ABS) spectrum. By
+exploiting the symmetries of the problem, we classify the solutions of the Bogoliubov-de Gennes
+equations into even and odd ABS with respect to the spatial inversion symmetry x → −x. We
+find the ABS spectrum of the device as a function of the different parameters of the model: the
+length L of the coexisting SU-FMI region, the induced Zeeman exchange field h0, and the induced
+superconducting coherence length ξ. In particular we analyze the evolution of the subgap spectrum
+as a function of the length L. Interestingly, we have found that depending on the ratio h0/∆, the
+emerging ABS can eventually cross below the Fermi energy at certain critical values Lc, and induce
+spin-and fermion parity-changing quantum phase transitions. We argue that this type of device
+constitute a promising highly-tunable platform to engineer subgap ABS.
+I.
+INTRODUCTION
+The interplay of superconductivity and magnetism at
+the microscopic scale has attracted a great deal of at-
+tention in recent years [1–4].
+For instance, the Yu-
+Shiba-Rusinov (YSR) states [5–7] arising from the ex-
+change interaction of an atomic magnetic moment in con-
+tact with a superconductor, have been proposed as fun-
+damental building blocks to engineer quantum devices
+with topologically non-trivial ground states. In partic-
+ular, the so-called “Shiba chains” (i.e., one-dimensional
+arrays of magnetic atoms deposited on top of a clean
+superconductor) are systems predicted to support Ma-
+jorana zero-modes at the ends of the chain [8–10], and
+could be used in topologically-protected quantum com-
+putation schemes. Low-temperature scanning-tunneling
+microscopy (STM) experiments have confirmed the pres-
+ence of intruiguing zero-energy end-modes [11–17].
+Other systems where the competition of superconduc-
+tivity and magnetism at the nanoscale generates ex-
+otic subgap states are superconductor (SU)- ferromag-
+net (FM) heterostructures, such as SU-FM-SU Josephson
+junctions and SU-FM proximity devices [18, 19]. Subgap
+states generated in these structures are usually referred
+to as Andreev bound states (ABS). More recently, a novel
+class of hybrid device, i.e., semiconductor (SE) nanowire
+systems combined with superconductors and ferromag-
+netic insulator (FMI) materials have been fabricated us-
+ing molecular-beam epitaxy techniques [20, 21]. These
+SE-SU-FMI hybrid structures allow to build nanostruc-
+tures with specific tailored properties which are impossi-
+ble to obtain with the isolated individual components.
+Despite the evident differences between the abovemen-
+x
+z
+SC Bulk
+FMI
+Semiconductor
+L
+x
+z
+L
+h0
+Magnetic profile
+FIG. 1. Schematic representation of the SC-FMI heterostruc-
+ture.
+tioned physical systems, from the theoretical perspective
+they can be described within the same unified theoretical
+model combining superconductivity and local exchange
+fields at the microscopic scale.
+The emerging subgap
+states (which can be referred to as either YSR states or
+ABS, depending on the context) appear symmetrically
+around the Fermi level EF , and localize spatially around
+the impurity or the FM region.
+Their energy-position
+within the gap depend on the value of the exchange
+field and on other experimental parameters.
+Interest-
+ingly, whenever one of these states crosses EF , a spin-
+and parity-changing quantum phase transition, usually
+arXiv:2301.03967v1  [cond-mat.supr-con]  10 Jan 2023
+
+2
+known as the “0 − π” phase transition, occurs [1, 22].
+In the case of atomic “Shiba impurities” or ultra-short
+SU-FM-SU junctions (i.e., junctions in which the length
+L of the FM region is much smaller than λF , the Fermi
+wavelength of the superconductor [23]), it is customary
+to consider the magnetic scatterer as a point-like classical
+spin S located at the point R0, interacting via a contact
+s-d exchange interaction HZ = J(r) S·s(r) with the host
+superconducting electrons [6]. Here J(r) = J0δ(r − R0)
+is the local exchange potential and s(r) is the spin den-
+sity vector of the electronic fluid. Subsequent theoretical
+works considered atomic-sized systems with finite- (al-
+beit short-ranged) exchange interactions with spherical
+symmetry [7, 24–26]. In that case, theory predicts the
+existence of multiple YSR states labelled by their orbital
+momentum ℓ, a prediction that has been recently ob-
+served in STM experiments [27–29].
+The behavior of subgap states and the associated 0−π
+quantum phase transitions has also been studied in the
+opposite limit L ≫ λF in the context of ballistic SU-
+FM-SU Josephson junctions with generic spin-dependent
+fields in the sandwiched region [30–32]. In this case the
+results differ from the well-known results of YSR states
+due to the finite extension of the magnetic profile. In
+particular, the subgap spectrum of long SU-FM-SU junc-
+tions with zero phase difference is known to be double
+degenerate [19, 31], showing the inherent complexity of
+these hybrid heterostructures. On the experimental side,
+the possibility to engineer and control the position of the
+subgap states by a modification of the fabrication para-
+maters (e.g., the length L or exchange field h0 via dif-
+ferent FM materials) opens interesting perspectives for
+potential electronic devices, where the precise knowledge
+of the subgap spectrum is crucial to control their trans-
+port properties.
+Motivated by the experimental developments men-
+tioned above, in this work we study the subgap states
+emerging in one-dimensional (1D) SE-SU-FMI het-
+erostructures where the SU and the FMI layers simul-
+taneously generate coexisting proximity-induced pairing
+and exchange interactions over a finite and arbitrary
+length L in the SE nanowire, as schematically shown in
+Fig. 1. This coexistence is a crucial aspect of this device,
+which makes it unique and different from the abovemen-
+tioned SU-FM-SU junctions, where such overlap occurs
+only at the SU-FM interface. Our main goal in this work
+is to study and understand the behavior of the subgap
+ABS in this device as a function of the experimentally rel-
+evant parameters of the model, i.e., the length L of the
+FMI region and the magnitude of the induced exchange
+field h0. As mentioned above, a device similar to that
+shown in Fig. 1 has been recently experimentally real-
+ized in SE nanowires with epitaxially-grown SU and FMI
+layers [20, 21]. While the main interest of that work was
+the fabrication of a device with non-trivial topological SU
+ground state hosting Majorana zero modes, here we will
+study the regime of parameters favoring a topologically-
+trivial ground state.
+As we will show below (see Sec.
+II), this case is already very complex and rich as a result
+of the antagonistic SU and FM interactions and, to the
+best of our knowledge, the detailed behavior of subgap
+states and the quantum phase diagram emerging in such
+a system have not been explicitly studied before.
+The article is organized as follows. In Section II, we
+introduce the model representing a 1D SE-SU-FMI hy-
+brid nanowire, discuss the solution to the Bogoliubov-
+de Gennes equations for the subgap states, and derive
+a generic equation for the subgap spectrum.
+In Sec-
+tion III, we analyze the results in two specific limits,
+where we recover well-known results: a) the semiclassical
+limit, where the superconducting coherence length ξ is
+much larger than the Fermi wavelength λF , and b) the
+atomic YSR limit, in which the exchange-field induced
+by the FMI region becomes a delta-function potential:
+i.e., infinitesimally narrow (L ≪ λF ), and infinitely deep
+(h0 ≫ EF ), in such a way that the product h0.L = J
+is kept constant.
+In both cases, well-known analytical
+solutions to the subgap spectrum can be recovered. In
+addition, we numerically solve the characteristic equation
+for the subgap states and provide a generic description
+of the subgap spectrum, not restricted to any of these
+limits.
+We find a rich behaviour of the subgap ABS,
+where the competing FM exchange and SU pairing inter-
+actions give rise to parity- and spin-changing quantum
+phase transitions. Finally, in Section IV, we present a
+summary and our conclusions.
+II.
+THEORETICAL MODEL
+We focus on the system schematically depicted in Fig.
+1, which represents a 1D SE-SU-FMI hybrid nanostruc-
+ture of total length Lw, similar to those fabricated in
+Refs. 20 and 21. We model this system with the Hamil-
+tonian H = Hw + H∆ + HZ, where
+Hw =
+�
+σ
+�
+Lw
+2
+− Lw
+2
+dx ψ†
+σ(x)
+�
+−ℏ2∂2
+x
+2m∗ − µ
+�
+ψ†
+σ(x),
+(1)
+H ∆ = ∆
+�
+Lw
+2
+− Lw
+2
+dx
+�
+ψ†
+↑(x)ψ†
+↓(x) + H.c.
+�
+,
+(2)
+HZ =
+�
+Lw
+2
+− Lw
+2
+dx h(x)
+�
+ψ†
+↑(x)ψ↑(x) − ψ†
+↓(x)ψ↓(x)
+�
+. (3)
+Here Hw is the Hamiltonian of a single-channel SE
+nanowire of length Lw, in which the fermionic operator
+ψσ(x) creates an electron at position x with spin projec-
+tion σ =↑, ↓ and effective mass m∗. The parameter µ is
+the chemical potential, which can be experimentally var-
+ied applying external gates beneath the nanostructure.
+The terms H∆ and HZ represent, respectively, the
+proximity-induced pairing interaction encoded by the pa-
+rameter ∆, and the Zeeman exchange interaction intro-
+duced by the FMI and described by a space-dependent
+exchange field h(x), which we assume oriented along the
+
+3
+z direction (see Fig. 1). Moreover, since these interac-
+tions are externally induced into the semiconductor, we
+make the additional assumption that ∆ is unaffected by
+the presence of h(x) (a renormalized value of ∆ does not
+change qualitatively our results). As mentioned before,
+these two terms can be effectively induced by the pres-
+ence of epitaxially-grown SU and FMI shells in contact
+with the SE nanowire [20, 21]. It has been experimen-
+tally confirmed [21] that the FMI shell (EuS in that case)
+consists of a single magnetic monodomain, and there-
+fore modelling this layer by the Hamiltonian HZ is a
+reasonable approximation. In addition, the epitaxially-
+generated interfaces are essentially disorder-free, a neces-
+sary condition to produce a proximity-induced hard-gap
+[33]. This feature allows to neglect the effects of disorder
+and considerably simplifies the theoretical description.
+The presence of both, a hard proximity-induced super-
+conductor gap and an effectively induced Zeeman field,
+in these nanowires have been reported in transport mea-
+surements in Refs.
+20 and 21.
+In addition, note that
+in the above model we have neglected the effect of the
+Rashba spin-orbit interaction. While this interaction is
+crucial for the emergence of a topologically non-trivial
+(i.e., D class) superconducting phase supporting Majo-
+rana zero-modes [34], here we will focus strictly on the
+topologically-trivial ground state. As we will show be-
+low, the competition of SU and FM interactions make
+this system already very complex and interesting in it-
+self.
+We note that since the total single-particle fermionic
+spin along z
+sz = 1
+2
+�
+Lw
+2
+− Lw
+2
+dx
+�
+ψ†
+↑(x)ψ↑(x) − ψ†
+↓(x)ψ↓(x)
+�
+,
+(4)
+is a conserved quantity which verifies [sz, H] = 0, we
+can label the electronic eigenstates of H with σ = {↑, ↓}.
+Therefore, we introduce the following Nambu spinors
+Ψ↑(x) =
+� ψ↑(x)
+ψ†
+↓(x)
+�
+,
+Ψ↓(x) =
+� ψ↓(x)
+ψ†
+↑(x)
+�
+,
+(5)
+related to each other via the charge-conjugation transfor-
+mation Ψ¯σ(x) = KτxΨσ(x), where τx is the 2 × 2 Pauli
+matrix, and K is the complex conjugation operator. In
+terms of these spinors the Hamiltonian writes
+H = 1
+2
+�
+σ
+�
+Lw
+2
+− Lw
+2
+dx Ψ†
+σ(x)HBdG,σ(x)Ψσ(x),
+(6)
+where the Bogoliubov-de Gennes (BdG) Hamiltonian is
+defined as
+HBdG,σ =
+�
+− ℏ2∂2
+x
+2m − µ + σh(x)
+σ∆
+σ∆
+ℏ2∂2
+x
+2m + µ + σh(x)
+�
+.
+(7)
+In this expression, the spin projection σ =↑ (↓) on
+the left-hand side corresponds to the + (−) sign in
+the definition of the BdG matrix.
+Using the above
+charge-conjugation transformation, we note that the
+BdG Hamiltonian Eq. (7) verifies the following symme-
+try transformation
+KτxHBdG,σ = −H∗
+BdG,¯σKτx,
+(8)
+and therefore, provided χσ(x) is a solution of the BdG
+eigenvalue equation
+HBdG,σ(x)χσ(x) = Eσχσ(x),
+(9)
+with eigenenergy Eσ, the transformed spinor χ¯σ(x) =
+Kτxχσ(x), is also a solution with eigenenergy E¯σ = −Eσ.
+In what follows, we assume for simplicity the thermo-
+dynamic limit Lw → ∞, and we focus on the features
+introduced by the magnitude and spatial dependence of
+h (x), which is crucial for the rest of this work. In addi-
+tion, we assume the following step-like spatial profile for
+the exchange field
+h(x) =
+�
+−h0
+if |x| < L
+2 ,
+0
+if |x| ⩾ L
+2 ,
+(10)
+which models a uniform FMI shell of length L in contact
+with the SE nanowire (see Fig. 1). This choice for h(x)
+allows to split the problem into regions with either |x| <
+L
+2 or |x| > L
+2 , with generic exponential solutions
+χσ(x) ∼
+�
+ασ
+βσ
+�
+eikx.
+(11)
+Linear combinations of Eq. (11), with appropriate coeffi-
+cients and with allowed values of k for each region, must
+be built so that continuity of the total wavefunction and
+its derivative at the interfaces is satisfied. With this re-
+quirement, the solution of Eq.(9) is finally obtained.
+Note that the BdG Hamiltonian (7) is even under space
+inversion x → −x, and therefore its eigenstates must be
+even or odd under this transformation of coordinates.
+This symmetry allows to reduce the number of unknowns
+of the problem (i.e., coefficients of the linear combinta-
+tion). Replacing the above ansatz Eq. (11) into the BdG
+eigenvalue Eq.
+(9), and looking for localized solutions
+with energy within the gap |Eσ| < ∆, we obtain the fol-
+lowing expressions for the eigenstates belonging to the
+even-symmetry subspace:
+
+4
+χe,σ
+�
+x > L
+2
+�
+= Ae
+1σ
+�
+1
+σe−iϕσ
+�
+e−κσx + Ae
+2σ
+�
+1
+σeiϕσ
+�
+e−κ∗
+σx,
+(12)
+χe,σ
+�
+−L
+2 ≤ x ≤ L
+2
+�
+= Be
+1σ
+�
+1
+σe−ησ
+�
+cos kσx + Be
+2σ
+�
+1
+σeησ
+�
+cos ¯kσx,
+(13)
+and the following expressions for the odd-symmetry eigenfunctions
+χo,σ
+�
+x > L
+2
+�
+= Ao
+1σ
+�
+1
+σe−iϕσ
+�
+e−κσx + Ao
+2σ
+�
+1
+σeiϕσ
+�
+e−κ∗
+σx,
+(14)
+χo,σ
+�
+−L
+2 ≤ x ≤ L
+2
+�
+= Bo
+1σ
+�
+1
+σe−ησ
+�
+sin kσx + Bo
+2σ
+�
+1
+σeησ
+�
+sin ¯kσx,
+(15)
+where the coefficients {Aν
+1σ, Aν
+2σ, Bν
+1σ, Bν
+2σ}, with ν =
+{e, o}, are unknowns to be fixed.
+In addition, in the
+above expressions we have introduced the parametriza-
+tion
+cos ϕσ = Eσ
+∆ ,
+(16)
+cosh ησ = Eσ + σh0
+∆
+,
+(17)
+where we fix the definition of ϕσ to the interval ϕσ ∈
+(0, π]. The phase variable ϕσ is associated to the An-
+dreev reflection taking place at the interface xb = L/2.
+Note that the parametrization in Eq. (17) makes sense
+whenever the right-hand side is positive. If this condi-
+tion is not satisfied, one can always use the symmetry
+Eq.(8) to send Eσ → −E¯σ and σ → ¯σ. In addition, note
+that whenever 1 ≤ (Eσ + σh0) /∆ the parameter ησ is
+purely real, while for 0 < (Eσ + σh0) /∆ < 1 it is purely
+imaginary. Finally, we have introduced the quantities
+κσ ≡ −ikF
+�
+1 + 2i
+kF ξ sin ϕσ,
+(18)
+kσ ≡ kF
+�
+1 +
+2
+kF ξ sinh ησ,
+(19)
+¯kσ ≡ kF
+�
+1 −
+2
+kF ξ sinh ησ,
+(20)
+and the definition of the coherence length of the
+(proximity-induced) 1D superconductor ξ = ℏvF /∆. No-
+tice also that the spatial dependence of the wavefunc-
+tions in the region x < −L/2 can be readily obtained by
+symmetry from the relations χe,σ (x) = χe,σ (−x), and
+χo,σ (x) = −χo,σ (−x).
+We can intuitively understand the form of the scatter-
+ing solutions in the regions x > L/2 and x < −L/2 in the
+limit kF ξ ≫ 1 (i.e., the semiclassical limit, see Sec.III A),
+where the momentum κσ in Eq. (18) can be expanded
+as κσ ≃ −ikF + sin ϕσ/kF ξ, and the eigenfunctions Eqs.
+(12) and (14) take the form
+χν,σ
+�
+x > L
+2
+�
+≈
+�
+Aν
+1σ
+�
+1
+σe−iϕσ
+�
+eikF x+
++Aν
+2σ
+�
+1
+σeiϕσ
+�
+e−ikF x
+�
+e− sin ϕσx
+ξ
+, (21)
+with ν = {e, o}. In this way, it becomes evident that the
+component proportional to Aν
+1σ corresponds to a right-
+moving particle ∼ eikF x while Aν
+2σ corresponds to a left-
+moving particle ∼ e−ikF x. In addition, the wavefunctions
+exponentially decay into the superconductor within a lo-
+calization length λloc = ξ/ sin ϕσ = ξ/
+�
+1 − (Eσ/∆)2.
+These results are in complete agreement with Ref. [32],
+where the spectrum of SU-FM-SU Josephson junctions
+has been recently studied as a function of the length L
+of the FM region. However, in our case, the presence of
+a finite pairing gap ∆ in the region −L/2 < x < L/2 (as
+opposed to the assumption ∆ = 0 in the FM region in
+that work), gives rise to important differences which we
+analyze below in Sec. III.
+A.
+Continuity conditions at the interface
+We now impose the continuity conditions on the wave-
+function and its derivative at the boundary xb = L/2:
+χν,σ
+�
+x−
+b
+�
+= χν,σ
+�
+x+
+b
+�
+(22)
+∂xχν,σ
+�
+x−
+b
+�
+= ∂xχν,σ
+�
+x+
+b
+�
+.
+(23)
+Note that the same equations are obtained by symme-
+try at the other boundary −xb. Inserting the solutions
+Eqs. (12)-(15), we can express the continuity equations
+in matrix form as
+
+5
+�
+1
+σe−iϕσ
+σe−iϕσ
+1
+� �
+aν
+1σ
+aν
+2σ
+�
+=
+�
+1
+σe−ησ
+σe−ησ
+1
+� �
+Fν
+� kσL
+2
+�
+0
+0
+Fν
+� ¯kσL
+2
+�
+� �
+bν
+1σ
+bν
+2σ
+�
+,
+(24)
+−
+�
+1
+σe−iϕσ
+σe−iϕσ
+1
+� �
+κσ
+0
+0
+κ∗
+σ
+� �
+aν
+1σ
+aν
+2σ
+�
+= −s(ν)
+�
+1
+σe−ησ
+σe−ησ
+1
+� �
+kσGν
+� kσL
+2
+�
+0
+0
+¯kσGν
+� ¯kσL
+2
+�
+� �
+bν
+1σ
+bν
+2σ
+�
+,
+(25)
+where we have conveniently redefined the unknown coef-
+ficients as
+Aν
+1σ → eκσL/2aν
+1σ
+Bν
+1σ → bν
+1σ
+(26)
+Aν
+2σ → σeκ∗
+σL/2e−iϕσaν
+2σ
+Bν
+2σ → σe−ησbν
+2σ,
+(27)
+in order to give these equations a more symmetric form.
+In addition, we have used the notation s(ν) = +1(−1) for
+ν = e(o), and Fe(x) = Go(x) ≡ cos(x), Ge(x) = Fo(x) ≡
+sin(x) for compactness.
+In each subspace (even or odd) we have four equa-
+tions and four unknowns.
+Eliminating the variables
+(bν
+1σ, bν
+2σ)T , and writing the equation for (aν
+1σ, aν
+2σ)T , we
+find from the nullification of the corresponding determi-
+nant the following equations:
+cosh ησ cos ϕσ − 1
+sinh ησ sin ϕσ
+=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+|κσ|2 −
+�
+Kσ + ¯Kσ
+�
+Re κσ + Kσ ¯Kσ
+� ¯Kσ − Kσ
+�
+Im κσ
+(even-symmetry subspace),
+|κσ|2 +
+�
+Qσ + ¯Qσ
+�
+Re κσ + Qσ ¯Qσ
+�
+Qσ − ¯Qσ
+�
+Im κσ
+(odd-symmetry subspace),
+(28)
+where we have defined the quantities
+Kσ = kσ tan
+�kσL
+2
+�
+,
+(29)
+¯Kσ = ¯kσ tan
+�¯kσL
+2
+�
+,
+(30)
+Qσ = kσ cot
+�kσL
+2
+�
+,
+(31)
+¯Qσ = ¯kσ cot
+�¯kσL
+2
+�
+.
+(32)
+From Eq. (28), the eigenvalue Eσ for each subspace is
+finally obtained. This equation summarizes our main the-
+oretical results. In the next Sec. III we analyze the nu-
+merical solution and different important limits.
+B.
+Spin-changing quantum phase transitions
+We now focus on the quantum phase transitions which
+occur whenever one of the subgap states crosses EF . To
+that end, let us analyze the spinors defined in Eq. (5),
+and consider the norm of the “up” spinor
+q↑ =
+� Lw/2
+−Lw/2
+dx
+�
+ψ†
+↑ (x) ψ↑ (x) + ψ↓ (x) ψ†
+↓ (x)
+�
+.
+Recalling the definition of the single-particle sz operator
+[see Eq. (4)], it is straightforward to associate these two
+quantities through the relation q↑ = 2sz − 1. Since sz
+is a conserved quantity, so is the norm q↑ of the “up”
+Nambu spinors. This connection allows to interpret q↑ as
+an effective “conserved charge”. Similar considerations
+allow to write the relation q↓ = −2sz − 1. Due to the
+particle-hole relation Eq.(8), the information about sz
+can be obtained with either q↑ or q↓. A more symmetric
+form involving both conserved charges is
+sz = q↑ − q↓
+4
+.
+(33)
+While redundant, this expression makes explicit that in
+the spin-symmetric case q↑ = q↓, the net spin sz must
+vanish (sz = 0).
+We now return to Hamiltonian Eq.
+(7), and let us
+separate the effect of the proximity-induced Zeeman field,
+by writing it as HBdG,σ = H0,σ + Vσ, where
+H0,σ =
+�
+− ℏ2∂2
+x
+2m − µ
+σ∆
+σ∆
+ℏ2∂2
+x
+2m + µ
+�
+,
+(34)
+Vσ =
+�
+σh(x)
+0
+0
+σh(x)
+�
+.
+(35)
+In this form, we can interpret the effect of the exchange
+field as a “perturbation” on an otherwise homogeneous
+
+6
+1D superconductor represented by H0,σ. Therefore, the
+full and the unperturbed single-particle Green’s functions
+in this problem are respectively defined as
+Gσ (z) = [z − H0,σ − Vσ]−1 ,
+(36)
+G0,σ (z) = [z − H0,σ]−1 ,
+(37)
+From here, the total number of effective “up” charges
+Q↑ induced in the ground state due to the potential Vσ,
+compared to the unperturbed homogeneous SU wire, can
+be computed as
+∆Q↑ = − 1
+π Im Tr
+� ∞
+−∞
+dϵ nF (ϵ) ∆G↑ (ϵ + iδ) .
+(38)
+where ∆Gσ (z) ≡ Gσ (z) − G0,σ (z). At T = 0, Eq. (38)
+can be easily computed from the well-known expression
+of the Friedel sum rule [35]
+∆Q↑ = 1
+π
+� 0
+−∞
+dϵ
+�∂η↑ (ϵ)
+∂ϵ
+− ∂η0,↑ (ϵ)
+∂ϵ
+�
+(39)
+= η↑ (0) − η0,↑ (0)
+π
+(40)
+where we have defined the phase shifts [32, 35]
+ησ (ϵ) = Im ln det Gσ (ϵ + iδ) ,
+(41)
+η0,σ (ϵ) = Im ln det G0,σ (ϵ + iδ) ,
+(42)
+and where we have used that the phase shifts vanish in
+the limit ϵ → ±∞.
+Since the system is non-interacting, the Green’s func-
+tion Eq. (36) can be written in terms of single-particle
+eigenstates |α, σ⟩, with α a generic label, as
+Gσ (z) =
+�
+α
+|α, σ⟩ ⟨α, σ|
+z − Eα,σ
+.
+(43)
+Therefore, after simple algebra, and using the above re-
+lations and the fact that in the absence of magnetic field
+sz = 0 [see Eq. (33)], the total Sz of the ground state is
+Sz = ∆Q↑
+2
+= 1
+2
+��
+α
+Θ (−Eα,↑) −
+�
+α′
+Θ
+�
+−E0
+α′,↑
+�
+�
+,
+(44)
+where Θ(ϵ) is the unit-step function. The above expres-
+sion allows to interpret the total Sz of the ground state
+as a function of the “up” Nambu spinors with energy be-
+low EF = 0, as compared to the (unperturbed) situation
+h0 = 0. Since the effective charges are quantized in inte-
+ger numbers, the total spin Sz can only change in discrete
+“jumps” of 1/2 whenever a subgap state with projection
+up crosses below EF (note that we have defined dimen-
+sionless spin operators). This interpretation makes sense
+since the ground state becomes spin-polarized when the
+exchange field h0 becomes large enough [i.e., the Zeeman
+energy of up-spin electron is decreased, see Eqs. (3) and
+(10)]. While the result of Eq. (44) has been obtained re-
+cently by the authors of Ref. [32], we note that here we
+have rederived it in a different physical situation which
+allows a more generic regime of parameters.
+III.
+RESULTS
+We start this section by analyzing different limits of
+the general result given in Eq. (28). In particular, in
+Sec. III A we focus on the semiclassical limit, and in Sec.
+III B we study the atomic limit, where we recover the
+YSR results. In both cases, Eq. (28) reduces to well-
+known analytical results. Finally in Sec. III C we show
+results corresponding to intermediate regimes, obtained
+by solving numerically Eq. (28).
+A.
+Semiclassical limit
+Generally speaking, the semiclassical limit is verified
+when EF is the largest scale of the problem [36]. In par-
+ticular, the condition EF ≫ ∆ (which is very well satis-
+fied in most experimental systems) can be expressed as
+kF ξ ≫ 1, recalling that after linearization of the normal
+quasiparticle dispersion, i.e., ϵk,σ ≃ ±ℏvF k, where the
++(−) sign corresponds to right-(left-)movers, the Fermi
+energy can be approximated as EF ≃ ℏkF vF . In this
+case, Eqs. (18)-(20) reduce to
+rσ ≡ κσ
+kF
+≃ −i + sin ϕσ
+kF ξ ,
+(45)
+ζσ ≡ kσ
+kF
+≃ 1 + sinh ησ
+kF ξ
+,
+(46)
+¯ζσ ≡
+¯kσ
+kF
+≃ 1 − sinh ησ
+kF ξ
+,
+(47)
+to leading order in O(kF ξ)−1, and Eq. (28) becomes
+cosh ησ cos ϕσ − 1
+sinh ησ sin ϕσ
+≃ s(ν)
+1 + tan
+�
+kF Lζσ
+2
+�
+tan
+�
+kF L¯ζσ
+2
+�
+tan
+�
+kF Lζσ
+2
+�
+− tan
+�
+kF L¯ζσ
+2
+� ,
+= s(ν) cot
+�L sinh ησ
+ξ
+�
+,
+(48)
+where
+we
+have
+used
+the
+trigonometric
+identity
+tan (x + y) = (tan (x) + tan y)/(1 + tan x tan y). In gen-
+eral this transcendental equation cannot be solved ana-
+lytically. However, in the regime of parameters EF ≫
+h0 ≫ ∆, where the exchange field h0 is much larger
+than ∆, we can write cosh ησ ≈ sinh ησ ≈
+�� h0
+∆
+�� ≫ 1
+[see Eqs.
+(16) and (17) ], and Eq.
+(48) reduces to
+cot ϕσ = s(ν) cot (Lh0/ℏvF ). Equivalently we can write
+this result as
+arccos
+�Eσ
+∆
+�
+=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+LEσ
+ℏvF
++ σ Lh0
+ℏvF
++ 2nπ,
+(even)
+LEσ
+ℏvF
++ σ Lh0
+ℏvF
++ (2n + 1) π.
+(odd)
+(49)
+This result can be interpreted as a semiclassical Bohr-
+Sommerfeld quantization condition for particles which
+
+7
+perform a complete a closed loop in the region −L/2 <
+x < L/2 [36]. In particular, it exactly coincides with the-
+oretical results obtained for SU-FM-SU Josephson junc-
+tions with a normal (i.e., ∆ = 0) FM region [30–32], the
+only difference being that within our theoretical treat-
+ment, we can distinguish the symmetry of the solutions.
+The similarity of these results can be rationalized noting
+that considering a normal sandwiched region in an SU-
+FM-SU junction corresponds to taking the limit h0 ≫ ∆
+in our Eq. (48) while keeping the ratio Eσ/∆ finite (since
+Eσ corresponds to a subgap state, it is always bounded
+by ∆), thus resulting in Eq. (49). This shows that our
+Eq. (28) is a generic relation describing different situa-
+tions regardless of the magnitude of the ratio h0/∆.
+B.
+YSR-impurity limit
+We now consider the atomic YSR (or simply Shiba)
+limit, in which the exchange profile becomes point-like,
+L → 0, while h0 → ∞, in such a way that the product
+Lh0 = J = const. Under these assumptions the magnetic
+barrier becomes a delta function and the Hamiltonian in
+Eq. (3) can be written as
+HZ ≈ −J
+� ∞
+−∞
+dx δ(x)
+�
+ψ†
+↑(x)ψ↑(x) − ψ†
+↓(x)ψ↓(x)
+�
+.
+(50)
+In this case, it is easy to see that the odd-symmetry solu-
+tions decouple from the above Hamiltonian (50), as they
+vanish at x = 0, and only even solutions can couple to
+the delta-potential.
+As in the previous section, note that the limit h0 → ∞
+implies cosh ησ ≈ sinh ησ ≈
+�� h0
+∆
+�� ≫ 1. However, the limit
+h0 → ∞ is not compatible with the semiclassical ap-
+proach, as it violates the requirement h0 ≪ EF . There-
+fore we cannot use here our previous Eq. (49). Instead,
+we must first take the limit ησ ≫ 1 together with the
+limit L → 0, which applied to Eqs. (19) and (20) yield
+kσ → kF
+�
+2h0
+ℏvF kF
+,
+(51)
+¯kσ → ikF
+�
+2h0
+ℏvF kF
+.
+(52)
+In addition Eqs. (29)-(32) become
+Kσ → kF h0L
+ℏvF
+= kF ρ0J,
+(53)
+¯Kσ → −kF h0L
+ℏvF
+= −kF ρ0J,
+(54)
+where the expressions for the density of states per spin
+of 1D quasiparticles at the Fermi energy ρ0 = 1/ℏvF ,
+and the exchange coupling J = h0L, have been used.
+Replacing these expressions into Eq. (28) for the even-
+symmetry solutions, we obtain
+σ
+Ee
+σ
+�
+∆2 − (Ee)2
+σ
+= 1 − (ρ0J)2
+(2Jρ0)
+.
+(55)
+From this expression, we can easily solve for Ee
+σ
+Ee
+σ
+∆ = σ 1 − (ρ0J)2
+1 + (ρ0J)2 ,
+(56)
+which is the well-known expression for the energy of YSR-
+impurity subgap level [1]. This result indicates that any
+finite value of J produces a YSR in-gap state. This type
+of subgap YSR states has been observed in several STM
+experiments on atomic magnetic adsorbates on supercon-
+ducing substrates [27, 37–41].
+For completeness, and in order to illustrate the general
+scope of Eq. (28), here we also show the result for the
+YSR odd states for a small (but finite) L. Using similar
+approximations, we obtain the expression
+Eo
+σ
+∆ = σ
+1
+�
+1 +
+�ρ0Jk2
+F L2
+6
+�2 ,
+(57)
+where it becomes evident that in addition to a finite value
+of J, a finite value of kF L is needed to observe an odd-
+symmetry subgap YSR state.
+C.
+Subgap ABS spectrum in generic cases
+As stated in Section II, Eq. (28) implicitly defines the
+energy of the subgap states as a function of the param-
+eters h0/∆ , kF ξ, and kF L. These parameters can be
+directly or indirectly controlled in experiments, i.e., the
+parameter h0 can be controlled by modifying the FMI
+material, the length L of the FMI region can be modified
+varying the length Lw of the semiconductor via vapor-
+liquid-solid (VLS) method and subsequent evaporation of
+the FMI material [20], and the parameter kF in the semi-
+conductor can be varied by changing the SE material or
+by introducing external gates to modify the chemical po-
+tential µ. Therefore, due to this high degree of tunability,
+hybrid heterostructures might offer a unique platform to
+produce and control engineered subgap states. Probably
+the easiest way to experimentally control the subgap elec-
+tronic structure is by producing different devices with the
+same FMI material and different lengths L. Therefore, in
+this section we show the numerical solutions of Eq. (28)
+with fixed parameters h0/∆ and kF ξ (which control the
+“operation regime” of the device), and calculate both the
+energy dependence of the even- and odd-symmetry ABS,
+and the total spin Sz of the device as a function of L (i.e.,
+dimensionless variable kF L).
+Generally speaking, the overall evolution of the ABS
+spectrum from L = 0 to L → ∞ is quite complex and de-
+serves a detailed explanation. As shown in Fig. 2, as the
+
+8
+parameter kF L increases, more and more subgap states
+emerge from the gap edges. This behavior is reminiscent
+of a quantum particle in a square-well potential, tipically
+taught in introductory quantum mechanics courses [42],
+where increasing the width L of the well increases the
+number of allowed bound states. In our case, the emer-
+gence of new ABS as kF L increases can be intuitively
+understood in terms of a competition between supercon-
+ductivity and magnetic field: the magnetic field tends
+to break Cooper-pairs and to locally disrupt supercon-
+ductivity in the magnetic region by introducing subgap
+states that become macroscopic in number for large L,
+eventually populating the whole gap.
+We note that for any finite L, even- and odd-symmetry
+states are generically non-degenerate (except at isolated
+points). However, as it is clear from Figs. 2 and 3, their
+energy difference (evidenced as oscillations of the blue
+and red lines around the semiclassical value) decreases
+very rapidly and the solutions become degenerate in the
+limit L → ∞. This transition from non-degenerate YSR
+states in the limit L → 0, to double degenerate ABS
+states for L → ∞ has been discussed in previous works
+on ballistic SU-FM-SU junctions [19, 30–32], and in the
+case of extended Shiba impurities in 1D nanowires [43].
+It is also clearly visible in Fig. 2, and more dramatically
+in Fig. 3 below. In our 1D geometry, this degeneracy
+in the limit L → ∞ can be intuitively understood by
+linearizing the spectrum around the Fermi energy, and
+expressing the original fermionic operators in terms of
+right- and left-moving fields slowly varying in the scale
+of k−1
+F
+[44], i.e., ψσ (x) ≈ eikF xψR,σ (x) + e−ikF xψL,σ (x).
+The slowly-varying fields ψR,σ(x) and ψL,σ(x) are two
+independent chiral fermionic fields obeying the usual an-
+ticommutation relations, in terms of which the original
+Hamiltonian becomes [43]
+Hw ≈
+�
+σ
+� ∞
+−∞
+dx
+�
+−iℏvF ψ†
+R,σ(x)∂xψR,σ(x)
++ iℏvF ψ†
+L,σ(x)∂xψL,σ(x)
+�
+(58)
+H ∆ ≈ ∆
+� ∞
+−∞
+dx
+�
+ψ†
+R,↑(x)ψ†
+L,↓(x) + ψ†
+L,↑(x)ψ†
+R,↓(x) + H.c.
+�
+,
+(59)
+HZ ≈ −
+� ∞
+−∞
+dx h0
+�
+ψ†
+R,↑(x)ψR,↑(x) − ψ†
+R,↓(x)ψR,↓(x)
++ ψ†
+L,↑(x)ψL,↑(x) − ψ†
+L,↓(x)ψL,↓(x)
+�
+,
+(60)
+where oscillating terms proportional to e±2ikF x have been
+neglected as they cancel out in the limit L → ∞ due to
+destructive interference. Defining the new chiral Nambu
+spinors
+Ψ1,σ(x) =
+� ψR,σ(x)
+ψ†
+L,¯σ(x)
+�
+,
+Ψ2,σ(x) =
+� ψL,σ(x)
+ψ†
+R,¯σ(x)
+�
+,
+(61)
+the Hamiltonian of the system can be expressed in terms
+of two decoupled chiral sectors
+H = 1
+2
+�
+σ=↑,↓
+�
+j=1,2
+� ∞
+−∞
+dx Ψ†
+j,σ(x)Hj,σ(x)Ψj,σ(x), (62)
+with the definitions of the chiral BdG Hamiltonians
+Hj,σ =
+�
+(−1)jivF ∂x − σh0
+σ∆
+σ∆
+(−1)j+1ivF ∂x − σh0
+�
+.
+(63)
+The Nambu spinors Eq. (61) define two independent chi-
+ral subspaces related by the inversion symmetry of the
+original Hamiltonian, i.e., under the space inversion op-
+eration x ↔ −x, the fermionic operators transform as
+ψL,σ(x) ↔ ψR,σ(x), and consequently we conclude that
+Ψ1,σ(x) ↔ Ψ2,σ(x), which must then be degenerate. In
+addition, the particle-hole symmetry Eq. (8) in this rep-
+resentation produces Ψ1,σ(x) → Ψ2,¯σ(x), and therefore
+H1,σ → −H2,¯σ, implying that the solutions verify the
+particle-hole symmetry property E1,σ = −E2,¯σ. More-
+over, notice that assuming periodic boundary conditions,
+the problem can be solved with the solutions ψR,σ(x) ∼
+eikx and ψL,σ(x) ∼ e−ikx, and the dispersion relation
+becomes E1,σ(k) = E2,σ(k) = ±
+�
+(ℏvF k)2 + ∆2 − σh0.
+From here, a renormalized quasiparticle gap 2∆ren =
+2 |∆ − h0| is obtained, consistent with our previous re-
+sult.
+In terms of the chiral Nambu spinors, the most general
+solution is the linear combination
+Ψσ(x) = AeikF xΨ1,σ(x) + Be−ikF xΨ2,σ(x).
+(64)
+This is exactly the same form that can be obtained by
+combining the degenerate even and odd solutions in Eqs.
+(13) and (15) in the semiclassical limit where kF ξ ≫ 1.
+From the analysis of the linearized Hamiltonian, we
+conclude that the degeneracy in the limit L → ∞
+arises from the absence of chirality-breaking terms, i.e.,
+terms ∼ Ψ†
+1,σ(x)Ψ2,σ(x) arising from, e.g., single par-
+ticle backscattering terms ψ†
+R,σ(x)ψL,σ(x) or Cooper-
+pairing channels ψ†
+R(L),↑(x)ψ†
+R(L),↓(x) carrying momen-
+tum ∓2kF .
+For this to occur, the magnetic FMI re-
+gion must be uniform and its length L must be much
+larger than k−1
+F
+in order to produce the required cancel-
+lation of the rapidly oscillating exponentials ∼ e±2ikF x.
+In other words, the product kF L must be kF L ≫ 1,
+consistent with our numerical results in Figs. 2 and 3.
+Only for small values of kF L, where this destructive in-
+terference is incomplete, residual couplings of the type
+∼ Ψ†
+1,σ(x)Ψ2,σ(x) remain, and the degeneracy is lifted.
+Finally, we stress that the degeneracy in the limit L → ∞
+is a robust property to the presence of interactions, as
+shown in previous works [43].
+On the other hand, in the limit L → 0 and for any
+finite value of the Zeeman field h0, both (even and odd)
+solutions converge to Eσ/∆ → ±1, indicating that the
+FMI region is no longer relevant (i.e., it physically drops
+
+9
+−1
+1
+0
+E/∆
+−1
+1
+0
+0
+10
+20
+30
+40
+50
+0
+1
+2
+3
+4
+5
+6
+kF L
+Sz
+0
+2
+4
+6
+8
+10
+12
+14
+kF L
+FIG. 2. Energy of the Andreev bound states (upper panel) and total spin Sz(lower panel) as a function of kF L, for kF ξ = 7.8
+and h0/∆ = 3.0 (left panel) and kF ξ = 3.4 and h0/∆ = 2.1 (right panel). Blue and red colors correspond to even and odd states
+respectively. Lines starting from the top gap edge at positive energy E/∆ = 1 (bottom gap edge at negative energy E/∆ = −1)
+correspond to up (down) spin projections of the states. For smaller values of kF L (right panel), plateaus corresponding to
+regions of integer and half-integer spin are more separated and might become easier to observe in experiments.
+from the description). However, the behavior near L = 0
+is quite different for each case: while the even-symmetry
+solution tends to E/∆ → 1 as [see Eq. (56)]
+Ee
+σ
+∆ ≈ σ
+�
+1 − 2
+�h0L
+ℏvF
+�2
+. . .
+�
+,
+(65)
+from Eq. (57) we conclude that the odd solution behaves
+as
+Eo
+σ
+∆ ≈ σ
+�
+1 − 1
+2
+�h0k2
+F L3
+6ℏvF
+�2
+. . .
+�
+,
+(66)
+therefore approaching the gap edge much faster as L → 0.
+Besides the general features of the spectrum discussed
+up to this point, its evolution as L increases is strongly
+affected by the values of the parameters kF ξ and h0/∆.
+In what follows, we analyze their effects on Figs. 2 and
+Fig. 3 respectively.
+1.
+Effect of varying the parameter kF ξ
+This parameter can be considered as a “knob” which
+tunes the device from the semiclassical behavior (kF ξ
+large, see left panel in Fig. 2) into a “quantum” regime
+(kF ξ small, see right panel) where the spectrum is dom-
+inated by quantum oscillations. The hybrid heterostruc-
+ture under study is promising in this sense since, due to
+the combination of materials (in particular, semiconduc-
+tors with a much smaller kF as compared to metals), it
+is in principle possible that kF ξ can be experimentally
+controlled. In addition, kF could be further modified by
+introducing external gating leads (through the modifica-
+tion of the chemical potential µ). To illustrate the dra-
+matic changes in the spectrum as kF ξ varies, in Fig 2 we
+show the numerically obtained subgap spectra as a func-
+tion of kF L for kF ξ = 7.8 and h0/∆ = 3.0 (left panel),
+for and kF ξ = 3.4 and h0/∆ = 2.1 (right panel). Solid
+blue (red) lines correspond to even(odd)-symmetry solu-
+tions. Moreover, since we always assume h0 > 0, solu-
+tions emerging from the top edge E/∆ = 1 (bottom edge
+E/∆ = −1) correspond to spin up (spin down) solutions.
+In addition, note the reflection symmetry of the solutions
+around the horizontal E = 0 axis, a consequence of the
+particle-hole symmetry of the BdG Hamiltonian, Eq. (8).
+Upon decreasing kF ξ, the subgap spectrum becomes
+much more intricate due to the enhanced even-odd
+energy-splitting, which results in an amplified oscillatory
+behavior of the ABS (we have reduced the range of kF L in
+the right panel for clarity in the figure). Unfortunately,
+in the regime kF ξ ∼ 1 no analytic expressions for the
+subgap ABS are possible, but qualitative considerations
+
+10
+can be provided. In fact, the amplified oscillations can
+be traced back to the larger energy dependence of the
+momenta Eq. (18)-(20) as kF ξ decreases. Then, whereas
+for large kF ξ all these quantities converge to a static (i.e.,
+energy-independent) value ∼ kF , the limit of small kF ξ
+produces a larger effect on the space-dependence of the
+wave functions through the exponential factors in Eqs.
+(12)-(15). This in turn produces larger interference ef-
+fects, and an enhanced lifting of the even-odd degener-
+acy.
+This phenomenological behavior enables interesting
+possibilities, such as the chance to observe half-integer
+spin (and fermion parity-switching) quantum phase tran-
+sitions in the ground state. To illustrate this effect, we
+show the ground-state Sz transitions in the bottom pan-
+els of Fig.
+2 in each case.
+While for larger kF ξ, the
+half-integer Sz steps are very narrow due to the almost-
+degenerate even-odd solutions (i.e., the even and odd so-
+lutions cross zero energy almost at the same value of
+kF L), for smaller kF ξ the Sz transitions occur in well-
+defined half-integer steps. This behavior is well explained
+by the enhanced lifting of the even-odd degeneracy, which
+allows to observe one ABS crossing zero energy at a time.
+2.
+Effect of varying the parameter h0/∆
+In Fig. 3 we show the evolution of the subgap spectrum
+as a function of kF L, for different values of the Zeeman
+field h0/∆ = 0.8, 1.54 and 2.2, and for a fixed relatively
+large value kF ξ = 8.2, allowing to interpret these results
+in terms of the semiclassical approximation. Here we can
+clearly distinguish three qualitatively different regimes:
+a) the “weak field” regime h0 < ∆ (top panel) where
+the ABS do not cross E = 0, b) the “intermediate field”
+regime ∆ < h0 < 2∆ (middle panel) where the ABS can
+evenually cross zero energy, and quantum phase transi-
+tions can be induced, and finally c) the “strong field”
+(2∆ < h0) regime (bottom panel), where the ABS can
+be found anywhere in the region −1 < Eσ/∆ < 1. In all
+cases, the value of h0 determines the asymptotic limit to
+which the ABS approach for large L (see dashed black
+lines in Fig. 3). Below we briefly discuss the main fea-
+tures of the spectrum in each regime.
+a.
+Weak-field regime 0 < h0 < ∆:
+This regime
+is characterized by a Zeeman field which is not strong
+enough to destroy the superconducting gap.
+In this
+case none of the ABS is able to cross E = 0 and in
+the limit L → ∞ they asymptotically approach the
+value Eσ/∆ → σ (1 − h0/∆) (see horizontal dashed black
+lines), and therefore a renormalized gap remains (see top
+panel in Fig. 3). More quantitatively, in the semiclassi-
+cal limit [Eq. (48)] they obey the asymptotic expression
+−1
+1
+0
+h0
+E/∆
+−1
+1
+0
+h0
+E/∆
+0
+20
+40
+60
+80
+100
+−1
+1
+0
+kF L
+E/∆
+FIG. 3. Energy of the Andreev bound states as a function of
+kF L for the three different values of h0 (h0/∆ = 0.8, 1.54, 2.2
+for the lower, middle and upper panels) and kF ξ = 8.2. Blue
+and red colors correspond to even and odd states respectively.
+Lines starting from negative (positive) energies correspond to
+down (up) spin projections of the state. Note that the value
+of h0/∆ sets the asymptotic limit for the Andreev states and
+is crucial to determine the overall subgap spectrum.
+valid for kF L → ∞
+Eν
+σ
+∆ ≃ σ
+�
+�1 − h0
+∆ + π2
+2
+� ξ
+L
+�2 �
+1 − s(ν)ξ
+L
+�
+2∆
+h0
+− 1
+�2�
+� ,
+(67)
+with s(ν) = 1(−1) for ν = e(o).
+From here, we can
+clearly see that whereas the even-odd averaged quanti-
+ties (i.e., the semiclassical values) approach the asymp-
+totic limit as L−2, the energy difference between even
+and odd solutions (i.e., the amplitude of the oscillation
+around the semiclassical limit) decreases as L−3, and the
+solutions become degenerate in the limit L → ∞. On the
+other hand, the quasiparticle gap in the limit L → ∞ is
+renormalized to 2∆ren = 2 |∆ − h0|. Note that this gap
+renormalization is quite specific to this setup, and is not
+present, for instance, in the case of Ref. [32], where the
+magnetic region is normal and not superconducting, and
+in addition the system corresponds to a “short” SU-FM-
+SU junction with L < ξ, and therefore only few subgap
+states are allowed.
+Another feature of the weak-field regime is that the
+ABS require a minimal length Lmin to emerge in the sub-
+
+11
+gap region. This can be easily understood in terms of Eq.
+49, where a minimal magnetic phase, represented by the
+product Lh0/ℏvF , must be accumulated in order to pro-
+duce an observable in-gap ABS. Finally, concerning the
+spin quantum number of the ground state, since none of
+the ABS cross EF , no quantum phase transitions are ex-
+pected according to the results of Sec. II B and the value
+of the ground state spin remains a spin-singlet Sz = 0.
+b. Intermediate field regime ∆ < h0 < 2∆: In this
+case the Zeeman field h0 is sufficiently strong to force the
+ABS to cross zero energy, eventually inducing quantum
+phase transitions (see middle panel in Fig. 3). The n-th
+critical value Lc,n can be obtained imposing the condition
+Eσ = 0 on the semiclassical approximation in Eq. (48),
+Lν
+c,n = ξ
+arctan
+�
+−s (ν) ∓
+�� h0
+∆
+�2 − 1
+�
++ nπ
+�� h0
+∆
+�2 − 1
+,
+(68)
+with s(ν) = 1(−1) for ν = e(o).
+In this regime, the ABS follow the same asymp-
+totic behavior as in Eq.
+(67), approaching Eσ/∆ →
+σ (1 − h0/∆), although the overall subgap spectrum is
+completely different due to the closing of the gap, and
+due to the overlap of the E↑ and E↓ spectrum as L
+increases beyond the first critical Lc,0. In fact, in the
+regime L > Lc,0 the quasiparticle gap becomes com-
+pletely populated (and washed away) by subgap states.
+Moreover, we predict an accumulation of levels in the re-
+gion −∆ + h0 < E < ∆ − h0, which can eventually form
+a peak structure in the total density of states.
+c. Strong field regime 2∆ < h0: Finally, in this regime
+(see bottom panel in Fig. 3), the asymptotic dashed lines
+fall within the continuum and it is no longer possible to
+obtain an analytic expression for the ABS behavior in
+the limit L → ∞. As a result, the subgap ABS can be
+found anywhere in the subgap region −1 < Eσ/∆ < 1.
+In addition, we note that the minimal length required to
+observe in-gap ABS has reduced to Lmin ≈ 0.
+IV.
+SUMMARY AND CONCLUSIONS
+In this work we have analyzed the subgap electronic
+structure in the one dimensional SE-SU-FMI heterostruc-
+ture schematically depicted in Fig. 1, a novel physical
+system recently fabricated using molecular beam epitaxy
+techniques (MBE). The main motivation to study this
+type of hybrid systems is that, via a careful combina-
+tion of different materials, the emergent characteristics
+can be completely different from those of the individ-
+ual components, providing a way to build devices with
+tailored properties and specific functionalities. In partic-
+ular, much of the experimental effort has focused on the
+realization of topological superconducting phases host-
+ing Majorana zero modes, with possible applications in
+topological quantum computing [20, 21]. A distinguish-
+ing feature of these heterostructures is the coexistence
+of antagonistic superconductor and ferromagnetic insu-
+lating layers over a finite and arbitrary length L in a
+semiconductor wire, a combination that confers unique
+spectral properties which cannot be found in elemental
+materials in nature.
+In particular, we have modelled the hybrid struc-
+ture
+assuming
+non-interacting
+fermions
+in
+a
+one-
+dimensional single-channel nanowire under the effect of
+two proximity-induced interactions: a SU pairing and
+a space-dependent Zeeman exchange coupling [see Eqs.
+(1)-(3)].
+We have solved the associated Bogoliubov-de
+Gennes equations and, by imposing standard continuity
+conditions on the wave functions, we have obtained an
+equation [Eq. (28)] defining the subgap ABS spectrum
+of the device.
+This single equation encodes our main
+theoretical results. We stress that our approach is equiv-
+alent to other works using the scattering-matrix formal-
+ism. We have analytically solved Eq. (28) in two paradig-
+matic limits: the semiclassical limit (Sec. III A) and the
+Yu-Shiba-Rusinov limit, typical of atomic magnetic mo-
+ments interacting with a superconductor (Sec. III B). In
+both cases, we have been able to recover well-known ana-
+lytical results, providing important sanity checks for our
+theoretical results. As a consequence of the symmetries
+of the Hamiltonian (i.e., inversion x → −x and sz spin
+symmetries), it was possible to classify the solutions into
+even- and odd-symmetry, and with sz labels σ =↑, ↓. In
+particular, we note that the even-odd classification, aris-
+ing in the present case due to the inversion symmetry of
+the Hamiltonian, is nothing but the 1D analog of the clas-
+sification in angular momentum eigenstates ℓ occurring
+in 3D spherically-symmetric Hamiltonians [7, 25, 26].
+We have studied the subgap spectrum of ABS as a
+function of different parameters, namely: the length of
+the magnetic region (through the dimensionless parame-
+ter kF L), the strength of the Zeeman exchange induced
+by the FMI (parameter h0/∆), and the superconducting
+coherence length (parameter kF ξ). We stress that each
+one of these parameters could in principle (directly or in-
+directly) be controlled in experiments. However, due to
+its potential relevance for on-going experimental efforts,
+we have in particular focused our study on the evolution
+of the subgap spectrum as a function of the length L (i.e.,
+as it is probably the easiest parameter to vary in experi-
+ments), for fixed parameters kF ξ and h0/∆. The parame-
+ter L can be controlled by, e.g., changing the experimen-
+tal growing conditions of the semiconductor nanowires
+using the VLS growth method. In Figs. 2 and 3 we have
+analyzed the evolution of the subgap spectrum in terms
+of the parameter kF L for different values of h0/∆ and
+kF ξ. Roughly speaking, while kF ξ controls the “semi-
+classical vs quantum” operation regime of the device,
+and the magnitude of the even-odd energy separation,
+the parameter h0/∆ essentially controls the energy sep-
+aration of the E↑ and E↓ solutions, eventually enabling
+many interesting physical phenomena such as the possi-
+
+12
+bility to observe multiple ABS crossing zero-energy, the
+existence of multiple spin- and parity-changing quantum
+phase transitions in the device, quasiparticle gap renor-
+malization ∆ → ∆ren = |∆ − h0| in the limit of large
+kF L, etc.. An important conclusion here is that in order
+to experimentally observe a quantum phase transition,
+the condition h0 > ∆ must be fulfilled.
+Interpreting L as a “tunable” parameter has another
+theoretical advantage, as it enables to address the in-
+teresting fundamental question of how to connect two
+paradigmatic limits in SU-FM hybrid devices: the atomic
+limit (kF L → 0), where the physics is that of the well-
+known non-degenerate YSR states, and the ballistic limit
+(kF L ≫ 1) where the spectrum of the subgap ABS be-
+comes double degenerate. Until very recently, these lim-
+its were treated as disconnected from each other. In Ref.
+[32] this issue was addressed in the particular case of SU-
+FM-SU junctions in the limit L < ξ. Here we have revis-
+ited this intriguing question for a different setup where
+such constraint does not exist, and have studied the evo-
+lution of the subgap spectrum as a function of L. The
+abovementioned symmetry classification into even and
+odd solutions is critically important to allow the inter-
+pretation of the degeneracy in the limit kF L → ∞ as an
+“even-odd degeneracy”. At the same time, it enables to
+explain the degeneracy lifting in the limit L → 0, where
+only even states prevail in the subgap region of ener-
+gies.
+Using an approximate model of one-dimensional
+fermions with linearized dispersion, we have provided a
+simple picture where the even-odd degeneracy naturally
+emerges as a consequence of destructive interferences of
+terms e±i2kF x arising from single-particle backscattering
+mechanisms.
+The continuous evolution of the subgap spectrum as a
+function of kF L allows a better understanding of previ-
+ous experimental STM results on atomic magnetic adsor-
+bates on superconducting substrates, where the subgap
+YSR states are usually interpreted in terms of a point-like
+magnetic moment [27, 37–41]. While the delta-function
+limit is obviously a mathematical idealization, in terms
+of our model the observed YSR states can be rationalized
+assuming a finite value of kF L and a (more physically ap-
+pealing) finite value of the atomic local field h0. This is
+precisely the case if we note that for magnetic impurities
+(e.g. Fe, Co or Mn atoms) deposited on top of bulk metal-
+lic S surfaces (e.g., Pb or Al), the spatial extension of the
+short-ranged Zeeman field can be estimated as the size of
+the d-shell orbitals L ∼ 1 ˚A, while the Fermi wavevector
+of bulk superconductors (e.g., Pb) is kF ∼ 1−2×1010m−1
+(see Ref. [45]). This type of adsorbate/substrate combi-
+nation yields a parameter kF L ∼ 1, which is within the
+regime where we recover observable subgap states (see
+Figs. 2 and 3). On the other hand, in 1D semiconduc-
+tor heterostructures as those of Refs. 20 and 21, kF is
+usually much smaller than in metallic superconductors.
+Measurements of the number of carriers from the Hall
+conductance RH in 2D InGaAl quantum wells [46] yield
+the estimated value kF ∼ 2.2 × 107m−1, three orders of
+magnitude smaller as compared to bulk Pb. This much
+smaller value of kF allows for much larger, experimentally
+accessible values of L, while keeping values of h0 also
+within experimental reach. All together, this combina-
+tion makes these hybrid materials a much more versatile
+platform to control the spectrum of YSR/ABS subgap
+states.
+To characterize the quantum phase transitions occur-
+ring in the device, we have computed the value of the to-
+tal Sz using a spin version of the Friedel sum rule [see Eq.
+(44) and also Ref. [32]. We stress that these transitions
+are a generalization of the well-known “0-π” transition
+occurring in atomic Shiba impurities [22, 47] or quantum
+dots coupled to superconductors [48–50]. From this per-
+spective, the difference with respect to atomic systems
+is that instead of a single transition, actually multiple
+transitions can occur due to the finite extension L of
+the “impurity” and the many ABS states with different
+symmetry which can eventually cross below EF . Inter-
+estingly, we stress that the ocurrence of these quantum
+phase transitions can be tuned varying the length L.
+We now briefly address the effect of the Rashba spin-
+orbit interaction, which has been neglected in our work.
+As mentioned previously, this interaction was neglected
+to simplify the theoretical description of this (already
+quite complex and rich) problem. This interaction can
+drive the system into the topological superconductor
+class D [51, 52], hosting Majorana zero modes at the ends
+(see e.g., Ref. 34 for a related setup), and in that case
+we expect qualitative changes with respect to the results
+presented here. Consequently our results apply to exper-
+imental SE-SU-FMI systems where the spin-orbit energy
+term ESOC = α2
+Rm∗/2, with αR the Rashba parameter,
+is negligible compared to ∆ and h0.
+Finally, we consider the effect of disorder in this setup.
+This might be a relevant effect as a random disorder po-
+tential will eventually break the inversion symmetry of
+the model and might lift the predicted even-odd degen-
+eracy in the limit kF L ≫ 1. However, we believe the
+energy-lifting effect might be weak in epitaxially-grown
+samples, where disorder is a relatively small effect.
+ACKNOWLEDGMENTS
+This work was partially supported by CONICET un-
+der grant PIP 0792, UNLP under grant PID X497, and
+Agencia I+D+i under PICT 2017-2081, Argentina. AML
+is grateful to Liliana Arrachea for pointing out crucial
+bibliographic references.
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+On the gate-error robustness of variational quantum algorithms
+Daniil Rabinovich,1 Ernesto Campos,1 Soumik Adhikary,1 Ekaterina
+Pankovets,1, 2 Dmitry Vinichenko,1, 3 and Jacob Biamonte4
+1Skolkovo Institute of Science and Technology, Moscow, Russian Federation
+2Moscow Institute of Physics and Technology, Moscow, Russian Federation
+3Moscow Engineering Physics Institute, Moscow, Russian Federation
+4Beijing Institute of Mathematical Sciences and Applications, Beijing, China
+Variational algorithms are designed to work within the limitations of contemporary devices and
+suffer from performance limiting errors.
+Here we identify an experimentally relevant model for
+gate errors, natural to variational quantum algorithms. We study how a quantum state prepared
+variationally decoheres under this noise model, which manifests as a perturbation to the energy
+approximation in the variational paradigm. A perturbative analysis of an optimized circuit allows
+us to determine the noise threshold for which the acceptance criteria imposed by the stability
+lemma remains satisfied. We benchmark the results against the variational quantum approximate
+optimization algorithm for 3-SAT instances and unstructured search with up to 10 qubits and 30
+layers. Finally, we observe that errors in certain gates have a significantly smaller impact on the
+quality of the prepared state. Motivated by this, we show that it is possible to reduce the execution
+time of the algorithm with minimal to no impact on the performance.
+I.
+INTRODUCTION
+Noisy Intermediate Scale Quantum (NISQ) quantum
+computing [1] suffers from limited coherence times and
+opeartion precision [2–5]. In practice we are severely lim-
+ited by the number of qubits and circuit depths that
+one may implement with reasonable fidelity.
+This has
+piratical implications in that it limits contemporary ex-
+perimental demonstrations. A host of theoretical results
+are now emerging, leading to improved understanding
+of the use of random circuit sampling as the basis of a
+scalable experimental violation of the extended Church-
+Turing thesis [6] and on the complexity analysis of NISQ
+[7]. The variational model of quantum computation is
+designed to work within these practical limitations [8–
+10]. More generally, the variational model is known to
+be computationally universal, yet these results are highly
+idealized and do not account for noise [11].
+Reminiscent of machine learning, a variational algo-
+rithm makes use of a short parameterized quantum cir-
+cuit, known as ansatz, in which parameters are itera-
+tively tuned to minimize a cost function in a quantum-to-
+classical feedback loop [12]. The cost function is typically
+given in the form of the expectation of a so called prob-
+lem Hamiltonian; where the ground state of the problem
+Hamiltonian encodes the solution of a given problem in-
+stance. Thus, by the way of cost function (energy) min-
+imization, a variational algorithm attempts to approx-
+imate the ground state of a given Hamiltonian.
+This
+strategy, however, does not provide us with a guarantee
+in regards to the quality of the approximate solution,
+where the latter is typically quantified as the overlap
+between the state prepared by the ansatz and the true
+ground state. Nevertheless, the overlap can be bounded.
+It has been shown using the stability lemma that the
+bounds can be directly related to the energy, thus allow-
+ing us to determine the energy threshold (upper bound)
+required to guarantee a fixed minimum overlap. We call
+this the acceptance threshold; a state with energy below
+this threshold is said to be accepted by the algorithm
+[11].
+Variational algorithms by their design alleviate the ef-
+fects of certain systematic limitations of NISQ devices.
+Nevertheless, variational algorithms are not immune to
+stochastic noise. While there exist some evidence that
+variational algorithms can in fact benefit from certain
+level of stochastic noise [13], in general, it is detrimental
+to the performance; stochastic noise leads to decoherence
+thus typically reducing solution quality.
+In this paper we study the extent to which errors, in the
+form of parameter alterations, affects the performance of
+variational algorithms.
+We analytically show that the
+shift in energy varies quadratically with the strength of
+noise (for small amounts of noise). We demonstrate this
+numerically for variational quantum approximate opti-
+misation in two common problems—3-SAT [14] and un-
+structured search [15, 16]. Furthermore we also found the
+performance to be more resilient to alterations in certain
+parameters. With that in mind we propose avenues to
+potentially improve performance and reduce the execu-
+tion time of variational quantum algorithms.
+II.
+PRELIMINARIES
+A.
+Variational Quantum Approximate
+Optimization
+The quantum approximate optimization algorithm
+(QAOA) [17], originally designed to approximately solve
+combinatorial optimization problems [14, 17–28], consists
+of ansatze circuits expressive enough to (in theory) emu-
+late any quantum cirucuit [19, 20].
+Consider a pseudo-Boolean function C : {0, 1}×n → R,
+the objective of the algorithm is to approximate a bit
+string that minimizes C. To accomplish this, C is first
+arXiv:2301.00048v1  [quant-ph]  30 Dec 2022
+
+2
+encoded as a problem Hamiltonian H, diagonal in the
+computational basis. The ground state H encodes the
+solution to the problem; in other words QAOA searches
+for a solution |g⟩ such that ⟨g|H|g⟩ = min H.
+The algorithm begins with an ansatz state |ψp(γ, β)⟩—
+prepared by a circuit of depth p — parameterized as:
+|ψp(γ, β)⟩ =
+p
+�
+k=1
+e−iβkHxe−iγkH |+⟩⊗n ,
+(1)
+with real parameters γk ∈ [0, 2π), βk ∈ [0, π).
+Here
+Hx = �n
+j=1 Xj is the standard one-body mixer Hamil-
+tonian with Pauli matrix Xj applied to the j-th qubit.
+The cost function is given by the expectation of the prob-
+lem Hamiltonian with respect to the ansatz state. The
+algorith minimizes this cost function to output:
+E∗ = minγ,β ⟨ψp(γ, β)| H |ψp(γ, β)⟩
+(2)
+γ∗, β∗ ∈ arg minγ,β ⟨ψp(γ, β)| H |ψp(γ, β)⟩
+(3)
+Here, |ψp(γ∗, β∗)⟩ is the approximate ground state of
+H and hence the approximate solution to C. Indeed, the
+quality of the approximation, quantified as the overlap
+between the true solution and the approximate solution,
+is not known a priori from (2).
+Nevertheless one can
+establish bounds on this quantity using the so called sta-
+bility lemma.
+B.
+Stability lemma
+The stability lemma states that if |g⟩ is the true ground
+state of H with energy Eg and ∆ is the spectral gap
+(the difference between the ground state energy and the
+energy of the first excited state) the following relation
+holds [11, 29]:
+1 − E∗ − Eg
+∆
+≤ |⟨ψp(γ∗, β∗)|g⟩|2 ≤ 1 − E∗ − Eg
+Em − Eg
+(4)
+where Em is the maximum eigenvalue of H.
+Thus to
+guarantee a non-trivial overlap one must ensure that
+E∗ ≤ Eg + ∆. We call the latter the acceptance con-
+dition.
+III.
+VARIATIONAL QUANTUM ALGORITHMS
+IN THE PRESENCE OF REALISTIC GATE
+ERRORS
+Implementation of unitary operations depends signif-
+icantly on the considered hardware. However, typically
+the implementation makes use of electromagnetic pulses,
+such as in superconducting quantum computers [30, 31],
+neutral atom based quantum computers [32, 33], and
+trapped ion based quantum computers [34, 35].
+Such
+pulses can change the population of the energy levels
+that constitute a qubit or introduce phases to the quan-
+tum amplitudes, thus controlling the state of the qubits.
+Consequently, the main contribution to gate errors comes
+from variation in pulse shaping, meaning that amplitude
+and timing of electromagnetic pulse can stochasticaly
+vary.
+In certain experimental setups, such as ground
+state ion qubits, where entangling operations are per-
+formed using the radial phonon modes [36], the variabil-
+ity in pulse shaping is the main source of gate errors.
+Angles of rotation in a typical gate operation depend
+on time averaged intensity I(t) of the electromagnetic
+pulse; θ ∝
+�
+I(t)dt. Thus, variations in the pulse shap-
+ing lead to stochastic deviations of the angles of rota-
+tions from the desired values. In other words, if a cir-
+cuit is composed of the parameterised gates {Uk(θk)}k;
+θ ∈ [0, 2π) and one tries to prepare a state |ψ(θ)⟩ =
+�
+k Uk(θk) |ψ0⟩, a different state
+|ψ(θ + δθ)⟩ =
+�
+k
+U(θk + δθk) |ψ0⟩ ,
+(5)
+is prepared instead due to the presence of errors. No-
+tice here that the perturbation δθ to the parameters is
+stochastic and is sampled with a certain probability den-
+sity p(δθ). This implies that the prepared state can be
+described by an ensemble {|ψ(θ + δθ)⟩ , p(δθ)}, which we
+can equivalently view as a density matrix
+ρ(θ) =
+�
+|ψ(θ + δθ)⟩⟨ψ(θ + δθ)|p(δθ)d(δθ).
+(6)
+Eq. (6) represents a noise model native to the vari-
+ational paradigm of quantum computing. For the rest
+of this paper we systematically study the effect of this
+noise model on the performance of QAOA for instances
+of 3-SAT and the unstructured search problem (see ap-
+pendix A for more details on the considered problems).
+In particular we study the energy perturbation around
+E∗ in different scenarios subsequently recovering the
+strength of noise under which the acceptance condition
+continues to be satisfied.
+IV.
+RESULTS
+A.
+Perturbative analysis in presence of gate errors
+Consider a problem Hamiltonian H and a variational
+ansatz |ψ(θ)⟩ = U1(θ1) . . . Uq(θq) |ψ0⟩ used to mini-
+mize H. Here the gates Uk(θk) have the form:
+Uk(θk) = eiAkθk, A2
+k = 1,
+(7)
+A typical example of such an ansatz is the checkerboard
+ansatz, with Mølmer-Sørensen (MS) gates as the entan-
+gling two qubit gates. Nevertheless, any quantum circuit
+can admit a decomposition in terms of operations that
+satisfy (7); this adds generality to this assumption.
+
+3
+In the presence of gate errors the prepared quantum
+state decoheres as |ψ(θ)��� → ρ(θ) as per (6). To obtain
+the analytic form of ρ(θ) we first note that
+Uk(θk + δθk) = Uk(θk)Uk(δθk)
+= cos δθkUk(θk) + sin δθkUk
+�
+θk + π
+2
+�
+.
+(8)
+This follows directly from (7). Therefore we get:
+|ψ(θ + δθ)⟩⟨ψ(θ + δθ)| =
+1
+�
+k1,...,kq,m1,...,mq=0
+(cos2 δθ1 tank1+m1 δθ1) . . . (cos2 δθq tankq+mq δθq)|ψk1...kq⟩⟨ψm1...mq|, (9)
+where
+|ψk1...kq⟩ = U1(θ1 + k1
+π
+2 ) . . . Uq(θq + kq
+π
+2 ) |ψ0⟩ .
+(10)
+Here we make three realistic assumptions—(a) pertur-
+bations to all the angles are independent, (b) average
+perturbation ⟨δθk⟩ = 0 and (c) the distribution p(δθk)
+vanishes quickly outside the range (−σk, σk); that is, the
+error is localized on the scale σk ≪ 1. Note that if as-
+sumption (b) does not hold, one can always shift the
+parameters as θ → θ + ⟨δθ⟩.
+Substituting (9) in (6) we arrive at the expression:
+ρ(θ) = |ψ(θ)⟩⟨ψ(θ)| + δρ,
+(11)
+where
+δρ ≈ −
+q
+�
+k=1
+ak|ψ(θ)⟩⟨ψ(θ)|+
+q
+�
+k=1
+ak|ψk⟩⟨ψk|+o(σ2
+k). (12)
+Here |ψk⟩ = |ψ00...1...00⟩ with 1 placed in the k-th posi-
+tion, and
+ak ≡ ⟨sin2 δθk⟩ =
+�
+sin2 δθkp(δθk)d(δθk) ∼ σ2
+k.
+(13)
+Notice that the derivation above does not require θ to
+be a minimum of the noiseless cost function.
+Let us
+now assume that θ∗ is a vector of parameters such that
+|ψ(θ∗)⟩ approximates the ground state of H. The noise
+induced energy perturbation around the optimal energy
+E∗ is given as:
+δE = Tr(ρ(θ∗)H) − ⟨ψ(θ∗)| H |ψ(θ∗)⟩
+≤ (Em − E∗)
+�
+k
+ak.
+(14)
+For the simplest case where each parameter is sampled
+from the same distribution (σk = σ) we can roughly es-
+timate:
+δE ≤ qσ2(Em − E∗).
+(15)
+Thus, requesting an energy threshold E ≤ Eg + ∆, we
+conclude that for σ <∼
+�
+∆ − (E∗ − Eg)
+q(Em − E∗)
+the acceptance
+condition is still satisfied.
+While our perturbative analysis holds for all varia-
+tional algorithms, we substantiate our findings numer-
+ically using QAOA. In particular we solve instances of
+3-SAT and unstructured search problems to study the
+behaviour of energy perturbation around E∗ caused by
+the presence of gate errors.
+1.
+Constant perturbation
+We begin with a simplified version of the noise model
+proposed in (6). We ran QAOA for 100 uniformly gen-
+erated 3-SAT instances of 6,8, and 10 variables with 26,
+34 and 42 clauses respectively.
+All the instances were
+selected to have a unique satisfying assignment. The in-
+stances were minimized by QAOA sequences of 15, 25
+and 30 layers respectively in order to obtain expected
+values well below the energy gap. In order to numeri-
+cally verify the behaviour of the energy perturbation, we
+vary all optimal parameters by a constant angle δ. Fig-
+ure 1 illustrates the shift in the energy for the minimized
+instances, which can be seen to have a quadratic depen-
+dence of the perturbed energy δE with respect to the
+shift δ. This is natural to expect since the parameters
+deviate from the local minimum, where linear contribu-
+tion must have vanished (a rigorous expression showing
+the quadratic behavior is derived in appendix B).
+Similar to the case of 3-SAT, for the problem of un-
+structured search we perturb optimal parameters of the
+circuit by an angle δ and plot corresponding energy in
+Fig. 2. Again, as expected, for small values of δ the en-
+ergy perturbation is quadratic which comes from the fact
+that the deviation happens around the minimum.
+
+4
+0.0000
+0.0025
+0.0050
+0.0075
+0.0100
+0.0125
+0.0150
+0.0175
+0.0200
+δ
+−0.02
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0.12
+δE
+6.0 qubits
+71.8δ2
+8.0 qubits
+160.5δ2
+10.0 qubits
+250.3δ2
+FIG. 1. Energy shift obtained by perturbing the ansatz state
+as |ψp(γ∗ + δ, β∗ + δ)⟩. The curves illustrate averages over
+100 uniformly generated 3-SAT instances of 6, 8 and 10 qubits
+with clause to variable ratio of 4.2 and unique satisfying as-
+signment. The error bars depict standard error. Polynomial
+fits of data indicates δ ∈ [0, 0.02] follow quadratic curves.
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+δ
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+δE
+6 qubits
+208.9δ2
+8 qubits
+1228.4δ2
+10 qubits
+4664.2δ2
+FIG. 2.
+Energy shift for the problem of unstructured
+search
+obtained
+by
+perturbing
+of
+the
+ansatz
+state
+as
+|ψp(γ∗ + δ, β∗ + δ)⟩.
+Polynomial fits for data points of 6,
+8 and 10 qubits follow quadratic curves in the ranges δ ∈
+[0, 0.02], [0, 0.01], [0, 0.008] respectively.
+2.
+Stochastic perturbation
+We now consider the complete noise model in (6) and
+verify our analytical prediction as shown in (15).
+For
+each 3-SAT instance, we randomly sample perturbations
+δ to each of the gates from a uniform distribution on the
+interval (−σ, σ) and average the obtained energy. Then
+we average energies over instances of the same number
+of qubits as depicted in Fig. 3. It is seen that for small
+values of σ the behaviour is quadratic as per (15). It is
+seen, that the value σ ∼ 0.075 could never violate the
+acceptance criteria, as corresponding energy error never
+exceeds the gap ∆ ≥ 1. For smaller number of qubits
+and gates the threshold value of σ increases.
+For unstructured search, we average the energy over
+δ sampled for each gate from the uniform distribution
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+σ
+0
+1
+2
+3
+4
+5
+δE
+6.0 qubits
+61.0σ2
+8.0 qubits
+118.5σ2
+10.0 qubits
+172.8σ2
+FIG. 3. Average energy shift of 100 uniformly generated 3-
+SAT instances of 6, 8 and 10 qubits with clause to variable
+ratio of 4.2 and unique satisfying assignment. The shifts are
+obtained by the perturbation of γ∗, β∗ by δ uniformly sam-
+pled from the range (−σ, σ). Error bars depict standard error.
+Polynomial fits of data indicates σ ∈ [0, 0.1] follow quadratic
+curves.
+(−σ, σ). We again recover quadratic behaviour in σ, as
+depicted in Fig. 4.
+It is seen that the same threshold
+σ ∼ 0.075 now increases energy by no more then 0.6,
+which guaranties 40% overlap with the target state.
+0.00
+0.04
+0.08
+0.12
+0.16
+0.20
+0.24
+0.28
+σ
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+δE
+6 qubits
+21.4σ2
+8 qubits
+60.3σ2
+10 qubits
+133.2σ2
+FIG. 4.
+Average energy for the problem of unstructured
+search obtained by the perturbation of γ∗, β∗ by δ uni-
+formly sampled from the range (−σ, σ).
+Error bars de-
+pict standard error.
+Polynomial fits of data points of 6,
+8 and 10 qubits follow quadratic curves in the ranges σ ∈
+[0, 0.1], [0, 0.07], [0, 0.05], respectively.
+B.
+Perturbation to individual parameters
+Here we consider a modified version of (6), where pa-
+rameters are perturbed one at a time while the rest are
+kept intact. Effect of this model on the energy is illus-
+trated in Figures 5 and 6. The results are numerical and
+are yet to be explained analytically. We observe that per-
+turbations to certain angles have a significantly smaller
+
+5
+tbh
+γ
+β
+n = 6
+p = 8
+1
+2
+3
+4
+5
+6
+7
+8
+k
+0.0105
+0.0110
+0.0115
+0.0120
+0.0125
+⟨H⟩
+δ=0.0
+δ=0.02
+δ=0.05
+δ=0.08
+δ=0.1
+1
+2
+3
+4
+5
+6
+7
+8
+k
+0.02
+0.04
+0.06
+0.08
+0.10
+0.12
+⟨H⟩
+δ=0.0
+δ=0.02
+δ=0.05
+δ=0.08
+δ=0.1
+n = 8
+p = 15
+2
+4
+6
+8
+10
+12
+14
+k
+0.0190
+0.0195
+0.0200
+0.0205
+0.0210
+⟨H⟩
+δ=0.0
+δ=0.02
+δ=0.05
+δ=0.08
+δ=0.1
+2
+4
+6
+8
+10
+12
+14
+k
+0.05
+0.10
+0.15
+0.20
+⟨H⟩
+δ=0.0
+δ=0.02
+δ=0.05
+δ=0.08
+δ=0.1
+n = 10
+p = 25
+0
+3
+6
+9
+12
+15
+18
+21
+24
+k
+0.0850
+0.0855
+0.0860
+0.0865
+0.0870
+⟨H⟩
+δ=0.0
+δ=0.02
+δ=0.05
+δ=0.08
+δ=0.1
+0
+3
+6
+9
+12
+15
+18
+21
+24
+k
+0.10
+0.15
+0.20
+0.25
+0.30
+⟨H⟩
+δ=0.0
+δ=0.02
+δ=0.05
+δ=0.08
+δ=0.1
+FIG. 5. Energy ⟨H⟩ = ⟨ψ(θ∗ + δθ)| H |ψ(θ∗ + δθ)⟩ from the unstructured search problem, where βk (right column) or γk (left
+column), from the k-th layer, are perturbed.
+effect on the energy.
+Thus we can infer that reducing
+the value of such angles would not have a significant ef-
+fect on performance but will reduce the execution time of
+the algorithm, that is texec = �p
+k=1 βk + γk. Conversely,
+we could limit the execution time as texec ≤ tmax and
+increase the number of layers, since
+min ⟨ψp| H |ψp⟩ ≥ min ⟨ψp+1| H |ψp+1⟩
+(16)
+for the same tmax.
+Reducing the execution time is important to quantum
+algorithms, since variational parameters are proportional
+to the time required to execute a gate experimentally.
+NISQ era devices suffers from limited coherence, thus
+reducing execution times can lead to more efficient hard-
+ware utilization [37, 38]. We test these ideas in the setting
+of unstructured search, as depicted in Fig. 7. Here we
+show the optimized QAOA energies for 6 qubits at mul-
+tiple depths with execution time limited to tmax. The
+highlighted green and orange rectangles depict the two
+groups of optimal angles that minimize the energy at
+each depth, as presented in [15]. Green rectangles also
+indicate the depth and texec at which an ansatz will not
+be able to decrease its energy by either increasing depth
+or tmax. Following the observations of Fig. 5, by slightly
+reducing tmax the optimizer will reduce the parameters
+to which the energy is less sensitive. This results in a
+
+6
+γ
+β
+n = 6
+p = 15
+2
+4
+6
+8
+10
+12
+14
+k
+0.08
+0.10
+0.12
+0.14
+⟨H⟩
+δ = 0.0
+δ = 0.02
+δ = 0.05
+δ = 0.08
+δ = 0.1
+2
+4
+6
+8
+10
+12
+14
+k
+0.075
+0.100
+0.125
+0.150
+0.175
+0.200
+⟨H⟩
+δ = 0.0
+δ = 0.02
+δ = 0.05
+δ = 0.08
+δ = 0.1
+n = 8
+p = 25
+0
+3
+6
+9
+12
+15
+18
+21
+24
+k
+0.08
+0.10
+0.12
+0.14
+0.16
+⟨H⟩
+δ = 0.0
+δ = 0.02
+δ = 0.05
+δ = 0.08
+δ = 0.1
+0
+3
+6
+9
+12
+15
+18
+21
+24
+k
+0.10
+0.15
+0.20
+0.25
+⟨H⟩
+δ = 0.0
+δ = 0.02
+δ = 0.05
+δ = 0.08
+δ = 0.1
+n = 10
+p = 30
+0
+4
+8
+12
+16
+20
+24
+28
+k
+0.10
+0.12
+0.14
+0.16
+0.18
+0.20
+⟨H⟩
+δ = 0.0
+δ = 0.02
+δ = 0.05
+δ = 0.08
+δ = 0.1
+0
+4
+8
+12
+16
+20
+24
+28
+k
+0.10
+0.15
+0.20
+0.25
+0.30
+⟨H⟩
+δ = 0.0
+δ = 0.02
+δ = 0.05
+δ = 0.08
+δ = 0.1
+FIG. 6.
+Average energy ⟨H⟩ = ⟨ψ(θ∗ + δθ)| H |ψ(θ∗ + δθ)⟩ of 100 uniformly generated 3-SAT instances where βk (right
+column) or γk (left column), from the k-th layer, are perturbed. The instances are of 6, 8 and 10 qubits with clause to variable
+ratio of 4.2 and unique satisfying assignment.
+slight energy increase as illustrated in Fig. 7 where to
+the left of the green rectangles we can observe darkening
+gradients.
+By contrast, orange rectangles highlight longer execu-
+tion times corresponding to different sets of angles that
+also minimize the energy for a given number of layers.
+Therefore if the optimization routine finds the a solu-
+tion corresponding to the orange rectangle, setting tmax
+to be slightly less than the texec of the orange rectangle
+will lead the optimizer to find angles corresponding to
+the green rectangle. This will amount to a considerable
+reduction in execution time.
+Alternatively increasing the number of layers while
+keeping tmax will reduce the energy. In general, for an
+arbitrary problem Hamiltonian we can not be sure if our
+optimization has returned the ideal set of angles (green
+ones in our example). For this reason, the best strategy
+would be to reduce tmax or increase depth while fixing
+tmax until performance stagnates.
+V.
+DISCUSSION
+In this study we considered a realistic noise model—
+one where the variational gate parameters are stochasti-
+cally perturbed—and demonstrated its effect on the per-
+
+7
+4.1
+6.1
+8.1
+10.1
+12.1
+14.1
+16.1
+18.1
+20.1
+22.1
+24.1
+26.1
+28.1
+30.1
+32.1
+34.1
+36.1
+38.1
+40.0
+42.0
+max execution time tmax
+8
+7
+6
+5
+4
+3
+2
+1
+depth p
+energy
+0.2
+0.4
+0.6
+0.8
+FIG. 7. Expected value for multiple combinations of depth for maximum execution times. Green and orange rectangles depict
+the two branches of angles that minimize expectation value for a given depth.
+formance of variational algorithms.
+Using a perturba-
+tive analysis we showed that the change in energy δE
+(from optimised energy E∗), caused due to the pres-
+ence of the considered gate errors, behaves quadratically
+with respect to the angle perturbations for small values
+of the perturbations. This allows us to establish upper
+bounds on the amount of perturbation such that the ac-
+ceptance condition continues to be satisfied. This guar-
+antees a fixed overlap between the target sate and the
+state prepared by the noisy variational circuit. We con-
+firm our analytical findings numerically in QAOA for two
+common problems—3-SAT and unstructured search, us-
+ing different modifications of the considered noise model.
+Moreover we observed form our numerical results that
+the algorithmic performance is more resilient to pertur-
+bations of certain variational parameters. Motivated by
+this observation we demonstrated that performance of
+QAOA with a total execution time texec = �
+k γk + βk
+is stable if retrained with a maximum execution time
+tmax = texec ± ϵ for ϵ ≪ texec. We also show that in
+some cases (a) reduction in tmax can lead to dramatic
+reductions in texec, and (b) increasing depth while fixing
+texec can lead to an energy reduction.
+While our study is primarily focused on energy pertur-
+bations around the noiseless optimum θ∗, in practice one
+has to train in the presence of noise. This would change
+optimal angles θ∗ → θ∗ + δθ∗, where shift δθ∗ increases
+with increase of the strength of the noise. Nevertheless,
+using perturbation theory around the noiseless optimum
+one can estimate δθ∗ = O(σ2), and the corresponding
+change in the energy is Tr(ρ(θ∗+δθ∗)H)−Tr(ρ(θ∗)H) =
+O(σ4). Therefore, working in the regime of weak noise
+one can safely use noiseless optimum θ∗. See appendix
+C for detailed calculation.
+VI.
+ACKNOWLEDGEMENT
+* D.R., E.C., S.A., E.P., D.V. acknowledge support
+from the research project, Leading Research Center on
+Quantum Computing (agreement No. 014/20).
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+
+9
+Appendix A: 3-SAT and unstructured search
+problems
+1.
+3-SAT
+Boolean satifyability, or SAT, is the problem of deter-
+mining weather a boolean formula written in conjunctive
+normal form (CNF) is satisfiable. It is possible to map
+any SAT instance via Karp reduction into 3-SAT, which
+are restricted to 3 literals per clause.
+In order to ap-
+proximate solutions to SAT we embed the instance into
+a Hamiltonian as
+HSAT =
+�
+j
+P(j),
+(A1)
+where j indexes clauses of an instance, and P(j) is the
+tensor product of projectors that penalizes bit string as-
+signments that do not satisfy the j-th clause.
+2.
+Unstructured search
+Consider an unstructured database S indexed by j ∈
+{0, 1}×n. Let f : {0, 1}×n → {0, 1} be a Boolean function
+(a.k.a. black box) such that:
+f(j) =
+�
+1
+iff j = t
+0
+otherwise.
+(A2)
+The task is to find t ∈ {0, 1}×n. The corresponding prob-
+lem Hamiltonian for QAOA is
+Ht = 1 − |t⟩⟨t|,
+(A3)
+thus the expected value is given by
+⟨H⟩ = 1 − |⟨t|ψp(γ, β)⟩|2.
+(A4)
+QAOA performance for unstructured search is not sen-
+sitive to the particular target state |t⟩ in the computa-
+tional basis. For any target state |t⟩ representing a binary
+string, there is a U = U † composed of X and 1 opera-
+tors such that U |0⟩⊗n = |t⟩. The overlap of an arbitrary
+state prepared by a QAOA sequence with |t⟩ is then:
+⟨t|ψp(γ, β)⟩ = ⟨t|
+p
+�
+k=1
+e−iβkHxe−iγk|t⟩⟨t| |+⟩⊗n
+= ⟨0|⊗n U
+p
+�
+k=1
+e−iβkHxe−iγkU(|0⟩⟨0|)⊗nU |+⟩⊗n
+= ⟨0|⊗n U
+p
+�
+k=1
+e−iβkHxUe−iγk(|0⟩⟨0|)⊗nU |+⟩⊗n
+= ⟨0|⊗n
+p
+�
+k=1
+e−iβkHxe−iγk(|0⟩⟨0|)⊗n |+⟩⊗n ,
+which is independent on t.
+Appendix B: Energy variation in presence of
+constant perturbations to gate parameters
+Using (9) one can calculate perturbation to the energy
+caused by a shift of the optimal angles by a constant δθ
+as
+δE = ⟨ψ(θ∗ + δθ)| H |ψ(θ∗ + δθ)⟩ − ⟨ψ(θ∗)| H |ψ(θ∗)⟩
+= −
+q
+�
+k=1
+δθ2
+kE∗ +
+q
+�
+m̸=k
+δθkδθm(⟨ψ(θ∗)| H |ψkm⟩ + h.c.)
++
+q
+�
+m,k
+δθkδθk ⟨ψm| H |ψk⟩ + o(δθkδθm)
+= 1
+2(δθ)T Hδθ + o(δθkδθm),
+(B1)
+where |ψmk⟩ = |ψ0...1...1...0⟩ with 1 placed only at m-th
+and k-th positions. H is the Hessian of the energy at
+noiseless optimum, Hij =
+∂2
+∂θi∂θj
+⟨ψ(θ)| H |ψ(θ)⟩ |θ=θ∗.
+Here we use the fact that at the optimal position linear
+contribution to the cost function necessarily vanishes. It
+is seen now that for the constant perturbation δθk = δ
+the energy changes as δE ∝ δ2.
+Appendix C: Optimal parameters variation in the
+presence of noise
+Let us use expressions (11) and (12) to estimate change
+in the energy if one accounts for shift of optimal param-
+eters θ∗ → θ∗ + δθ∗:
+Tr(ρ(θ∗ + δθ∗)H) = (1 −
+q
+�
+k=1
+ak) ⟨ψ(θ∗ + δθ∗)| H |ψ(θ∗ + δθ∗)⟩ +
+q
+�
+k=1
+ak ⟨ψk(θ∗ + δθ∗)| H |ψk(θ∗ + δθ∗)⟩ + o(σ2
+k)
+(C1)
+We introduce gradients of the noisy terms Bk
+=
+∂
+∂θ ⟨ψk(θ)| H |ψk(θ)⟩ |θ=θ∗. Notice that gradients of the
+
+10
+noiseless function ⟨ψ(θ)| H |ψ(θ)⟩ vanish at optimum.
+Then,
+Tr(ρ(θ∗ + δθ∗)H) ≈ (1 −
+q
+�
+k=1
+ak)E∗ + 1
+2(δθ∗)T Hδθ∗
++
+q
+�
+k=1
+ak[⟨ψk(θ∗)| H |ψk(θ∗)⟩ + (δθ∗)T Bk].
+(C2)
+Minimizing it with respect to δθ∗ one gets δθ∗ =
+�q
+k=1 akH−1Bk. Thus, if we account for the change of
+optimal parameters in the presence of noise, the energy
+shifts by
+Tr(ρ(θ∗ + δθ∗)H) − Tr(ρ(θ∗)H) ≈
+(δθ∗)T Hδθ∗ +
+q
+�
+k=1
+ak(δθ∗)T Bk = O(σ4).
+(C3)
+

6NFJT4oBgHgl3EQflSzb/content/tmp_files/2301.11583v1.pdf.txt

@@ -0,0 +1,1151 @@
+Tunable Strong Magnon-Magnon Coupling in Two-
+Dimensional Array of Diamond Shaped Ferromagnetic 
+Nanodots 
+Sudip Majumder1, Samiran Choudhury1, Saswati Barman2, Yoshichika Otani3, 4, 
+Anjan Barman1,* 
+1Department of Condensed Matter Physics and Material Sciences, S. N. Bose National Centre for 
+Basic Sciences, Block JD, Sector III, Salt Lake, 700106, Kolkata, India 
+2Institute for Engineering and Management, Sector V, Salt Lake, 700091, Kolkata, India 
+3 CEMS-RIKEN, 2-1 Hirosawa, Saitama, 3510198, Wako, Japan 
+4Institute for Solid State Physics, University of Tokyo, 515 Kashiwanoha, Chiba, 277 8581, 
+Kashiwa, Japan 
+*Email: abarman@bose.res.in 
+ 
+ 
+Abstract 
+Hybrid magnonics involving coupling between magnons and different quantum particles have 
+been extensively studied during past few years for varied interests including quantum 
+electrodynamics. In such systems, magnons in magnetic materials with high spin density are 
+utilized where the “coupling strength” is collectively enhanced by the square root of the number 
+of spins to overcome the weaker coupling between individual spins and the microwave field. 
+However, achievement of strong magnon-magnon coupling in a confined nanomagnets would 
+be essential for on-chip integration of such hybrid systems. Here, through intensive study of  
+interaction between different magnon modes in a Ni80Fe20 (Py) nanodot array, we demonstrate 
+that the intermodal coupling can approach the strong coupling regime with coupling strength 
+up to 0.82 GHz and cooperativity of 2.51. Micromagnetic simulations reveal that the 
+intermodal coupling is mediated by the exchange field inside each nanodot. The coupling 
+strength could be continuously tuned by varying the bias field (Hext) strength and orientation 
+(), opening routes for external control over hybrid magnonic systems. These findings could 
+greatly enrich the rapidly evolving field of quantum magnonics. 
+ 
+1. Introduction 
+Hybrid quantum systems [1] have recently attracted great attention due to their fundamental 
+importance and potential applications. It provides a new paradigm for the coherent transfer of 
+
+quantum states from one platform to another to execute quantum information processing [2,3]. 
+This significantly facilitates the research on the fundamental physics of coupling between 
+different platforms which may lead to varied applications of quantum technologies, such as: 
+quantum computing [4,5], quantum communications [6,7], and quantum sensing [8]. The 
+introduction of magnons in hybrid systems was initiated from the exploration of spin ensembles 
+coupled to microwave photons [8-10]. The higher densities of spin in magnetic materials and 
+their collective dynamics as magnons, provide ultra-strong coupling with cooperativity up to 
+103-104 [11,12]. During the last decade, extensive research has been done on magnon-magnon 
+coupling [13-19]. However, on-chip integration of hybrid systems requires downscaling the 
+dimensions of the systems to the nanometer range. The microwave cavity usually has the 
+dimension of millimeters. The coupling strength (g) is proportional to the square root of the 
+number of spins present in the magnetic material [20,21]. To increase the coupling strength the 
+number of spins in the magnetic material is usually required to be large enough (N  1013), 
+thereby restricting the size of the microwave cavity and magnet and the ensuing device 
+miniaturization towards CMOS integration. 
+To overcome this geometrical limitation of a microwave cavity, it becomes imperative to 
+search for different systems to act as nanometric resonators. In this context, the recent 
+development of interlayer magnon coupling or exchange-driven magnon-magnon coupling in 
+the magnetic systems has opened a new avenue for quantum magnonics [22-24]. In the last 
+decade, extensive studies have been done using both confined and propagating magnons in the 
+field of magnonics, which emerged as an exciting field of research. To this end single 
+nanomagnets have been studied extensively due to their geometrically confined rich volume 
+and localized magnetic modes [25-29] in nanometer dimension and their tunability with 
+different external parameters. Therefore, such systems possess great potential in quantum 
+magnonics with the possibility of developing magnon-based on-chip quantum information 
+processing systems in the GHz and THz frequency range with high energy efficiency. Recently 
+magnon-magnon coupling has been observed experimentally in ferromagnetic nanowire 
+array[15] and in single nanomagnet using micromagnetic simulation[30]. Furthermore, 
+moderate to strong magnon-magnon coupling have also been observed in Ni80Fe20 (Py) 
+nanocross array mediated by dynamic dipolar interaction [31] and anisotropic dipolar 
+interaction[32]. These studies have opened a new approach for executing and controlling this 
+phenomenon in a large variety of systems by tailoring the geometric and material parameters 
+of these artificially patterned systems and the external bias field. This leads the quest for 
+
+optimal solutions for applications in magnon-based quantum information technology. 
+ 
+Here, we have explored magon-magnon coupling in diamond-shaped Py nanodot array with 
+the aid of a broadband ferromagnetic resonance (FMR) spectrometer[33,34] and 
+micromagnetic simulations. Remarkably, we observe an avoided crossing (anticrossing) of 
+magnon modes [1] characteristic of the formation of hybrid system. Anticrossing gap of up to 
+0.82 GHz and the ensuing cooperativity value as high as 2.51 are observed. Micromagnetic 
+simulations reveal that the coupling between two magnon modes is mediated by the exchange 
+field within each nanodot. Furthermore, the coupling strength is found to be highly dependent 
+on the orientation and strength of the bias magnetic field, leading towards the possibility of 
+externally controlled hybrid magnonic devices. 
+ 
+ 
+2. Experimental Details 
+The 20-nm-thick diamond shaped Py nanodots, arranged in an array of dimensions 25 μm × 
+200 μm, were prepared on self-oxidized Si [100] substrate by using electron beam evaporation 
+(EBE), electron beam lithography (EBL), and Ar+ ion milling tools. A coplanar waveguide 
+(CPW) made of Au, having 150 nm thickness, 30 μm wide central conducting (signal) line and 
+50 Ω characteristic impedance (Fig. 1(a)) was deposited on top of each array for broadband 
+FMR measurements. The CPW is separated from the nanodot array by a 60-nm-thick insulating 
+Al2O3 layer. The fabrication details are described in section S1 of the Supplementary Materials. 
+Fig. 1(b) exhibits the scanning electron microscope (SEM) image of the diamond nanodot array 
+arranged in a square lattice having width and height of the nanodots as 325 nm (dx) and 350 
+nm (dy) and lattice constant of 400 nm. The nanomagnet’s lateral dimensions and pitch are 
+shown in the SEM image of Fig. 1(b). The SEM image shows that the fabricated structures 
+suffer from slight edge deformations and rounded corners. All these deformations have been 
+incorporated in the micromagnetic simulations as described later. The applied bias magnetic 
+
+field orientation is shown in the inset of Fig. 1(b). The spin-wave (SW) spectra from the 
+samples were measured using a broadband FMR spectrometer, consisting of a high-frequency 
+Vector Network Analyzer (VNA, Agilent PNA-L, model no.: N5230C, frequency range: 10 
+MHz to 50 GHz) and a homemade high-frequency probe station equipped with nonmagnetic 
+ground-signal-ground (GSG)-type picoprobe (GGB Industries, model no.: 40A-GSG-150-
+EDP) and a coaxial cable. One end of the CPW is shorted and the back-reflected signal is 
+collected and fed back to the VNA by the same GSG probe and the coaxial cable. From the 
+frequency dependent real part of the S-parameter in the reflection geometry (Re (S11)), different 
+SW frequencies are identified, which results in the characteristic SW spectrum of the sample. 
+Additional details of the experimental setup are given in section S2 of the Supplementary 
+Materials. 
+ 
+ 
+ 
+FIG. 1. (a) Schematic of the experimental geometry. The directions of the bias magnetic field 
+(Hext) and rf magnetic field (hrf) are shown in the schematic. (b) SEM image of diamond-shaped 
+Ni80Fe20 (Py) nanodots arranged in a square lattice having lattice constant a = 400 nm and nanodot 
+width dx = 325 nm, height dy = 350 nm. The inset again shows the orientation of Hext with respect 
+to hrf. (c) Real parts of the forward scattering parameter (S11) representing the FMR spectra at Hext 
+= 400 Oe applied at an azimuthal angle  = 0°. The observed spin-wave (SW) modes are marked 
+by down arrows. (d) Bias field (Hext) dependent SW absorption spectra of Py nanodots is shown 
+at  = 0°. The surface plots correspond to the experimental results, while the symbols represent 
+the simulated data. The color map for the surface plots and the schematic of Hext are given at the 
+bottom right corner of the figure. 
+ 
+3
+6
+9
+0.0
+0.5
+1.0
+ 
+ 
+ 
+0.0
+0.3
+0.6
+0.9
+1.2
+3
+6
+9
+12
+ M1
+ M2
+ M3
+ 
+Frequency (GHz)
+Hext (kOe)
+Frequency (GHz)
+Re S11 (Normalized) 
+M1
+M2
+M3
+Hext= 400 Oe
+(a)
+(b)
+(c)
+(d)
+500 nm
+x
+y
+Hext
+
+dx
+a
+dy
+Re S11
+Normalised
+1
+0
+(b)
+
+G
+s
+G 
+3. Results and Discussion 
+3.1. Experimental Result 
+3.1.1. Field Dependence of SW 
+       The SW absorption spectra (Re (S11)) are acquired from FMR measurements for a broad 
+range of bias magnetic field. Fig. 1(c) shows representative raw spectra at Hext = 400 Oe.  At 
+first, the magnetization of the samples are saturated along the +x direction by applying Hext = 
+1800 Oe, followed by gradual reduction of the field from 1600 Oe to 0 Oe at steps of 20 Oe in 
+a single trace. The surface plot in Fig. 1(d) displays the bias-field-dependent of SW absorption 
+spectra with their maximum power normalized to 1.0. These surface plots are generated from 
+the individual Re (S11) spectra acquired at a given applied magnetic field. Here, the bright 
+regions represent the experimental data while the symbols represent the micromagnetic 
+simulation results. The normalized surface plots help to identify three separate branches of SW, 
+among which the lowest frequency branch M1 shows maximum intensity in the entire field 
+regime. As we decrease the bias field M1 shows a dip (minimum) in f-Hext at Hext ≈ 300 Oe, 
+which indicates a mode softening due to transition in magnetization state of the nanomagnet 
+array. Other two SW modes M2 and M3 do not show any such transition and monotonically 
+decrease with the reduction in the bias field. 
+ 
+Fig. 2 shows the magnetic field dependences of the frequencies at different bias field angles. 
+The variation of magnetic field orientation creates some remarkable changes. First, the dip in 
+M1 occurring at ~300 Oe gradually disappears. Fig. 2(a) shows the f-Hext plot at  = 5, where 
+the dip shows an upward shift. At  = 15, the dip completely disappears and the M1 shows a 
+monotonic variation of frequency with the field, as shown in Fig. 2(b). Secondly, the relative 
+intensity of M2 and M3 shows a clear variation with the bias field orientation. For 5 ≤   ≤ 
+15, M2 gradually losses its intensity at the expense of gradual increment of intensity of M3, 
+which starts to dominate over M2 at  = 15. With further increment of angle, M2 further loses 
+its intensity and at  = 23 it completely disappears. Fig. 2(c) shows the f-Hext plot at  = 23 
+where a clear anticrossing between the branches representing modes M1 and M3 is observed 
+at Hext = 1060 Oe. The vertical dotted line represents the anticrossing field (Hac) in the f-Hext 
+plot. The value of Hac gradually shifts towards the lower field regime as we keep increasing . 
+
+Fig. 2(d) shows the magnetic field dispersion of SW frequencies at  = 30 where an 
+anticrossing is observed at Hext = 920 Oe in between the SW modes M1 and M3. Here, the mid 
+frequency SW mode M2* reappears, though the intensity of this mode is low. With further 
+increment of , this mode becomes more prominent and two different anticrossings are now 
+observed instead of one. One of those appears in between M1 and M2* and another one in 
+between M2* and M3. At  = 45, both of the anticrossings are observed at Hext = 475 Oe as 
+shown in Fig. 2(e). With further increment of , the first anticrossing shifts towards lower bias 
+magnetic field values, whereas the second one appears in higher bias field values. Fig. 2(f) 
+shows the magnetic field dispersion of SW frequencies at  = 60 where the first anticrossing 
+in between M1 and M2* appear at Hext = 410 Oe and second one at Hext = 600 Oe. 
+ 
+3.1.2. Angular Dependence of SW 
+ 
+The variation of SW modes and their mutual interactions show high dependence on the in-
+plane magnetic field orientation. For this reason, -dependence of SW spectra were acquired 
+at a constant bias field magnitude Hext in the range 0º ≤  ≤ 360º. In Fig. 3(a-d), we have 
+ 
+ 
+ 
+FIG. 2. Bias field (Hext) dependent SW absorption plots of Py diamond shaped nanodot array are shown 
+for the bias field orientation () of (a) 5°, (b) 15°, (c) 23°, (d) 30°, (e) 45° and (f) 60°. The surface plots 
+correspond to the experimental results, while the symbols represent the simulated data. The color map 
+for the surface plots and the schematic of the external applied field (Hext) are given at the bottom right 
+corner of the figure. 
+ 
+0
+400
+800
+1200
+3
+6
+9
+12
+ M1
+ M2*
+ M3
+ 
+0
+500
+1000
+1500
+ M1
+ M3
+ 
+ 
+ 
+Frequency (GHz)
+Hext (kOe)
+0
+400
+800
+1200
+ M1
+ M2
+ M3
+ 
+ = 15 
+ = 30 
+0
+400
+800
+1200
+ M1
+ M2*
+ M3
+ 
+ = 45 
+ = 23 
+0
+400
+800
+1200
+ M1
+ M2*
+ M3
+ 
+ = 60 
+0
+400
+800
+1200
+3
+6
+9
+12
+ M1
+ M2
+ M3
+ 
+ = 5 
+x
+y
+Hext
+
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+Re S11
+Normalized
+1
+0
+
+presented the -dependence at Hext = 200, 400, 600 and 800 Oe. To show the anticrossing points 
+we have magnified the relevant regions of the -dependent SW spectra. In the Supplementary 
+Information figure S4, we have shown the full range of -dependence. At a lower field value 
+like Hext = 200 Oe, only M1 shows angular dispersion as shown in Fig. 3(a). With an increment 
+in Hext, two more modes start to show angular dispersion. Here, mode M1 shows a sharp 
+variation of frequency with a minimum at  = 0, corresponding to the minimum observed in 
+Fig. 1(d). As we increase the field this sharp modulation gradually transforms into a continuous 
+angular variation. Fig. 3(b) shows the angular dispersion at Hext = 400 Oe. For   between 50 
+and 55, an anticrossing gap appears in between M1 and M2* which is shown by a white dotted 
+line. At a higher field of Hext = 600 Oe instead of one, two different anticrossings are observed. 
+The first one appears in between M1 and M3 at  = 40 while the 2nd one appears in between 
+M2* and M3 at  = 60. With an increment of magnetic field (e.g., 800 Oe) the first anticrossing 
+shifts towards lower angle (e.g. 35), while the second one gradually disappears as shown in 
+Fig. 3(d). Due to four fold symmetry[35] of diamond shaped nanodot array these anticrossing 
+also appear in other three quadrants of angular variation spectra of SW, which is shown in 
+section S4 of supplementary section.  
+ 
+
+   
+ 
+3.1.3. Anticrossing Strength 
+Fig. 4(a) shows the power spectrum measured at Hext = 1060 Oe, which is the anticrossing field 
+(Hac) for  = 23 configuration. The blue line represents the FMR spectra whereas the red line 
+represent the fitted spectra using an antisymmetric lorentzian function. Other FMR spectra for 
+varying anticrossing fields are presented in section S5 of Supplementary Information. The 
+magnon–magnon coupling strength g is defined as half of the peak-to-peak frequency spacing 
+at the anticrossing field, which is shown in Fig. 4(a). In order to estimate the strength of 
+interaction between these two modes, we have extracted the value of g13 and the corresponding 
+dissipation rates 1, 3 as shown in Fig. 4(a). Here, 1 and 3 are defined as half-width at half-
+maximum of the FMR peak of SW mode M1 and M3, respectively.  
+ 
+ 
+ 
+ 
+FIG. 3. Variation of SW frequency as a function of the azimuthal angle () varying from 0° to 360° for 
+bias field value fixed at (a) Hext = 200 Oe, (b) 400 Oe, (c) 600 Oe and (d) 800 Oe. The surface plots 
+correspond to the experimental results, while the symbols represent the simulated data. The colour map 
+for the surface plots and the schematic of Hext are shown on the right side of the figure.  
+ 
+ 
+ 
+6
+9
+ M2*
+ M3
+ M1
+ M2
+ 
+ 
+ 0
+-60
+60
+3
+6
+9
+ M1
+ M2
+ M3
+ M1
+ M2
+ M3
+ M2*
+ 
+0
+-60
+60
+Frequency (GHz)
+x
+y
+Hext
+
+6
+9
+ M2*
+ M3
+ M1
+ M2
+ 
+0
+-60
+60
+Azimuthal Angle,  (Degree)
+3
+6
+9
+ M2
+*
+ M3
+ M1
+ M2
+ 
+0
+-60
+60
+(a)
+(b)
+(c)
+(d)
+Re S11
+Normalized
+1
+0
+200 Oe
+400 Oe
+600 Oe
+800 Oe
+
+ 
+ 
+ 
+ 
+At  = 23 the extracted value of g13 is 0.592 GHz, while the values of 1 and 3 are found to 
+be 0.60 GHz and 0.711 GHz, respectively. Since g13  1 and 3, therefore the interaction 
+between M1 and M3 can be considered as weak coupling. In the opposite case, i.e. when g13 > 
+1 and 3 it will be considered as strong coupling between two SW branches. We have also 
+calculated magnon–magnon cooperativity (C), which is defined as C = g2/() (, = 1, 2, 
+3) and obtained C13 = 0.821 for the coupling between M1 and M3. The extracted value of g, 
+k, k, and the estimated value of C for anticrossing points corresponds to different bias field 
+ 
+ 
+g13 
+(GHz) 
+g12 
+(GHz) 
+g23 
+(GHz) 
+1(GHz) 
+2(GHz) 
+3(GHz) 
+C13 
+C12 
+C23 
+23o 
+0.592 
+- 
+- 
+0.60 
+- 
+0.711 
+0.821 
+ 
+ 
+30o 
+0.82 
+- 
+- 
+0.423 
+- 
+0.660 
+2.515 
+ 
+ 
+45o 
+- 
+0.745 
+0.255 
+0.426 
+0.645 
+0.645 
+- 
+2.019 
+0.113 
+60o 
+- 
+0.915 
+0.205 
+1.35 
+0.69 
+0.707 
+- 
+0.878 
+0.675 
+ 
+Table 1 The extracted values of coupling strength (g), FWHM (2k) and calculated cooperativity factor 
+(C) for different orientation of bias field at the anticrossing points. Values of g and k are extracted 
+from the FMR spectra). 
+ 
+ 
+FIG. 4. Real part of S11 parameter as a function of frequency to highlight the anticrossing field are 
+shown for  = (a) 23°. The frequency gap in the anticrossing mode reveals the coupling strength g. (b) 
+Variation of cooperativity factor with the orientation of bias field. It shows that coupling strength is 
+stronger at  = 30 and 45. The schematic of Hext are shown on the right side of the figure.  
+ 
+ 
+ 
+ 
+ 
+8
+10
+0.0
+0.3
+0.6
+0.9
+ 
+ 
+ = 23 
+1062 Oe
+Frequency (GHz)
+Re S11  (Normalized)
+x
+y
+Hext
+
+2k1
+2k3
+2g
+(a)
+20
+40
+60
+0
+1
+2
+3
+ C13
+ C23
+ C12
+ 
+ 
+ (Degree)
+Cooperativity 
+(b)
+
+angles are listed in Table 1. At  = 30  obtained value for g13, 1, 3 and C13 are estimated 
+0.82, 0.423, 0.66, and 2.515, respectively and here this magnon-magnon coupling falls in the 
+strong coupling regime. From Table 1, we can see that first anticrossing at  = 45 also shows 
+strong magnon-magnon coupling with C = 2.019, while the second one shows weak interaction. 
+At  = 60  both the interactions are in the weak coupling regime. Fig. 4(b) shows the -
+dependence of the C where it shows the tunability of coupling strength with the in-plane 
+magnetic field orientation. It also exhibits that the interaction between different SW branches 
+show strong coupling in-between 30 to 45 orientation.  
+ 
+3.2.  Micromagnetic Simulation 
+3.2.1. Static Magnetic Configuration 
+ 
+In Fig. 1(d) at  = 0, a sharp minimum is observed which gradually vanishes for higher values 
+of . The answer to this lies in the nanodot structure and its rich and flexible spin configurations 
+which we have simulated using OOMMF software[36]. Details of the micromagnetic 
+simulations are given in section S3 of the Supplementary Materials. The simulations reproduce 
+important features of the experimental SW spectra with nearly identical frequencies and 
+number of modes besides their relative intensity variations. The simulated static spin textures 
+within the nanomagnet array for different bias field magnitudes Hext at  = 0 and 45 are shown 
+in Fig. 5. At  = 0, the nanodot structure shows drastic variation in spin configurations with 
+Hext. It shows the formation of an S-state at the lower field regime (Hext = 100 Oe) as shown in 
+Fig. 5. At larger bias fields (e.g., Hext = 800 Oe), the spins are nearly aligned along the bias-
+field direction (x-axis) and switch to a leaf-state (Fig. 5). This transformation from S- to leaf-
+state occurs for 250 Oe ≤ Hext ≤ 350 Oe, where the SW frequency shows a minimum as a 
+function of Hext. At   = 45 , this transformation is not observed. Here, for the entire field 
+range, the static magnetic configuration shows a leaf state.  
+
+ 
+ 
+3.2.2. SW mode Characterization 
+ 
+To interpret the nature of the SW modes, we have further simulated the spatial profiles of power 
+and phase of each SW mode by using a home-built MATLAB based code Dotmag[37]. 
+OOMMF simulation provides magnetization (M (r, t)) information of each rectangular prism-
+like cell at different simulation times. By performing discrete Fourier transformation with 
+respect to time in each of these cells and subsequently extracting the power and phase of the 
+dynamic magnetization for a desired frequency gives rise to the spatial distribution of the power 
+phase profile for that particular mode. In Fig. 6, we have shown the power distribution profile 
+of SW mode at  = 45 orientation for five different fields, Hext = 200 Oe (Hext << Hac), 400 
+Oe (Hext < Hac), 475 Oe (Hac), 600 Oe (Hext  Hac) and 1000 Oe (Hext >> Hac), while the phase 
+profile for each case is shown in the inset. The power profile at Hext = 1000 Oe indicates that 
+at high bias field only existing mode is M3, which is boosted by all the available energy. With 
+a gradual decrement of bias field, two additional modes M1 and M2 appear and the power of 
+  
+ 
+ 
+FIG. 5. Simulated static magnetic configurations for Py nanodot array at four different bias magnetic-
+field magnitude (Hext) at  = 0 and  = 45. We have shown here a single nanodot from the center of 
+the array for clarity in spin configurations. The nanodot structure shows a drastic variation in spin 
+configurations with bias magnetic-field strength. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+100 Oe
+250 Oe
+800 Oe
+350 Oe
+
+x
+y
+Hext
+
+0 
+45 
+-Y
++Y
+0.0
+0.3
+0.6
+0.9
+1.2
+M1
+M2
+M3
+M4
+M5
+M'
+ 
+ 
+0.0
+0.3
+0.6
+0.9
+1.2
+ M1
+ M2
+ M3
+ M4
+ M5
+ M6
+ M7
+ 
+ 
+0.0
+0.3
+0.6
+0.9
+1.2
+3
+6
+9
+12
+ M1
+ M2
+ M3
+ M4
+ M5
+ M6
+ 
+ 
+Frequency (GHz)
+H1
+H2
+0.0
+0.3
+0.6
+0.9
+1.2
+ M*
+ M1
+ M2
+ M3
+ M4
+ M5
+ M6
+ M7
+ 
+ 
+H3
+0.0
+0.3
+0.6
+0.9
+1.2
+ M '
+  M1
+ M2
+ M3
+ M4
+ M5
+ M6
+ 
+ 
+Applied Field Hext (kOe)
+0.0
+0.3
+0.6
+0.9
+1.2
+3
+6
+9
+12
+ M1
+ M2
+ M3
+ M4
+ 
+ 
+O1
+O2
+O3
+H2
+Fig 2
+Hext
+x
+y
+1
+0
+Re (S11)
+Normalised
+
+M3 is gradually transferred to these two modes. At the anticrossing field, Hext = 475 Oe, M2 
+appears as the most intense mode although M1 and M3 have significant power at this field. At 
+lower fields, this power is gradually transferred to M1, and at 200 Oe, barring M1 other modes 
+  
+ 
+ 
+FIG. 6. Simulated spatial distribution of power and phase (in the inset) profiles corresponding to 
+different SW modes at five different bias field values for  = 45 for the Py nanodot array. The 
+applied field direction is shown at the bottom left of the figure. Symbols with different colors 
+represent different SW modes. The color map is shown at the upper right side of the figure. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+200 Oe
+400 Oe
+475 Oe
+600 Oe
+1000 Oe
+M1
+M2*
+M3
+20
+0
+Power 
+(dB)
+Phase
+(rad)
++
+-
+x
+y Hext
+
+ = 45 
+
+disappear. Similar to this energy exchange, the phase profiles also exhibit interchange of mode 
+behavior. At high bias fields (e.g., 1000 Oe), M3 shows quantized nature in BV-like geometry 
+with a quantization number n = 3. With a decrease in the field, this mode gradually transforms 
+into higher-order quantized mode and M2* is transformed into a quantized mode with n = 3. 
+At Hext = 475 Oe the quantization number of M1, and M3 are n = 5, and 7, respectively, while 
+for M2*, n = 3, which is identical to the quantization number of M3 at Hext = 1000 Oe. This 
+transformation of mode quantization number is also seen in-between M1 and M2* as we further 
+reduce the bias field and finally at Hext = 200 Oe, M1 shows a quantized behavior with n = 3. 
+This transformation of power as well as mode property from one branch of SW mode to another 
+at the anticrossing region indicates a strong interaction between these modes. For other 
+orientation like  = 23, 30 and 60, similar kind of behavior are observed, which are shown 
+in section S6 of the Supplementary Materials. 
+ 
+3.2.3. Distribution of Exchange field 
+To understand the origin of the magnon-magnon coupling and its modulation with bias 
+magnetic field, we have simulated the spatial distribution of the dipole-exchange field 
+(Exchange field distribution of each dot, which is modulated by dipolar interaction of nanodot 
+array) lines at the equilibrium for different bias field orientations. Fig. 7 shows the exchange 
+field map of nanodots array at eight different fields for  = 45 orientation. Due to inter-dot 
+dipolar interactions, a dynamic variation of exchange field line with the bias field amplitude 
+(for better viewing purpose, we just present a single nanodot) is observed. The Supplementary 
+Movie A1 shows the dynamics of this exchange field in more detail. At lower bias fields (Hext 
+<< Hac), due to dominating effect of demagnetizing field, spins take a configuration such that 
+at equilibrium condition the exchange field lines create three different regions within a single 
+dot. The field lines of center and edge regions are configured in opposite direction as denoted 
+with yellow and green arrows in Fig. 7(a). As we increase the bias field, the region around the 
+edge of the dot start to vanish and the center region gradually expands. At a very high bias field 
+(Hext   Hac), e.g., Hext = 1000 Oe, only the central region with unidirectional field lines are 
+observed inside a dot. This transformation from three mutually opposite (antiparallel) field-line 
+configuration to uniform (parallel) configuration occurs for 450 Oe ≤ Hext ≤ 500 Oe, which is 
+exactly the anticrossing field region for  = 45 orientation. This change in exchange field 
+profile can be observed much more clearly if we take a linescan along the bias field direction 
+(white dotted line in Fig. 7(a)) as shown in Fig. 7(b). In the inset, we have magnified the end 
+
+part of the linescan. Here, it is clearly visible that below the anticrossing field (Hext = Hac = 475 
+Oe) the linescan has two different local maxima  
+ 
+which transform into one maximum as we increase Hext. The exchange field profile for other 
+values of   are shown in section S7 of the Supplementary Materials, where similar 
+transformation is observed in the anticrossing field region. Our observation of correlation 
+between these two phenomena indicates that the anticrossing gap appears only when such a 
+variation of exchange field occurs due to the bias field strength as well as its orientation. The 
+internal field distribution in presence and absence of the exchange field leads to similar 
+conclusion, which we have described in section S8 of the Supplementary Materials. 
+ 
+4. Conclusion 
+In summary, the interaction between magnons confined in a sole magnonic cavity has been 
+realized in the strong coupling regime. We have investigated a bias field strength and angle-
+dependent magnetization dynamics in diamond-shaped Py nanodot arrays using the broadband 
+ferromagnetic resonance technique. Our study has demonstrated that the coupling between two 
+magnon modes is mediated by the exchange coupling inside individual nanodot. Furthermore, 
+the coupling strength is found to be highly dependent on the orientation and strength of 
+the bias magnetic field, leading towards the possibility of externally controlled hybrid 
+ 
+ 
+ 
+ 
+FIG. 7. Exchange field distributions for (a) single nanodot for eight different bias field values at  
+= 45 . Yellow and green arrows represent the direction of exchange field at the center and edge 
+position of the nanodot. We have shown here a single nanodot from the center of the array for clarity 
+in spin configurations. The color bars are shown at the right side of the figure. (b) Linescan of the 
+simulated exchange field for nanodot array along the field direction. In the inset magnified portion 
+of simulated exchange field is shown. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+360
+420
+480
+540
+0
+250
+500
+  1200 Oe
+  550 Oe
+  450 Oe
+  200 Oe
+ 
+ 
+Exchange Field (Oe)
+Distance (nm)
+x
+y
+Hext
+
+(a)
+(b)
+200 Oe
+300 Oe
+450 Oe
+500 Oe
+550 Oe
+700 Oe
+800 Oe
+1000 Oe
+-8
+-6
+-4
+-2
+  1200 Oe
+  550 Oe
+  450 Oe
+  200 Oe
+ 
+ 
+Power(dB)
+0.0
+0.3
+0.6
+0.9
+1.2
+M1
+M2
+M3
+M4
+M5
+M'
+ 
+ 
+0.0
+0.3
+0.6
+0.9
+1.2
+ M1
+ M2
+ M3
+ M4
+ M5
+ M6
+ M7
+ 
+ 
+0.0
+0.3
+0.6
+0.9
+1.2
+3
+6
+9
+12
+ M1
+ M2
+ M3
+ M4
+ M5
+ M6
+ 
+ 
+Frequency (GHz)
+H1
+H2
+0.0
+0.3
+0.6
+0.9
+1.2
+ M*
+ M1
+ M2
+ M3
+ M4
+ M5
+ M6
+ M7
+ 
+ 
+H3
+0.0
+0.3
+0.6
+0.9
+1.2
+ M '
+  M1
+ M2
+ M3
+ M4
+ M5
+ M6
+ 
+ 
+Applied Field Hext (kOe)
+0.0
+0.3
+0.6
+0.9
+1.2
+3
+6
+9
+12
+ M1
+ M2
+ M3
+ M4
+ 
+ 
+O1
+O2
+O3
+H2
+Fig 2
+Hext
+x
+y
+1
+0
+Re (S11)
+Normalised
+
+800 Oe7000e600 0e200 0e500 0emagnonic devices. The experimental results have been well reproduced by micromagnetic 
+simulation. The power and phase profiles of the resonant modes have been numerically 
+calculated to gain insight into the spatial nature of the dynamics. The transformation of power 
+as well as mode property from one branch of SW to another, apparently support the strong 
+interaction in-between these modes. Numerical study shows that the anticrossing gap appears 
+when the symmetry of exchange configuration inside each nanodot is broken due to the applied 
+bias magnetic field. We have also observed mode softening phenomena when the static 
+magnetic configuration switches from the S-state to the leaf state and with the variation of bias 
+field angle it gradually disappears.  Our findings offer a new approach toward tunable magnon-
+magnon coupling in ferromagnetic nanostructures for applications in quantum transduction 
+using magnons. 
+ 
+ 
+ 
+5. Acknowledgements 
+ 
+AB gratefully acknowledges the financial support from S. N. Bose National Centre for 
+Basic Sciences, India (Grant No. SNB/AB/18-19/211). SB acknowledges Science and 
+Engineering Research Board (SERB), India for funding (Grant no. CRG/2018/002080). SM 
+and SC acknowledge S. N. Bose National Centre for Basic Sciences for senior research 
+fellowship 
+ 
+ 
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+

79E0T4oBgHgl3EQfwQGR/content/tmp_files/2301.02630v1.pdf.txt

@@ -0,0 +1,1079 @@
+arXiv:2301.02630v1  [math.AG]  15 Aug 2022
+HEIGHT PAIRING AND NEARBY CYCLES
+A. Beilinson
+To Yuri Ivanovich Manin with deepest gratitude
+Abstract. We prove that, as was conjectured by Spencer Bloch, the Hodge period
+of some limit Hodge structures equals the height pairing of algebraic cycles on the
+resolution of singularities of the singular fiber.
+§1. Introduction: the theorem and the idea of the proof
+1.1. The Hodge period.
+Suppose we have a Q-Hodge structure E with weights
+in [−2, 0] equiped with isomorphisms ι0 : grW
+0 E = Q(0), ι−2 : grW
+−2E = Q(1).
+One defines the Hodge period ⟨E⟩ = ⟨E, ι0, ι−2⟩ ∈ R as follows.
+Consider the
+R-Hodge structure E ⊗ R. Since the weight filtration on any R-Hodge structure
+with two consequitive weights (canonically) splits one has E ⊗ R = G ⊕ grW
+−1E ⊗ R
+where G is an extension of R(0) by R(1). Our ⟨E⟩ is the class of this extension in
+Ext1(R(0), R(1)) = R.
+Remark. One computes ⟨E⟩ explicitly as follows. Let ER be E ⊗ R viewed a plain
+R-vector space, EC be its complexification. Let 1F 0 ∈ F 0 ⊂ EC be any lifting of
+ι−1
+0 (1). Then ⟨E⟩ is the image of 1F 0 in (ER + W−1EC)/(ER + (F 0 ∩ W−1EC))
+∼
+←
+W−2EC/W−2ER = C/2πiR
+∼
+← R.
+1.2. A geometric example. Let Y be a smooth proper equidimensional algebraic
+variety over C. We denote by Hi(Y ) the homology of Hi(Y (C), Q) seen as an object
+of the category of Q-Hodge structures; ditto for relative homology, etc. Let Zm(Y )
+be the group of algebraic m-cycles on Y with Q-coefficients, Zm(Y )0 := Ker(cl :
+Zm(Y ) → H2m(Y )(−m)) be the subgroup of cycles homologically equivalent to
+zero. For a closed subset P ⊂ Y let Zm(P) ⊂ Zm(Y ) be the subgroup of cycles
+supported on P, Zm(P)0 := Zm(P) ∩ Zm(Y )0. For an m-cycle A on Y we denote
+by |A| its support (which is a closed subset of Y ).
+Suppose m + m′ = dim Y − 1 and we have A ∈ Zm(Y )0, B ∈ Zm′(Y )0 such that
+|A| ∩ |B| = ∅. Set E|A|,|B| := H2m+1(Y ∖ |B|, |A|)(−m). Notice that E|B|,|A| =
+E∗
+|A|,|B|(1) by the Poicar´e duality.
+Lemma. E|A|,|B| has weights in [−2, 0]. One has grW
+−2E|A|,|B| = Zm′(|B|)∗
+0(1),
+grW
+−1E|A|,|B| = H2m+1(Y )(−m), grW
+0 E|A|,|B| = Zm(|A|)0.
+Proof. Notice that H2m(|A|)(−m) = Zm(|A|) and H>2m(|A|) = 0. By the Poincar´e
+duality Hi(Y, Y ∖|B|)(− dim Y ) = H2 dim Y −i(|B|)∗, hence H2m+2(Y, Y ∖|B|)(−m) =
+1991 Mathematics Subject Classification. Primary 14C25; Secondary 14D07.
+Key words and phrases. height pairing, nearby cycles, Hodge periods.
+Typeset by AMS-TEX
+1
+
+2
+A. BEILINSON
+(H2m′(|B|)(−m′))∗(1) and H<2m+2(Y, Y ∖ |B|) = 0. Now use the long exact ho-
+mology sequences for (Y ∖ |B|, |A|) and (Y, Y ∖ |B|).
+□
+Denote by EA,B the Hodge structure obtained from H|A|,|B| by pullback by A
+and pushforward by B:
+(1.2.1)
+Zn(|B|)∗
+0(1)
+֒→
+E|A|,|B|
+։
+Zm(|A|)0
+B ↓
+↑ A
+Q(1)
+֒→
+EA,B
+։
+Q(0)
+Our EA,B is as in 1.1, so we have ⟨EA,B⟩ ∈ R.
+1.3. The height pairing (cf. [B], [Bl1]). Let k be a subfield of C and suppose that
+Y comes from a variety Yk over k, Y = Yk ⊗ C. Let Zm(Yk) ⊂ Zm(Y ) be the
+group of algebraic cycles with Q-coefficients on Yk, Zm(Yk)0 := Zm(Yk) ∩ Zm(Y )0,
+and let CHm(Yk)0 ⊂ CHm(Yk) be their quotients modulo the rational equivalence
+relation. One checks (see §2) that if A, B as above are cycles on Yk then the class
+of ⟨EA,B⟩ in R/Q log |k×| depends only on linear equivalence classes of A and B,
+and so one has a bilinear height pairing
+(1.3.1)
+⟨ , ⟩Yk : CHm(Yk)0 ⊗ CHm′(Yk)0 → R/Q log |k×|.
+Namely ⟨a, b⟩Yk = ⟨EA,B⟩ where A, B are any cycles on Yk of classes a, b such that
+|A| ∩ |B| = ∅.
+Remark. If k = Q and we assume some motivic rationality conjectures (see (2.2.1),
+(2.2.3) of [B]) then ⟨EA,B⟩ can be corrected (by adding a finite sum of corrections
+log(p)⟨EA,B⟩p where p is a prime, ⟨EA,B⟩p is defined using the Gal(Qp)-action on
+EA,B ⊗ Qℓ) so that the resulting real number depends only on rational equivalence
+classes of A and B. In this manner (1.3.1) lifts naturally to an R-valued pairing.
+1.4. Finding elements of Chow groups that are homologically equivalent to zero
+is an art. Spencer Bloch described one situation where they naturally arise, and
+conjectured that the height pairing of his cycles can be computed in s different way,
+namely, as Hodge periods of some nearby cycles. We start with preliminaries.
+Let X be a smooth variety over C of pure dimension n ≥ 2, S be a smooth curve,
+0 ∈ S be a closed point, and f : X → S be a proper map which is smooth otside
+a finite subset {xα} of the fiber X0 = f −1(0). Let Zα be the projectivized tangent
+cone to X0 at xα; this is a hypersurface in the projectivization Pα := P(TxαX) of
+the tangent space; denote by dα its degree. We assume the next condition:
+(∗) All hypersurfaces Zα are smooth.
+Let π : Y → X0 be the blowup of X0 at {xα}. Condition (∗) implies that Y is
+a smooth variety, and Zα are pairwise disjoint divisors on Y . Set Z := ⊔Zα and
+K := Ker(Hn−2(Z) → Hn−2(Y )) = Im(Hn−1(Y, Z) → Hn−2(Z)). If n = 2 then let
+K0 ⊂ K be the subgroup of those elements A = ΣAα that deg Aα = 0 for every
+α. One has a natural map Hn−1(X0) → Hn−1(Y, Z) defined as the composition
+Hn−1(X0) → Hn−1(X0, {xα})
+∼
+← Hn−1(Y, Z).
+Lemma. (i) The map Hn−1(X0) → Hn−1(Y, Z) is an isomorphism if n > 2. If
+n = 2 it is injective and its image equals the preimage of K0 in Hn−1(Y, Z).
+(ii) Hn−1(Y, Z) has weights 1 − n and 2 − n, and grW
+2−nHn−1(Y, Z) = K. The map
+
+HEIGHT PAIRING AND NEARBY CYCLES
+3
+Hn−1(Y ) → Hn−1(Y, Z) has image W1−nHn−1(Y, Z). If n is even then Hn−1(Y )
+∼
+→
+W1−nHn−1(Y, Z).
+Proof. (i) Replace Hn−1(Y, Z) by Hn−1(X0, {xα}) and use the long exact homology
+sequence. (ii) The first assertions follow from the exact homology sequence and
+purity of weights on H·(Y ), H·(Z). The last one comes because Hn−1(Z) = 0 if n
+is even (since Zα are hypersurfaces).
+□
+1.5. Consider a variation of Q-Hodge structures V on S ∖ {0} with fibers Vs =
+Hn−1(Xs). One has a nondegenerate intersection pairing ( , ) : V ⊗ V → Q(n − 1).
+Choose a parameter t at 0 ∈ S and consider the limiting (a.k.a. nearby cycles)
+Hodge structure ψtV. Let ψun
+t V be its direct summand where the monodromy acts
+unipotently. Since ψun
+t
+commutes with duality, ( , ) yields self-duality pairing on
+it that we denote again by ( , ). One has the log of monodromy morphism N =
+NV : ψun
+t V(1) → ψun
+t V and the specialization morphism sp : ψun
+t V → Hn−1(X0).
+Let (ψun
+t V)N := Coker(NV) be the monodromy coinvariants. The next assertion
+follows from the local invariant cycles theorem, see 3.5 for a detailed proof:
+Proposition. sp factors through the isomorphism (ψun
+t V)N
+∼
+→ Hn−1(X0).
+Corollary. ψun
+t V has weights in [−n, 2 − n]. One has grW
+2−nψun
+t V = K if n > 2
+and grW
+2−nψun
+t V = K0 if n = 2. By self-duality, grW
+−nψun
+t V = (grW
+2−nψun
+t V)∗(n − 1).
+If n is even then grW
+1−nψun
+t V = Hn−1(Y ).
+Proof. Since ψun
+t V is self-dual and N is nilpotent, the claim follows from the propo-
+sition and the lemma in 1.4.
+□
+1.6. Bloch cycles. We are in the setting of 1.4; suppose n is even, n = 2m + 2. Let
+A = ΣAα be an m-cycle on Z. We say that A is a Bloch cycle if it is homologically
+equivalent to zero on Y , i.e., cl(A) lies in K(−m) ⊂ Hn−2(Z)(−m). If m = 0 then
+we demand, in addition, that cl(A) ∈ K0 ⊂ K.
+Lemma. If A is a Bloch cycle then each cl(Aα) ∈ Hn−2(Zα)(−m) is primitive.
+Proof. The composition Hn−2(Z)(−m) → Hn−2(Y )(−m) → Hn−4(Zα)(−m + 1),
+where the second arrow is the pullback by Zα ֒→ Y , sends any class c = Σcα to
+cα ∩ c1(O(−1)) (for O(−1) is the normal bundle to Zα in Y ). This composition
+kills cl(Aα) since the first arrow does.
+□
+If A, B are two Bloch cycles then we denote by Eψ
+A,B = Eψ
+A,B,t the Hodge struc-
+ture obtained from ψun
+t V(−m) by pullback by cl(A) and pushforward by cl(B)∗:
+(1.6.1)
+K∗(m + 1)
+→
+ψun
+t V(−m)
+→
+K(−m)
+cl(B)∗ ↓
+↑ cl(A)
+Q(1)
+֒→
+Eψ
+A,B
+։
+Q(0)
+Our Eψ
+A,B is as in 1.1 so we have ⟨Eψ
+A,B⟩ ∈ R.
+1.7. Examples. Consider the case when we have single singular point x0 ∈ X0 of
+f and the singularity at x0 is quadratic. Then the monodromy action on ψtV is
+unipotent, the only possible Bloch cycle is the difference A of the rulings of the
+quadric Z0, and it is actually a Bloch cycle if and only if the monodromy action on
+ψtV is nontrivial or, equivalently, the Hodge structure on Hn−1(X0) is not pure.
+
+4
+A. BEILINSON
+Lemma. (i) If m = 0 then the curve X0 can have either 1 or 2 irreducible compo-
+nents, and A is a Bloch cycle if and only if X0 is irreducible.
+(ii) If X/S is a family of quadratic hypersurfaces in Pn then A is not a Bloch cycle.
+(iii) If X/S is a family of hypersurfaces of degree d on a given smooth projective
+variety P then A is a Bloch cycle if d is large enough.
+Proof. (i) is clear. (ii) follows since the global monodromy for quadratic hypersur-
+faces is ±1, and so it can’t contain non-trivial unipotent local monodromy.
+(iii) Consider the corresponding map r : S → B := {hypersurfaces of degree
+d on P}. Since X is smooth r is transversal to the locus D ⊂ B of degenerate
+hypersurfaces. Replacing S by a germ of another transversal to D that intersects
+D near r(0) would not change the topology of X over a small disc around 0. So we
+can assume that S is a Zariski open subset of the base of a Lefschetz pencil on P.
+Then, since local monodromies of a Lefschetz pencil are all conjugate, triviality of
+one local monodromy amounts to triviality of the global monodromy. Thus A is a
+Bloch cycle if and only if the global monodromy on V is not trivial. Let us check
+that this happens for large enough d.
+If R ⊂ P is the axis of our pencil then H·(X) = H·(P) ⊕ H·−2(R)(−1), and so
+hn−1,0(P) = hn−1,0(X) which equals hn−1,0(Xs) if the global monodromy is trivial.
+Thus the monodromy is not trivial when hn−1,0(Xs) > hn−1,0(P). To finish the
+argument it remains to notice that hn−1,0(Xs) ≥ dim(H0(P, Ωn
+P (d))/H0(P, Ωn
+P )),
+and so it tends to ∞ when d → ∞.
+□
+1.8. Statement of the theorem. Now suppose we have a subfield k ⊂ C and our
+datum is defined over k, i.e., there is Xk/Sk, a closed point 0 of Sk, a parameter t
+on Sk at 0, and Bloch cycles A, B on Zk such that X/S, etc., come by base change
+k → C. Let a ∈ CHm(Yk)0, b ∈ CHm(Yk)0 be the classes of A and B. The next
+result was conjectured by Spencer Bloch:
+Theorem. One has ⟨a, b⟩Yk = ⟨Eψ
+A,B⟩ mod Q log |k×|.
+In case n = 1 the theorem was proven in [BlJS].
+Remark. Suppose we are in the situation of Remark in 1.3. If ⟨Eψ
+A,B⟩ is corrected
+in the same way as was discussed there, then the theorem lifts to an equality of real
+numbers. The proof does not change; we will not discuss it below.
+1.9. Reformulation of the theorem that discards Hodge periods; the idea of the
+proof. Let A′, B′ be cycles on Yk of classes a, b such that |A′| ∩ |B′| = ∅ (no-
+tice that they are, most probably, not supported on Zk). We want to show that
+⟨EA′,B′⟩ = ⟨Eψ
+A,B⟩ (see 1.2, 1.6). Let us compare the Hodge structures E = EA′,B′
+and Eψ = Eψ
+A,B themselves. Their weights lie in [−2, 0], and one has a canonical
+identification grW
+· E = grW
+· Eψ. Indeed, grW
+0 E(ψ) = Q(0), grW
+−2E(ψ) = Q(1) by the
+constructions, and grW
+−1E = H2m+1(Y )(−m) = grW
+−1E(ψ) by the lemma in 1.2, and
+the one in 1.4 combined with the corollary in 1.5. This identification lifts (uniquely)
+to W−1E = W−1Eψ and E/W−2E = Eψ/W−2Eψ. Indeed, the classes of extensions
+0 → H2m+1(Y )(−m) → E(ψ)/W−2E(ψ) → Q(0) → 0 both equal Deligne cohomol-
+ogy class clD(A) (a.k.a. Griffiths’ Abel-Jacobi periods) of A; by duality, the classes
+of (the duals to) extensions 0 → Q(1) → W−1E(ψ) → H2m+1(Y )(−m) → 0 both
+equal to clD(B) (see loc.cit.).
+
+HEIGHT PAIRING AND NEARBY CYCLES
+5
+Now suppose we have a Q-Hodge structure H of weight −1 and two classes
+a ∈ Ext1(Q(0), H), b ∈ Ext1(H, Q(1)).
+Consider the set EH
+a,b = EH(H)a,b of
+all Hodge structures E with weights in [−2, 0] and equipped with identifications
+grW
+0 E = Q(0), grW
+−1E = H, grW
+−2E = Q(1) such that the extensions E/W−2E and
+W−1E have classes a and b. The group Ext1(Q(0), Q(1)) = C× ⊗ Q acts on EH
+a,b by
+the Baer sum action, and EH
+a,b is a C× ⊗ Q-torsor. Notice that for q ∈ C× one has
+⟨q · E⟩ = log |q| + ⟨E⟩. Applying this format to H = H2m+1(Y )(−m), a = clD(A),
+b = clD(B) and EA′,B′, Eψ
+A,B ∈ EH
+a,b we get EA′,B′ − Eψ
+A,B ∈ C× ⊗ Q. Now the
+theorem in 1.8 follows immediately from the next result (notice that the Hodge
+periods and the height pairing play no role here):
+Theorem. One has EA′,B′ − Eψ
+A,B ∈ k× ⊗ Q ⊂ C× ⊗ Q.
+The theorem would be an immediate corollary of the motivic formalism if all
+the above constructions could be spelled in motivic world: Indeed, we would have
+then a motivic version EM of EH which is an Ext1
+M(Q(0), Q(1)) = k× ⊗ Q-torsor
+equipped with the Hodge realization embedding EM ֒→ EH; our EA′,B′, Eψ
+A,B
+would come from elements of EM, and so their difference lies in k× ⊗ Q. The only
+problem is that in the present day formalism of motives, due to Voevodsky, Ayoub,
+and Cisinski-D´eglise, the t-structure is not available, so we do not have the motivic
+version of separate homology groups like Hi(Y ). The actual proof is an exercise in
+spelling out the constructions in a way that makes the t-structure redundant.
+I am very grateful to Spencer Bloch for explaining me his conjecture and stimu-
+lating discussions (pity Spencer refused to coauthor the article), to Volodya Drinfeld
+for valuable comments and discussions, and to Luc Illusie for calling my attention
+to the construction of [I] which helped to clearify and simplify the argument.
+§2. The height pairing and the construction of EM
+a,b ∈ EM
+a,b ⊂ EH
+a,b
+This section is a variation on the theme of [Bl2] and [G].
+2.1. Let C be a stable dg category. It yields two other dg categories C(1) and C(2)
+constructed as follows:
+An object of C(1) is a closed morphism α : M → N of degree 0 in C. One has
+Hom((M, N, α), (M ′, N ′, α′))i = Hom(M, M ′)i ×Hom(N, N ′)i ×Hom(M, N ′)i+1 ⊂
+Hom(Cone(α), Cone(α′))i, and the differential is defined so that the latter embed-
+ding is a morphism of complexes; the composition of morphisms is defined in a sim-
+ilar way. There are three dg functors C(1) → C which send (M, N, α) to M, N, and
+Cone(α) respectively. We can view C(1) as the category of distinguished triangles,
+and the rotation yields an autoequivalence ρ : C(1) → C(1) which sends α : M → N
+to ρ(α) : N → Cone(α); the inverse autoequivalence is ρ−1(α) : Cone(��)[−1] → M.
+An object of C(2) is a datum (P, M, Q, α, β, κ) where P, M, Q are objects of C,
+α ∈ Hom(P, M)1, β ∈ Hom(M, Q)1 are closed maps, and κ ∈ Hom(P, Q)1 is such
+that d(κ) = βα; we sometimes abbreviate it to (α, β, κ). One can assign to such a
+datum an object E = E(α, β, κ) ∈ C which equals P ⊕ M ⊕ Q with α, β, and −κ
+added as the components to the differential.1 There is a filtration Q ⊂ Cone(β :
+M[−1], Q) ⊂ E, and morphisms in C(2) are the same as morphisms between the
+1Thus E = Cone((α, κ) : P [−1] → Cone(β : M[−1] → Q)) = Cone((κ, β) : Cone(α : P [−2] →
+M[−1]) → Q).
+
+6
+A. BEILINSON
+corresponding objects E that preserve this filtration.
+We have two dg functors
+C(2) → C(1) which send to (α, β, κ) to α : P[−1] → M and β : M → Q[1], and
+six dg functors C(2) → C which send (α, β, κ) to P, M, Q, Cone(α : P[−1] → M),
+Cone(β : M[−1] → Q), and E(α, β, κ) respectively.
+The dg category C(3) carries a natural involution σ which sends (P, M, Q, α, β, κ)
+to the object (Q[−1], E(α, β, κ), P[1], ασ, βσ, 0) where ασ and βσ are the evident
+embedding and projection.
+Remark. One can view an object (α, β, κ) ∈ C(2) as an object of C equipped with
+a 3-step filtration in two different ways. Namely, this could be E(α, β, κ) equipped
+with an evident filtration with successive quotients Q, M, and P. Or this could be
+M equipped with a filtration whose successive quotients are P[−1], E(α, β, κ), and
+Q[1]. The involution σ exchanges the two perspectives.
+2.2. For C as above we denote by C× the ∞-groupoid of its homotopy equivalences,
+by C×τ the corresponding 1-truncaded plain groupoid, and by HC the homotopy
+category of C. For S, T ∈ C set Exti(S, T ) := HiHom(S, T ) = HomHC(S, T [i]).
+Denote by Ext(S, T ) the plain Picard groupoid of extensions that corresponds to
+the two-term complex τ [0,1]Hom(S, T ).
+For M, N ∈ C let C(1)×
+M,N be the ∞-groupoid of collections (α′ : M ′ → N ′, ιM, ιN)
+where (α′ : M ′ → N ′) ∈ C(1) and ιM : M → M ′, ιN : N → N ′ are homotopy equiv-
+alences. It is equivalent to the Picard ∞-groupoid that corresponds to the complex
+τ ≤0Hom(M, N). The 1-truncated plain Picard groupoid C(1)×τ
+M,N
+corresponds to the
+two-term complex τ [−1,0]Hom(M, N).
+Similarly, for three objects P, M, Q ∈ C we have the ∞-groupoid C(2)×
+P,M,Q whose
+objects are data (P ′, M ′, Q′, α′, β′, κ′, ιP , ιM, ιQ) where (P ′, M ′, Q′, α′, β′, κ′) ∈ C(2)
+and ιP : P → P ′, ιM : M → M ′, ιQ : Q → Q′ are homotopy equivalences.
+The 1-truncated plain groupoid C(2)×τ
+P,M,Q contains a normal subgroup Ext0(P, Q) =
+HomHC(P, Q).
+Let by E = E(M) = E(P, M, Q) be the quotient groupoid.
+It
+is equivalent to the groupoid of triples (α, β, κ) where α ∈ Hom(P, M)1, β ∈
+Hom(M, Q)1 are closed maps, and κ ∈ Hom(P, Q)1/d(Hom(P, Q)0) is such that
+d(κ) = βα; a morphism (α, β, κ) → (α′, β′, κ′) in E is a pair (φ, ψ) where φ ∈
+Hom(P, M)0/d(Hom(P, M)−1), ψ ∈ Hom(M, Q)0/d(Hom(M, Q)−1) are such that
+α′ − α = d(φ), β′ − β = d(ψ), κ′ − κ = βφ + ψα + ψd(φ).
+The projection C(2)
+P,M,Q → C(1)
+P [−1],M × C(1)
+M,Q[1] yields a map of plain groupoids
+E(P, M, Q) → C(1)×τ
+P [−1],M × C(1)×τ
+M,Q[1] = Ext(P, M) × Ext(M, Q), (α, β, κ) �→ (α, β).
+The group Ext1(P, Q) acts on E by translations of κ, and non-empty fibers Eα,β
+are Ext1(P, Q)-torsors.
+Remark. E(P, M, Q) is naturally functorial with respect to P and Q: every pair
+of closed morphisms µ : P1 → P and ν : Q → Q1 yields a map E(P, M, Q) →
+E(P1, M, Q1), (α, β, κ) �→ (αµ, νβ, νκµ); is compatible with the Ext1(P, Q)-action
+via the map (µ∗, ν∗) : Ext1(P, Q) → Ext1(P1, Q1).
+Suppose Ext2(P, Q) = 0.
+Then Eα,β are non-empty, and the addition maps
+Eα1,β × Eα2,β → Eα1+α2,β, Eα,β1 × Eα,β2 → Eα,β1+β2 define on E the structure of
+an Ext1(P, Q)-biextension of (Ext(P, M), Ext(M, Q)).
+2.3. In our first example C is the dg category whose homotopy category is the
+bounded derived category DH of the category H of Q-Hodge structures, and
+
+HEIGHT PAIRING AND NEARBY CYCLES
+7
+P = Q(0), Q = Q(1). We denote the corresponding E by EH = EH(M). Then
+Ext̸=1
+DH(P, Q) = 0 and Ext1
+DH(P, Q) = C× ⊗ Q, so EH is a C× ⊗ Q-biextension of
+(Ext(Q(0), M), Ext(M, Q(1))).
+Let Ext1
+0(Q(0), M) ⊂ Ext1(Q(0), M), Ext1
+0(M, Q(1)) ⊂ Ext1(M, Q(1)) be the
+subgroups of those elements a, b that the maps H0a : Q(0) → H1M, H−1b :
+H−1M → Q(1) vanish. Let Ext0(Q(0), M) ⊂ Ext(Q(0), M), etc., be the Picard
+groupoids of such extensions.
+Lemma. Suppose that Hom(Q(0), H0M) = Hom(H0M, Q(1)) = 0.
+(i) The restriction of EH to (Ext0(Q(0), M), Ext0(M, Q(1))) descends to the C×⊗Q-
+biextension of (Ext1
+0(Q(0), M), Ext1
+0(M, Q(1))).
+(ii) EH is naturally functorial with respect to M: if ϕ : M → M ′ is a morphism,
+and we have a′ ∈ Ext1
+0(Q(0), M ′), b′ ∈ Ext1
+0(M ′, Q(1)) with ϕ∗(a) = a′, ϕ∗(b′) = b
+then there is a canonical identification EH(M)a,b = EH(M ′)a′,b′.
+(iii) The isomorphisms Ext1
+0(Q(0), M)
+∼
+→ Ext1(Q(0), H0M), Ext1
+0(M, Q(1))
+∼
+→ Ext1
+(H0M, Q(1)) which assign to an extension its zero cohomology, lifts naturally to an
+isomorphism of biextensions H0 : EH(M)
+∼
+→ EH(H0M). One has EH0 = H0E.
+Proof. Let us prove (i); the rest is clear. We need to check that for every closed α ∈
+Hom1
+0(Q(0), M), β ∈ Hom1
+0(M, Q(1)) the action of Aut(α)×Aut(β) = Hom(Q(0), M)
+×Hom(M, Q(1)) on EH
+α,β is trivial.
+Since H has homological dimension 1 our M is isomorphic to the direct sum of its
+homologies and so Aut(α) = Ext1(Q(0), H−1M), Aut(β) = Ext1(H1(M), Q(1)) by
+the condition on M. The action of (e, h) ∈ Ext1(Q(0), H−1M)×Ext1(H1(M), Q(1))
+on EH
+α,β is the translation by H−1(β)e + hH0(α) which is 0 since α, β ∈ Ext1
+0.
+□
+2.4. Lemma. Suppose that H0M is pure of weight −1 (which implies the condition
+of the lemma in 2.3). Then the function EH(M) → R, (α, β, κ) �→ ⟨E(α, β, κ)⟩ :=
+⟨H0E(α, β, κ)⟩, see 1.1, is a natural trivialization of the R-biextension log |EH(M)|.
+Proof. Everything said in 2.3 works for the category HR of R-Hodge structures. The
+extension of scalars functor H → HR, ? �→? ⊗ R, yields a morphism of our biex-
+tensions EH(M) → EHR(M ⊗ R). The map Ext1(Q(0), Q(1)) → Ext1(R(0), R(1))
+equals log | | after the standard identifications of the Ext groups with, respectively,
+C× ⊗ Q and R.
+Since Ext1(R(0), H0M ⊗ R) = Ext1(H0M ⊗ R, R(1)) = 0 by
+the condition on M, one has EHR(M ⊗ R) = EHR(H0M ⊗ R) = R.
+The map
+EH(M) → EHR(M ⊗ R) = R is ⟨ ⟩ of 1.1.
+□
+2.5. Let k ⊂ C be a subfield. Denote by DM(k) the dg category of geometric
+Voevodsky Q-motives over k. We have the Hodge realization dg functor DM(k) →
+DH, M �→ M H.
+Consider the story of 2.2 for C = DM(k) with P = Q(0),
+Q = Q(1). As before one has Ext̸=1
+DM(k)(Q(0), Q(1)) = 0, and there is a canonical
+identification Ext1(Q(0), Q(1)) = k× ⊗ Q such that the Hodge realization map
+between the Ext1’s is the embedding k× ⊗ Q ֒→ C× ⊗ Q. So for any M ∈ DM(k)
+we get a k× ⊗ Q-biextension of (Ext1(Q(0), M), Ext1(M, Q(1))) together with the
+Hodge realization morphism EM(M) → EH(M) := EH(M H) of the biextensions.
+Remark. Since the homomorphism k× ⊗ Q ֒→ C× ⊗ Q is injective, the maps of
+torsors EM(M)α,β → EH(M)α,β := EH(M)αH,βH are injective too.
+We define Ext1
+0(Q(0), M) ⊂ Ext1
+0(Q(0), M) and Ext1
+0(M, Q(1)) ⊂ Ext1(M, Q(1))
+as preimages of the Ext1
+0 subgroups of the Hodge setting by the Hodge realiza-
+
+8
+A. BEILINSON
+tion maps.
+Assume that H0M H is pure of weight −1.
+Then (i) and (ii) of
+the lemma in 2.3 remain true in the DM(k) setting (with C× replaced by k×):
+this follows from loc.cit. by Remark above. Thus we have a k× ⊗ Q-biextension
+EM(M) of (Ext1
+0(Q(0), M), Ext1
+0(M, Q(1))) together with a map of biextensions
+EM(M) → EH(M), so the lemma in 2.4 provides a natural trivialization of the
+R-biextension log |EM(M)|. The image of EM
+a,b in R/Q log |k×| depends only on
+a, b ∈ Ext1
+0(M, Q(1)) × Ext1
+0(Q(0), M), and we denote it by ⟨a, b⟩M. It is clearly
+biadditive with respect to a, b.2 We have defined a canonical height pairing
+(2.5.1)
+⟨ ⟩M : Ext1
+0(Q(0), M) × Ext1
+0(M, Q(1)) → R/Q log |k×|.
+2.6. We return to the situation of 1.3 and set M := M(Yk)(−m)[−1 − 2m] where
+M(Yk) is the motive of Yk. One has Ext1(Q(0), M) = CHm(Yk), Ext1(M, Q(1)) =
+CHm′(Yk) by the Poincar´e duality, and Ext1
+0 are the subgroups CH(Yk)0 of cycles
+homologically equivalent to zero. Therefore we get a k× ⊗ Q-biextension EM of
+(CHm(Yk)0, CHm′(Yk)0), the map of biextensions EM → EH, the trivialization of
+log |EM|, and the height pairing ⟨ , ⟩M : CHm(Yk)0 × CHm′(Yk)0 → R/Q log |k×|.
+By (iii) of the lemma in 2.3 one has H0 : EH(M)
+∼
+→ EH(H2m+1(Y )(−m)). For
+a ∈ CHm(Yk)0, b ∈ CHm′(Yk)0 pick, as in 1.3, cycles A, B that represent them
+such that |A| ∩ |B| = ∅.3 Let us construct (a, b, κA,B) ∈ EM
+a,b such that the Hodge
+realization EH
+A,B of EM
+A,B := E(a, b, κA,B) (see 2.1) has zero cohomology H0EH
+A,B
+equal to the Hodge structure EA,B from 1.3. This would imply that for our M the
+height pairing (2.5.1) equals (1.3.1).
+The composition of the maps M(|A|)
+α→ M(Yk)
+β→ M(Yk, Yk ∖ |B|) is naturally
+homotopic to 0: indeed, M(Yk, Yk ∖ |B|) := Cone(M(Yk ∖ |B|) → M(Yk)), and the
+homotopy κ|A|,|B| is M(|A|) → M(Yk ∖|B|) ⊂ Cone. Thus we have (α, β, κ|A|,|B|) ∈
+DM(2) (see 2.1). Notice that E(α, β, κ|A|,|B|) = M(Yk ∖ |B|, |A|).
+One has Ext−2m(Q(m), M(|A|)) = Zm(|A|) := the group of m-cycles supported
+on |A| (recall that dim |A| = m), and Ext2m+2(M(Yk, Yk ∖ |B|), Q(m + 1)) =
+Zm′(|B|) by the Poincar´e duality.
+Therefore we have (αA, Bβ, Bκ|A|,|B|A) =
+(Q(m)[2m+1], M(Yk), Q(m)[2m+2], αA, Bβ, Bκ|A|,|B|A) ∈ DM(2). The promised
+(a, b, κA,B) ∈ EM
+a,b is (αA, Bβ, Bκ|A|,|B|A)(−m)[−1 − 2m]. The fact that H0EH
+A,B
+equals the Hodge structure EA,B from 1.3 follows from the construction.
+§3. The unipotent nearby cycles in the Hodge setting
+3.1. A nearby cycles reminder. In this section we play with algebraic varieties over
+C. For an algebraic variety X we denote by H(X) the abelian category of perverse
+Hodge Q-sheaves of M. Saito on X, by DH(X) its bounded derived category. It sat-
+isfies the usual Grothendieck six functors formalism. Below ∗ is the Verdier duality.
+Every object of H(X), hence of DH(X), carries a canonical weight filtration.
+For F ∈ DH(X) let Γ(X, F), Γc(X, F) ∈ DH be the complex of chains, resp.
+chains with compact support, with coefficients in F equipped with the natural
+Hodge structure, H·
+(c)(X, F) := H·Γ(c)(X, F) ∈ H; set Γ(c)(X) := Γ(c)(X, Q(0)X),
+H·
+(c)(X) := H·
+(c)(X, Q(0)), and denote by ( , ) the Poincar´e duality pairing. Simi-
+larly for a closed subvariety A ⊂ X we set ΓA(X) := ΓA(X, Q(0)) ∈ DH, etc.
+2Indeed, a morphism from a biextension by a trivial group to a trivialized biextension amounts
+to a biadditive pairing.
+3Recall that |A|, |B| ⊂ Yk are supports of the cycles.
+
+HEIGHT PAIRING AND NEARBY CYCLES
+9
+Let g : X → A1 be a function on X; set X0 := g−1(0), and let v : X ∖ X0 ֒→ X,
+iX0 : X0 ֒→ X be the open and closed embeddings. One has the unipotent nearby
+cycles functor ψun
+g
+: DH(X ∖ X0) → DH(X0) that carries a natural logarithm
+of monodromy morphism N = Ng = NF : ψun
+g (F)(1) → ψun
+g (F) where F ∈
+D(X ∖ X0). It has ´etale local origin with respect to X0. For sheaves on X there
+is a natural morphism of functors ι : i∗
+X0 → ψun
+g v∗.
+There are basic canonical
+identifications:
+(i) Compatibility with Verdier duality: One has ψun
+g (F∗) = (ψun
+g F)∗(1)[2].
+(ii) Compatibility with proper direct images: Suppose h : X → T is a proper map
+and t is a function on T such that g = th; then one has ψun
+t h∗F = h∗ψun
+g F.
+(iii) One has Cone(NF) = i∗
+X0v∗F(1)[1].
+(iv) For every n > 0 one has ψun
+gnF
+∼
+→ ψun
+g F.
+These identifications are mutually compatible; (i) and (ii) are compatible with
+the action of N, and (iv) identifies Ngn with nNg. Finally, one has
+(v) ψun[−1] is t-exact for the perverse t-structure.
+Examples. Suppose that X is smooth of dimension n and F = Q(0)X∖X0. Then
+F∗ = F(n)[2n] hence ψun
+g (F)∗ = (ψun
+g F)(n − 1)[2n − 2].
+(a) If g is smooth then ιQ(0)X : Q(0)X0
+∼
+→ ψun
+g F, NF = 0.
+(b) Suppose g is semi-stable and X0 has two irreducible components Y and Y ′. By
+(a) one has natural morphisms jY ′∖Y !QY ′∖Y → ψun
+g F → jY ∖Y ′∗QY ∖Y ′ compatible
+with the N-action (we take it that on the left and right object N acts trivially).
+They form an exact triangle; its Verdier dual is the same triangle with Y and Y ′
+interchanged.
+3.2. We are in the setting of 1.4 and follow the notation there.
+Let j : U := X0 ∖ {xα} ֒→ X0 ←֓ {xα} : ⊔ixα be the complementary open
+and closed embeddings. Let I be the intersection cohomology sheaf j!∗Q(0)U =
+τ ≤n−2j∗Q(0)U 4 on X0; set I+ := π∗Q(0)Y . One has natural self-duality isomor-
+phisms I∗ = I(n − 1)[2n − 2], I+∗ = I+(n − 1)[2n − 2] (recall that Y is smooth of
+dimension n − 1 and π is proper).
+The decomposition theorem for π is easy and explicit:
+Proposition. There is a natural orthogonal direct sum decomposition
+(3.2.1)
+I+ = I ⊕ ⊕αixα∗τ [2,2n−4]Γ(Pα)
+compatible with the self-dualities.
+Proof. One has a natural orthogonal direct sum decomposition
+(3.2.2)
+Γ(Zα) = Hn−2
+prim(Zα)[2 − n] ⊕ τ ≤2n−4Γ(Pα)
+defined as follows. Consider the embedding Zα ֒→ Pα. The pullback and Gysin
+maps Γ(Pα) → Γ(Zα) → Γ(Pα)(1)[2] are mutually dual for the Poincar´e duality
+pairings, and their composition in either direction equals to the multiplication by
+c1(O(dα)).5 Thus the composition of τ ≤2n−4Γ(Pα) → Γ(Zα) → τ ≥0(Γ(Pα)(1)[2]) is
+an isomorphism. This yields a direct sum decomposition Γ(Zα) =?⊕τ ≤2n−4Γ(Pα).
+4Below τ is the usual truncation, pτ is the perverse one.
+5Since O(dα) is the normal bundle to Zα in Pα.
+
+10
+A. BEILINSON
+Since multiplication by c1(O(dα)) preserves the direct sum decomposition, the only
+nonzero cohomology of ? is Hn−2
+prim(Zα) ⊂ Hn−2(Zα), q.e.d.
+Consider the embeddings of smooth divisors iZα : Zα ֒→ Y . One has i!
+ZαQ(0)Y =
+Q(−1)[−2]Zα, i∗
+ZαQ(0)Y = Q(0)Zα, and the composition of the adjunction maps
+iZα∗i!
+ZαQ(0)Y → Q(0)Y → iZα∗i∗
+ZαQ(0)Y equals the multiplication by c1(O(−1))
+map Q(−1)[−2]Zα → Q(0)Zα.6 Apply π∗; then i!
+xαI+ = Γ(Zα)(−1)[−2], i∗
+xαI+ =
+Γ(Zα) by base change, and the composition of the adjunctions ixα∗i!
+xαI+ → I+ →
+ixα∗i∗
+xαI+ is multiplication by c1(O(−1)) map ixα∗Γ(Zα)(−1)[−2] → ixα∗Γ(Zα).
+Composing the maps τ ≤2n−6Γ(Pα) ֒→ Γ(Zα) and Γ(Zα) ։ τ [2,2n−4]Γ(Pα) that
+come from decomposition (3.2.2) from the left and from the right with the latter ad-
+junctions, we get the maps ixα∗(τ ≤2n−6Γ(Pα))(−1)[−2] → I+ → ixα∗τ [2,2n−4]Γ(Pα).
+Their composition is an isomorphism, which yields a decomposition I+ = I? ⊕
+ixα∗τ [2,2n−4]Γ(Pα). Since the adjunctions are mutually dual, the decomposition is
+orthogonal.
+By (3.2.2) one has i!
+xαI? = Hn−2
+prim(Zα)(−1)[−n] ⊕ Q(n − 1)[2 − 2n], i∗
+xαI? =
+Hn−2
+prim(Zα)[2 − n] ⊕ Q(0).
+Thus I?[n − 1] is a perverse sheaf which equals Q(0)[n − 1]U on U and has no
+subquotients supported on {xα}, and so I? = I. We are done.
+□
+Remarks. (i) The adjunction map Q(0)X0 → π∗Q(0)Y = I+ takes value in I ⊂ I+
+since Hom(Q(0)X0, ixα∗τ [2,2n−4]Γ(Pα)) = 0.
+(ii) Set B := ⊕ixα∗Hn−2
+prim(Zα)[1 − n]. By the formula for i∗
+xαI at the end of the
+previous paragraph, one has an exact triangle Q(0)X0 → I → B[1].
+3.3. As in 1.5, t is a local coordinate at 0 ∈ S; shrinking S we can assume that
+t is defined and invertible on S ∖ {0}, so X0 = (tf)−1(0). Consider the functor
+ψun
+tf : DH(X ∖ X0) → DH(X0) (see 3.1). Set R := ψun
+tf Q(0)X∖X0. By 3.1(i) one
+has a canonical self-duality identification R∗ = R(n − 1)[2n − 2] and the mutually
+dual maps Q(0)X0
+ι→ R
+ι∗
+→ Q(0)∗
+X0(1 − n)[2 − 2n] which are isomorphisms over U.
+The next result is due to Illusie [Il]; we will need it in 4.5. The reader can skip
+it at the moment and jump directly to section 3.4.
+Proposition. For every critical point xα one has canonical isomorphisms
+(3.3.1)
+i!
+xαR = Γc(Pα ∖ Zα),
+i∗
+xαR = Γ(Pα ∖ Zα)
+interchanged by the duality. The N-action on i!
+xαR, i∗
+xαR is trivial.
+Proof. (a) The claim is local at xα, so for the proof we remove from X the rest
+of critical points, and still call it X by the abuse of notation. Let S♭ → S be the
+covering of degree dα obtained by adding t♭ = t1/dα to the sheaf of functions; its
+Galois group is µdα. Set X♭ := X×SS♭ and let f ♭ : X♭ → S♭ be the projection. Our
+X♭ is a hypersurface {(x, t♭) : (tf)(x) − t♭dα = 0} in X × A1; its only singular point
+is (xα, 0). The projectivized tangent cone Qα of X♭ at (xα, 0) is a hypersurface in
+P +
+α := P(T(xα,0)X × A1). The Galois group µdα acts on X♭ hence on Qα.
+(b) Let us check that Qα is a µdα-covering of Pα completely ramified along Zα
+and ´etale over its complement, and Qα is smooth. To see this, consider the leading
+term [tf]dα(x) (of the Taylor expansion) of tf at xα; then the leading term of
+6Since O(−1) is the normal bundle to Zα in Y .
+
+HEIGHT PAIRING AND NEARBY CYCLES
+11
+(tf)(x) − t♭dα at (xα, 0) is [tf]dα(x) − t♭dα. The zeros of [tf]dα is Zα ⊂ Pα, of
+[tf]dα(x) − t♭dα is Qα ⊂ P +
+α , and so the projection Qα → Pα (x, t♭) �→ x, is as
+claimed. The smoothness of Qα follows from that of Zα.
+(c) Let π+ : X+ → X♭ be the blowup of X♭ at (xα, 0). By (b) X+ is smooth
+and the map f + := f ♭π+ : X+ → S♭ has semistable reduction at 0 ∈ S♭. The
+fiber X+
+0 has two irreducible components: one equals Y and the other Qα, and
+their intersection equals Zα. The action of µdα on X♭ yields one on X+. The
+µdα-action on X+
+0
+fixes Y and acts on Qα as described in (b).
+The projection
+π+
+0 : X+
+0 → X♭
+0 = X0 contracts Qα to xα.
+Set R+ := ψun
+tf +Q(0)X+∖X+
+0 , R♭ := ψun
+tf ♭Q(0)X♭∖X♭
+0. These are sheaves on X+
+0
+and X♭
+0 = X0 respectively that are naturally µdα-equivariant.
+By 3.1(ii) (with
+h = π+) one has a natural identification π+
+0∗R+ = R♭ compatible with the µdα-
+actions. Since the projection p : X♭ → X is a µdα-torsor over X ∖ X0 one has
+Q(0)X∖X0 = (p∗Q(0)X♭∖X♭
+0)µdα , and so, by 3.1(ii) with h = p, one has R = R♭µdα .
+Therefore R = (π+
+0∗R+)µdα .
+(d) By 3.1(iv) with g = t♭f +, n = dα, one has ψun
+tf + = ψun
+t♭f +. Our t♭f + is semi-
+stable, so we have the exact triangle jY ∖Zα!QY ∖Zα → R+ → jQα∖Zα∗QQα∖Zα
+as in Example (b) in 3.1. Applying π+
+0∗ we get an exact triangle j!QU → R♭ →
+ixα∗Γ(Qα∖Zα). Passing to µdα-invariants we get, by (b), an exact triangle j!QU →
+R → ixα∗Γ(Pα ∖ Zα); here we use the identification Γ(Qα ∖ Zα)µdα
+∼
+→ Γ(Pα ∖ Zα)
+defined as the composition Γ(Qα ∖ Zα)µdα ⊂ Γ(Qα ∖ Zα)
+tr
+→ Γ(Pα ∖ Zα). Thus
+we get the isomorphism i∗
+xαR
+∼
+→ Γ(Pα ∖ Zα) in (3.3.1). The second isomorphism
+there comes in the dual manner from the dual exact triangle jQα∖Zα!QQα∖Zα →
+R+ → jY ∖Zα∗QY ∖Zα. Since π+
+0∗ commutes with duality, the two isomorphisms are
+mutually dual, and we are done.
+□
+Let αR be the composition B
+∂→ Q(0)X0
+ι→ R where ��� is the boundary map of the
+triangle from Remark (ii) in 3.2, so I = Cone(∂). Let us compute the map i!
+xα(αR).
+Consider the standard triangle Hn−2
+prim(Zα)[1−n]
+δ→ Γc(Pα ∖Zα)
+tr
+→ Q(1−n)[2−2n]
+that comes from (3.2.2).
+Lemma. −i!
+xα(αR) equals the composition δR of the maps Hn−2
+prim(Zα)[1 − n]
+δ→
+Γc(Pα ∖ Zα)
+(3.3.1)
+=
+i!
+xαR.
+Proof. Consider the exact triangle
+(3.3.2)
+jQα∖Zα!Q(0)Qα∖Zα ⊕ jY ∖Zα!Q(0)Y ∖Zα → Q(0)X+
+0 → Q(0)Zα.
+Let (δQ, δY ) : Q(0)Zα[−1] → jQα∖Zα!Q(0)Qα∖Zα ⊕ jY ∖Zα!Q(0)Y ∖Zα be the bound-
+ary map. Its composition with the map to Q(0)X+
+0 , and hence with the further
+composition with Q(0)X+
+0
+ι→ R+, is 0.
+Therefore the sum of the compositions
+Q(0)Zα[−1]
+δQ
+−→ jQα∖Zα!
+ι→ R+ and Q(0)Zα[−1]
+δY
+−→ jY ∖Zα!
+ι→ R+ is 0.
+Ap-
+ply i!
+xαπ+
+∗ and consider the restriction of our compositions to Hn−2
+prim(Zα)[1 − n] ⊂
+Γ(Zα)[−1]. For the first one it is δR, for the second one it is i!
+xα(αR), and we are
+done.
+□
+
+12
+A. BEILINSON
+3.4. Set P := R[n − 1] = ψun
+tf Q(0)X∖X0[n − 1]; this is a perverse sheaf on X0; one
+has a canonical self-duality identification P∗ = P(n − 1). Consider the perverse
+sheaves PN := Ker(N : P → P(−1)), PN := Coker(N : P(1) → P).
+Lemma. (i) Q(0)X0[n − 1] is a perverse sheaf of weights n − 1 and n − 2 with
+grW
+n−1 = I[n − 1], grW
+n−2 = ⊕α ixα∗Hn−2
+prim(Zα).
+(ii) One has PN = Q(0)X0[n − 1], PN = (Q(0)X0[n − 1])∗(1 − n).
+(iii) P has weights in [n − 2, n]. One has Wn−1P = Q(0)X0[n − 1], P/Wn−2P =
+(Q(0)X0[n − 1])∗(1 − n), grW
+n−2P = ⊕α ixα∗Hn−2
+prim(Zα), grW
+n−1P = I[n − 1], grW
+n P =
+(grW
+n−2P)∗(1 − n).
+Proof. (i) The exact triangle from Remark (ii) in 3.2 amounts to an exact triangle
+⊕ixα∗Hn−2
+prim(Zα) → Q(0)X0[n − 1] → I[n − 1], and we are done since its left and
+right terms are pure perverse sheaves of weights n − 2 and n − 1 respectively.
+(ii) For any sheaf A on X one has a canonical exact triangle i∗
+X0A → i∗
+X0v∗v∗A →
+i!
+X0A[1]: Indeed, the map v!v∗A → v∗v∗A factors as composition v!v∗A → A →
+v∗v∗A, and so one has an exact triangle Cone(v!v∗A → A) → Cone(v!v∗A →
+v∗v∗A) → Cone(A → v∗v∗A) which is supported on X0.
+The promised exact
+triangle is its restriction to X0.
+Now take for A the perverse sheaf Q(0)X[n]. The first term of the triangle is
+Q(0)X0[n] which is perverse sheaf shifted by 1, its third term is (Q(0)X0[n−1])∗(−n)
+which is a perverse sheaf. Therefore they equal, respectively, pH−1 and pH0 of
+i∗
+X0v∗v∗Q(0)X[2n], i.e., of Cone(N : P → P(−1)) by 3.1(iii), and we are done.
+(iii) Since N is nilpotent, the weights of P are bounded from below by the
+minimum of weights of PN, which is n − 2 by (ii) and (i). By self-duality of P they
+are bounded then from above by n, and we have the first assertion. It implies that
+Wn−2P ⊂ PN. The rest follows directly from (i), (ii), and self-duality of P.
+□
+3.5. Proof of the proposition in 1.5. We use the notation in loc.cit. Injectivity of
+sp : (ψun
+t H)N → Hn−1(X0) follows from the local invariant cycles theorem. Let us
+check the surjectivity. By 3.1(ii) applied to h = f (recall that f is proper) and 3.1(v)
+applied to ψun
+t , one has ψun
+t H = H0(X0, P)(n−1). By 3.4 we have exact sequence of
+perverse sheaves 0 → ⊕α ixα∗Hn−2
+prim(Zα)(n−1) → P(n−1) → (Q(0)X0[n−1])∗ → 0.
+Its left term has finite support, and so has no cohomology in degrees ̸= 0. Therefore
+the map H0(X0, P)(n − 1) → H0(X0, (Q(0)X0[n − 1])∗) = Hn−1(X0) is surjective.
+This map equals sp, and we are done.
+□
+§4. The motivic setting and the construction of EψM
+a,b
+∈ EM
+a,b
+4.1. We are in the setting of 1.8 so k ⊂ C is a subfield and we play with varieties
+over k. Changing slightly the notation of 1.3 and 1.8, for a variety Z = Zk we set
+ZC := Z ⊗k C. The notation of §3 is preserved except that we equip from now on
+all Hodge sheaves and Hodge structures met previously with extra upper index H.
+We play with motives (a.k.a. motivic sheaves) over varieties, see [A1] and [CD].
+For a variety Z the category of constructible Q-motives over Z is denoted by
+DM(Z).
+We use Grothendieck’s six functors formalism for DM as developed
+in [CD]. Recall that DM(Spec k) = DM(k) is the category of Voevodsky’s geo-
+metric Q-motives over k.
+For a variety Z one has M(Z) = πZ!π!
+ZQ(0) where
+πZ : Z → Spec k is the structure map. For a motivic sheaf F on Z set Γ(Z, F) :=
+πZ∗F, Γc(Z, F) := πZ!F ∈ DM(k); we write Γ(c)(Z) := Γ(c)(Z, Q(0)Z). There is
+
+HEIGHT PAIRING AND NEARBY CYCLES
+13
+a Hodge realization functor DM(Z) → DH(ZC), F �→ FH, compatible with the
+six functors and the Verdier duality ∗. For a smooth Z of dimension d one has
+π!
+ZQ(0) = Q(d)Z[2d], and so M(Z) = Γc(Z)(d)[2d].
+The formalism of unipoteny nearby cycles in the setting of motivic sheaves was
+developed in §§3.4, 3.6 of [A2]. The motivic version of everything said in 3.1 holds
+except property (v) (for the t-structure is not available). The Hodge realization
+functor commutes with the nearby cycles functors.
+4.2. Notation: Notice that Hom(Q(i)[2i], Q(j)[2j]) is 0 if i ̸= j and Q for i = j,7
+and so every object M ∈ M(k) which is isomorphic to a direct sum of motives
+Q(i)[2i], i ∈ Z, can be written in a unique manner as ⊕i Vi(i)[2i] where Vi is a
+vector space (then Vi = Hom(Q(i)[2i], M)). Set τ ≤2aM := ⊕i≥−a Vi(i)[2i], etc.
+We are in the situation of 3.2 in the setting of k-varieties. As in loc.cit., I+ :=
+π∗Q(0)Y ∈ DM(X0) (so I+H is the corresponding Hodge sheaf from loc.cit.) Since
+Y is smooth and π is proper one has a natural self-duality I+∗ = I+(n−1)[2n−2].
+The t-structure in DM is not available, so we define the motivic intersection
+cohomology sheaf I using a motivic version of decomposition (3.2.1):
+Proposition. There is a natural orthogonal direct sum decomposition in DM(X0)
+(4.2.1)
+I+ = I ⊕ ⊕αixα∗τ [2,2n−4]Γ(Pα)
+whose Hodge realization is (3.2.1)
+Proof. It repeats the proof in 3.2 (minus its last paragraph). Namely, we first define
+a natural orthogonal decomposition
+(4.2.2)
+Γ(Zα) = Hn−2
+prim(Zα)[2 − n] ⊕ τ ≤2n−4Γ(Pα)
+in DM(xα) = DM(kxα) whose Hodge realization is (3.2.2).8 The construction in
+loc.cit. uses only basic six functors functoriality, so we can repeat it literally in the
+motivic setting. Then we proceed to define (4.2.1) as in loc.cit.
+□
+Set B := ⊕α ixα∗Hn−2
+prim(Zα)[1 − n] ∈ DM(X0). The self-dualities of Γ(Zα) and
+of I+, and the above orthogonal decompositions yield natural self-dualities
+(4.2.3)
+B∗ ∼
+→ B(n − 2)[2n − 2],
+I∗ ∼
+→ I(n − 1)[2n − 2].
+4.3. Lemma. (i) The adjunction χ : Q(0)X0 → π∗Q(0)Y = I+ takes values in
+I ⊂ I+.
+(ii) One has Cone(χ : Q(0)X0 → I) = B[1].
+Proof. (i) Follows since Hom(Q(0)X0, ixα∗τ [2,2n−4]Γ(Pα)) = Hom(Q(0), τ [2,2n−4]Γ(Pα))
+= 0.
+(ii) Since χ|U = idQ(0)U the cone Cone(χ) is supported on {xα}. Now i∗
+xαCone(χ) =
+7This follows since M(Pn) = ⊕i∈[0,n]Q(i)[2i] and End(M(Pn)) = CHn(Pn × Pn) = Q[0,n].
+8So Hn−2
+prim(Zα) is a notation for a motive whose Hodge realization is the primitive cohomology
+of Zα; its definition does not involve any cohomology. To construct it explicitly, choose a k-point
+z in Pα ∖ Zα. Let πz : Zα → Pn−2 be the corresponding projection; this is a finite map of degree
+dα. Then Hn−2
+prim(Zα) is the kernel of the projector d−1
+α πt
+zπz acting on M(Zα)(2 − n)[4 − 2n].
+
+14
+A. BEILINSON
+Cone(i∗
+xα(χ)) equals Hn−2
+prim(Zα)[2 − n] by (4.2.2) and the construction of I, q.e.d.
+□
+Remark. Since Exti(Q(0)X0, Q(0)∗
+X0(1−n)[2−2n]) = Exti(Q(0), M(X0)(1−n)[2−
+2n]) = CHn−1(X0, −i) we see that Ext0 = Zn−1(X0) and Ext̸=0 = 0, i.e., one has
+Hom(Q(0)X0, Q(0)∗
+X0(1 − n)[2 − 2n]) = Zn−1(X0) = Zn−1(U).
+Example. One has χ∗χ = ǫ where ǫ : Q(0)X0 → Q(0)∗
+X0(1 − n)[2 − 2n] is the map
+that corresponds to the sum of irreducible components cycle (it is enough to check
+the assertion on U where it is obvious).
+4.4. We are in the situation of 3.3 in the setting of k-varieties. Consider the functor
+ψun
+tf : DM(X ∖ X0) → DM(X0). There is a canonical morphism ι : i∗
+X0 → ψun
+tf v∗
+of functors on DM(X) and its Verdier dual ι∗ : ψun
+tf v∗ → i!
+X0. Therefore we have
+a motivic sheaf R := ψun
+tf Q(0)X∖X0 equipped with a natural self-duality R∗
+∼
+→
+R(n − 1)[2n − 2] and mutually dual maps Q(0)X0
+ι→ R
+ι∗
+→ Q(0)∗
+X0(1 − n)[2 − 2n]
+that are isomorphisms over U.
+Let ∂ : B → Q(0)X0 be the boundary map of the triangle from 4.3(ii).
+Set
+αR := ι∂ : B → R, and let βR be α∗
+R combined with the self-duality identifications
+for R and B, so we have
+(4.4.1)
+B
+αR
+−→ R
+βR
+−→ B(−1).
+Lemma-construction. The composition βRαR is homotopic to zero.
+In fact,
+there is a canonical up to a homotopy κR such that d(κR) = βRαR.
+Proof. By Remark and Example in 4.3 one has βRαR = ∂∗ι∗ι∂ = ∂∗ǫ∂ = ∂∗χ∗χ∂ =
+(χ∂)∗χ∂. Notice that χ∂ is homotopic to 0; choose a homotopy λ, d(λ) = χ∂. Now
+set κR := λ∗χ∂.
+Independence of κR up to a homotopy from the choice of λ: if λ′ is another
+homotopy as above, i.e., d(λ) = d(λ′), then κ′
+R = λ′∗χ∂ = κR + (λ′ − λ)χ∂ =
+κR + d((λ − λ′)λ).
+□
+Remark. Our κR is self-dual up to homotopy: Indeed, one has κ∗
+R = (χ∂)∗λ =
+κR + d(λ∗λ).
+4.5. Below we use the notation from 2.1, 2.2. We have defined (αR, βR, κR) ∈
+DM(X0)(2). It yields the objects ER := E(αR, βR, κR) ∈ DM(X0) and (αI, βI, κI)
+:= σ(αR, βR, κR) ∈ DM(X0)(2). As follows from Remark in 4.4 and the defini-
+tions, the above three objects are naturally self-dual.
+Proposition. There is a homotopy equivalence θ : I
+∼
+→ ER such that the maps
+βIθ : I → B[1], θ−1αI : B(−1)[−1] are a morphism of the triangle in 4.3(ii) and
+its dual. Our θ is unitary, i.e., θ∗ = θ−1.
+Proof. Recall that we have a natural homotopy equivalence (λ, χ) : Cone(∂ : B →
+Q(0)X0)
+∼
+→ I (see 4.3(ii)), and ER is the direct sum B[1] ⊕ R ⊕ B(−1)[−1] with
+(αR, −κR, βR) added to the differential (see 2.1). Our θ is the composition I
+∼
+←
+Cone(∂)
+θ′
+→ ER where θ′ is the next morphism: its restriction to B[1] ⊂ Cone(∂)
+identifies it with the first summand in ER, and its restriction to Q(0)X0 ⊂ Cone(∂)
+is (0, ι, −λ∗χ).
+
+HEIGHT PAIRING AND NEARBY CYCLES
+15
+One has θ∗θ = idI: we need to check that θ′∗ρθ′ = (λ, χ)∗(λ, χ) : Cone(∂) →
+Cone(∂∗) where ρ : ER
+∼
+→ E∗
+R(1 − n)[2 − 2n] is the self-duality for ER. As follows
+from Remark in 4.4, ρ is the matrix with the self-dualities for R and B’s on the
+diagonal and the only non-zero off-diagonal entry being λ∗λ : B → B∗(1−n)[2−2n].
+The rest is an immediate calculation.
+The assertion that βIθ is the morphism of the triangle in 4.3(ii) means that
+βIθ′ is the projection Cone(∂) → B[1] which is evident from the construction. The
+assertion that αIθ is dual to βIθ follows from the unitarity of θ once we know that
+θ is a homotopy equivalence. Let us check it.
+Our θ′ is a morphism Cone(B → Q(0)X0) → Cone(B → Cone(βR)[−1]) com-
+patible with the projections to B, and so it is enough to check that the map
+(ι, −λ∗χ) : Q(0)X0 → Cone(βR)[−1] is a homotopy equivalence. Since ι is a homo-
+topy equivalence on U, it is enough to check our claim after applying i∗
+xα.
+The story of section 3.3 uses only the six functors formalism and basic facts
+from 3.1, so it remains literally true in the motivic setting. Consider the canonical
+homotopy equivalence a : i∗
+xαR
+∼
+→ Γ(Pα ∖ Zα) of (3.3.1). By the Verdier dual
+assertion to the lemma in 3.3, a identifies i∗
+xα(βR) with minus the residue map
+r : Γ(Pα ∖ Zα) → Hn−2
+prim(Zα)(−1)[1 − n] ⊂ Γ(Zα)(−1)[−1]. By (4.2.2) we have a
+split exact triangle Q(0) → Γ(Pα ∖ Zα)
+r→ Hn−2
+prim(Zα)(−1)[1 − n], so a identifies
+i∗
+xαCone(βR)[−1]) with Q(0) ⊂ Γ(Pα∖Zα). It follows directly from the construction
+of a that ai∗
+xα(ι) coincides with the latter embedding, and we are done.
+□
+4.6. Proof of the theorem in 1.9.
+We have (αI, βI, κI) ∈ DM(X0)(2), hence
+Γ(αI, βI, κI) ∈ DM(2). For two Bloch cycles A, B of classes clA, clB ∈ Hom(Q(0),
+Hn−2
+prim(Zα)(m)) we have (cl∗
+A, clB∗)Γ(αI, βI, κI) ∈ EM(Γ(I)(m − 1)[1 − n]) =
+EM(Γ(I+)(m − 1)[1 − n]) = EM(M) where M := M(Y )(−m)[−1 − 2m]. By the
+construction the Hodge realization embedding EM(M) ֒→ EH(M) = EH(Hm(Y ))
+identifies it with Eψ
+A,B from 1.6, and we are done.
+□
+References
+[A1]
+J. Ayoub, Les six op´erations de Grothendieck et le formalisme des cycles ´evanescents
+dans le monde motivique (I), Ast´erisque 314, SMF, 2007.
+[A2]
+J. Ayoub, Les six op´erations de Grothendieck et le formalisme des cycles ´evanescents
+dans le monde motivique (II), Ast´erisque 315, SMF, 2007.
+[B]
+A. Beilinson, Height pairing between algebraic cycles, K-theory, Arithmetic and Geom-
+etry, Yu. I. Manin (Ed.), Lect. Notes in Math. 1289, Springer, 1987.
+[Bl1]
+S. Bloch, Height pairings for algebraic cycles, Journal of Pure and Applied Algebra 34
+(1984), 119–145.
+[Bl2]
+S. Bloch, Cycles and biextensions, Contemporary Mathematics 83 (1989), 19–30.
+[BlJS]
+S. Bloch, R. de Jong, E. Can Sert˜oz, Heights on curves and limits of Hodge structures,
+arXiv:2206.01220 (2022).
+[CD]
+D.-C. Cisinski, F. D´eglise, Triangulated categories of mixed motives, Springer Mono-
+graphs in Mathematics, Springer, 2019.
+[G]
+S. Gorchinskiy, Notes on the biextension of Chow groups, Motives and algebraic cycles,
+Fields Institute Commun., vol. 56, Amer. Math. Soc., 2009, pp. 111–148.
+[Il]
+L. Illusie, Sur la formule de Picard-Lefschetz, Algebraic geometry 2000, Azumino,
+Advanced Studies in Pure Math, vol. 36, Mathematical Society of Japan, 2002, pp. 249–
+268.
+

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+A Lightweight Blockchain and Fog-enabled Secure Remote Patient
+Monitoring System
+Omar Cheikhrouhoua,e,g,∗, Khaleel Mershadb, Faisal Jamilc, Redowan Mahmudd, Anis
+Koubaae, Sanaz Rahimi Moosavif
+aCES Laboratory, University of Sfax, Tunisia
+bComputer Science and Mathematics Department, Lebanese American University, Beirut, Lebanon
+cDepartment of Computer Engineering, Jeju National University, Korea
+dSchool of Electrical Engineering, Computing and Mathematical Sciences, Curtin University, Perth,
+Australia
+eRobotics and Internet of Things Lab, Prince Sultan University, Riyadh, Saudi Arabia.
+fCalifornia State University, Dominguez Hills (CSUDH)
+gISIMA, Mahdia, University of Monastir, Tunisia
+Abstract
+IoT has enabled the rapid growth of smart remote healthcare applications. These IoT-based
+remote healthcare applications deliver fast and preventive medical services to patients at risk
+or with chronic diseases. However, ensuring data security and patient privacy while exchang-
+ing sensitive medical data among medical IoT devices is still a significant concern in remote
+healthcare applications. Altered or corrupted medical data may cause wrong treatment and
+create grave health issues for patients. Moreover, current remote medical applications’ ef-
+ficiency and response time need to be addressed and improved. Considering the need for
+secure and efficient patient care, this paper proposes a lightweight Blockchain-based and Fog-
+enabled remote patient monitoring system that provides a high level of security and efficient
+response time. Simulation results and security analysis show that the proposed lightweight
+blockchain architecture fits the resource-constrained IoT devices well and is secure against
+attacks. Moreover, the augmentation of Fog computing improved the responsiveness of the
+remote patient monitoring system by 40%.
+Keywords:
+IoT, Healthcare monitoring, Lightweight Blockchain, Fog computing,
+consensus protocol.
+1. Introduction
+Healthcare IoT networks are evolving from centralized to distributed systems to con-
+nect with each other to provide patients with high-quality healthcare. According to pre-
+∗I am corresponding author
+Email addresses: omar.cheikhrouhou@isetsf.rnu.tn (Omar Cheikhrouhou),
+khaleel.mershad@lau.edu.lb (Khaleel Mershad), faisal@jejunu.ac.kr (Faisal Jamil),
+mdredowan.mahmud@curtin.edu.au (Redowan Mahmud), akoubaa@psu.edu.sa (Anis Koubaa),
+srahimimoosavi@csudh.edu (Sanaz Rahimi Moosavi)
+Preprint submitted to Elsevier
+January 10, 2023
+arXiv:2301.03551v1  [cs.CR]  9 Jan 2023
+
+dictions, the current hospital-centered healthcare monitoring systems will develop first to
+hospital–home-balanced in 2025 and then ultimately to home-centered in 2030 [1]. New
+system architectures, technologies, and computing paradigms are needed to realize such
+evolution, specifically in the Healthcare Internet of Things (HIoT) [2].
+Emerging tech-
+nologies like IoT, blockchain, and artificial intelligence have made deploying smart remote
+patient monitoring systems a fact. Indeed, IoT devices permit them to sense and moni-
+tor patients’ physiological parameters, hence exempting them from a long waiting queue at
+a doctor’s visit. All necessary physiological parameters needed by doctors can be sensed
+by the biomedical IoT devices (also known as the Internet of Medical Things devices) and
+sent remotely to the doctor, allowing the latter to decide the appropriate treatment for the
+patient [3].
+The evolution of sophisticated security attacks and the rising need for individualized
+healthcare has made it essential for medical institutions to embrace blockchain technology.
+The arrival of the blockchain provides solutions to several problems that the healthcare sys-
+tem has been facing for a long time. The growing numbers of healthcare data breaches, pa-
+tient privacy violations, counterfeit drugs, and many other issues are major reasons for steer-
+ing the blockchain market’s growth in the healthcare industry. In general, the blockchain
+brings a large number of opportunities to smart healthcare, which can be summarized as
+follows:
+• Secure access to personal health records: the decentralized blockchain system offers the
+power of controlling data access to the owner of the data itself. Smart contracts register
+and authorize users to access the patient’s data according to the patient consent policy.
+• Patient Consent Management: the fundamental features of the blockchain, such as
+transparency and immutability, enables healthcare applications to build trust among
+patients and verify compliance with consent management policies.
+• Traceability of remote treatment: the blockchain permits healthcare applications to
+create immutable and coherent electronic records (EHRs) that can be viewed by all
+stakeholders. The transparency and consistency of blockchain EHRs aid in tracing the
+medical history of patients to offer the appropriate treatment.
+• Traceability of in-home medical kits and devices: the blockchain provides immutable
+and transparent record transactions to the ownership and performance of medical kits.
+Reputation scores of medical devices and kits are saved in the blockchain using smart
+contracts.
+• Reputation-aware specialist referral services: during the treatment of a remote pa-
+tient, medical referrals and expert suggestions are acquired through smart contracts.
+Blockchain enables healthcare providers to store these referral documents on an Inter-
+Planetary File System (IPFS) server, such that an IPFS hash of the document is stored
+securely in the blockchain. The hash prevents the alteration of the stored document
+and maintains its integrity.
+2
+
+• Automated payments: blockchain provides digitally signed automatic payments to
+guarantee non-repudiated secure transactions.
+A complete discussion on the blockchain benefits to smart healthcare applications can
+be found in [4].
+Ensuring the security of the remote patient monitoring (RPM) system is a must. Since a
+vulnerability in such a system could enable attackers to steal/modify sensitive information
+and endanger the patient’s life. The blockchain has emerged as a promising technology that
+can store and secure assets through a transparent and distributed ledger. In healthcare,
+where patient data is a critical asset that needs to be securely managed, the blockchain could
+become the right technology to address this challenge and provide a secure, transparent, and
+tamper-proof management of patient healthcare data. However, the blockchain is a heavy
+system requiring much processing and communication. Lightweight IoT devices would face
+problems if they were to act as full blockchain nodes. Hence, a solution should be adopted to
+enable IoT devices to participate in the blockchain network without affecting their limited
+resources.
+The lightweight blockchain [5, 6] has been proposed to achieve this purpose.
+Here, the blockchain architecture and processes are modified to assign light roles to the IoT
+devices while allowing them to benefit from the blockchain services.
+In traditional RPM systems, patient healthcare data is stored in an Electronic Healthcare
+Record (EHR) and saved in the cloud.
+Cloud computing provides ubiquitous access to
+patients’ data through a user-centric access control model, where the user chooses which
+data and to whom he/she should give access. However, a cloud computing system presents
+the disadvantages of high latency and, therefore, cannot fit critical healthcare application
+requirements where immediate intervention is needed. More precisely, real-time detection
+and notification of abnormal situations must be implemented in the context of a heart disease
+use case. Otherwise, the patient’s life will be at risk.
+To overcome the high latency limits of cloud computing and to fit the real-time require-
+ments of most healthcare applications, we propose leveraging fog computing technology in
+this paper. In our proposed architecture, fog computing will not replace cloud computing
+but will cooperate via the lightweight blockchain to provide real-time and efficient service.
+More precisely, we introduce the fog computing layer that will host a lightweight blockchain
+application with low latency requirements. On the other hand, complex AI algorithms can
+be executed at the cloud computing layer.
+Currently, smart cities are moving towards adopting blockchain technology in many smart
+city applications. In healthcare, and especially in remote patient monitoring, the blockchain
+can change the methods in which the application is executed and managed. Integrating the
+blockchain allows healthcare managers to guarantee the transparency of public healthcare
+data and removes the need to apply trust-based mechanisms and systems to achieve this
+target. In addition, the blockchain guarantees the privacy of patients’ personal data through
+smart contracts. Moreover, the blockchain allows for fast and direct connectivity between
+healthcare officials, providers, staff, and patients. Issuing blockchain transactions allows
+these entities to communicate securely via the blockchain without intermediaries. Finally,
+the blockchain allows healthcare and smart city officials to know the origin and destination
+3
+
+of each medical resource. They can also find out how healthcare services are being used
+without compromising people’s privacy.
+To sum up, we propose a smart and secure remote patient monitoring system based
+on three technology pillars: IoT, fog computing, and blockchain. More precisely, the key
+contributions of this paper are as follows:
+• We propose the architecture of a smart and secure remote patient monitoring system.
+The proposed architecture uses IoT for patient vital signs collection and blockchain to
+guarantee the privacy and security of the patient-collected data.
+• The efficiency of the proposed architecture is achieved through the introduction of the
+fog computing layer to provide real-time response and aggregate the patients’ collected
+data.
+• To reduce the heavy demands of traditional blockchain, we modify the blockchain
+structure to include a local blockchain within the IoT ecosystems and a global chain
+at the cloud layer. Each IoT ecosystem saves the block headers of all blockchain blocks,
+the bodies of the blocks of interest to the local chain, and the smart contract functions
+needed within the local chain. On the other hand, the global chain comprises whole
+blocks and smart contracts.
+• We propose a lightweight consensus model that enables the fog nodes to participate
+in the consensus protocol without consuming a lot of processing and energy resources
+and allows IoT nodes to store only the information they need to verify the legitimacy
+and integrity of the blockchain data that they obtain from fog nodes and cloud servers.
+The remainder of this paper is as follows.
+Section 2 outlines the existing literature
+on the remote patient monitoring system using blockchain and Fog Computing. Section 3
+gives an overview of the proposed remote patient monitoring architecture with its different
+components. Section 4 describes the details of the proposed lightweight blockchain model.
+Section 5 describes the fog computing layer functions and properties.
+The performance
+evaluation of the system is discussed in Section 6. Section 7 analyses the security of the
+proposed system. Finally, we conclude and give future directions in Section 8.
+2. Related Work
+As our work is based on three technologies, namely the IoT, fog computing, and blockchain,
+in this section, we present relevant work that uses one or more of these technologies to deploy
+a healthcare solution. The discussed works are summarized in Table 1.
+4
+
+Table 1: Summary of related work
+Ref Contribution
+Use case
+Used Technologies
+Pros(+)/Cons(-)
+IoT BC
+FC
+[7]
+Cloud based remote health
+monitoring system with sig-
+nal watermarking
+ECG-based
+health monitor-
+ing
+✓
+✓
++Providing sig-
+nal
+authentica-
+tion using water-
+marking
+[8]
+A
+hierarchical
+fog-
+computing-assisted
+ar-
+chitecture for IoT health
+monitoring system
+Arrhythmia de-
+tection
+✓
+✓
++
+Map
+the
+IBM’s
+MAPE-
+K
+computing
+model
+to
+the
+healthcare
+ap-
+plication
+[1]
+They developed a smart e-
+Health gateway localized at
+the edge.
+Heart disease
+✓
+✓
++Full-system
+implementation
+[9]
+Improved the energy con-
+sumption of sensor nodes
+during
+data
+transmission
+and processing.
+Migraine disease
+✓
+✓
++Energy
+con-
+sumption reduc-
+tion
+[10]
+An
+Edge-Based
+Architec-
+ture for IoT-Healthcare ap-
+plication.
+Detect
+high-
+stress conditions
+for workers and
+athletes.
+✓
+✓
+-security
+is-
+sues
+are
+not
+addressed.
+[11]
+Used retraining of SDA in
+the
+testing
+phase
+of
+ar-
+rhythmia
+classification
+to
+add or merge features in
+the
+anomaly
+detector
++
+Blockchain for access con-
+trol
+arrhythmia clas-
+sification
+✓
++High accuracy
+[12] Used blockchain to secure
+remote patient monitoring
+General
+✓
+✓
+-Time issue
+-Key
+manage-
+ment issue
+Continued on next page
+5
+
+Table 1: Summary of related work
+Ref Contribution
+Use case
+Used Technologies
+Pros(+)/Cons(-)
+IoT BC
+FC
+[13]
+Remote health monitoring
+system using Tor to min-
+imize the latency of the
+blockchain network.
+Cardiac
+Pa-
+tients.
+Sleep
+Apnoea
+Pa-
+tients. Epileptic
+Patients
+✓
+✓
+-The accuracy of
+the system is not
+tested.
+[14]
+HealthFog: A heart disease
+analysis system based on
+ensemble deep learning and
+using integrated IoT and
+Fog computing
+Heart disease
+✓
+✓
+-Security
+is-
+sue
+are
+not
+addressed.
+[15]
+They developed an intelli-
+gent e-Health architecture
+integrating
+AI,
+IoT,
+and
+cloud computing.
+ECG-based
+arrhythmia
+detection
+✓
+✓
++Hardware
+im-
+plementation of
+AI algorithms
+[16]
+An IoT and fog comput-
+ing architecture with par-
+allelization and core allo-
+cation capabilities to accel-
+erate healthcare processor-
+intensive services
+ECG-based
+arrhythmia
+detection
+✓
+✓
++Response
+Time
+was
+im-
+proved.
+[17]
+Lightweight identity man-
+agement and access control
+scheme for IoT devices us-
+ing IOTA.
+General
+✓
+✓
+-Does not sup-
+port smart con-
+tracts
+[18]
+Blockchain based architec-
+ture to provide patient cen-
+tric data access
+Healthcare
+✓
+✓
++
+Use
+cluster-
+ing techniques to
+improve system
+scalability
+BC: Blockchain, FC: Fog Computing or any cloud computing related technologies
+Hossain et al. [7] proposed a cloud-based architecture for ECG signal monitoring. To
+authenticate the captured ECG signal, the authors add a watermark that will be checked on
+the cloud side. Moreover, the authors proposed additional services, including ECG signal
+enhancement, classification, and analysis. Azimi et al., [8] proposed a hierarchical computing
+architecture leveraging fog and cloud computing technologies.
+The authors proposed a
+methodology to partition the existing machine learning methods for fog-enabled healthcare
+IoT systems. The authors in [1] developed a smart e-Health gateway localized at the edge to
+provide several functions, including local storage, real-time local data processing, embedded
+data mining, etc. By releasing the small IoT devices from these functions, a considerable
+6
+
+amount of energy can be saved by outsourcing some loads from sensor nodes to these smart
+gateways.
+In [9], the authors also proposed the integration of the IoT and cloud computing tech-
+nologies to predict migraine disease.
+The authors’ main contributions are the design of
+low-power techniques in the radio and data processing for the sensor nodes. Moreover, the
+authors proposed workload-balancing policies for cloud computing servers.
+The authors in [10] proposed BodyEdge: an edge-based architecture for IoT-healthcare
+applications. The system was implemented in two examples of edge gateway: Raspberry Pi3
+and Zotac CI540 NANO Pc, and its performance was compared to cloud systems. Moreover,
+as a validation example, the authors have implemented the system to detect high-stress
+conditions for users in two different scenarios, namely Workers in a factory and Athletes
+training in a fitness center. In [11], authors used blockchain to secure access control to patient
+EHR. The authors proposed to store patient data in an off-chain database to overcome the
+storage constraint of the blockchain.
+The authors in [12] used a consortium blockchain based on the IBM Hyperledger platform
+to secure remote patient monitoring.
+In their proposed system, sensors interact with a
+gateway (such as a mobile phone) that implements smart contracts for data analysis and
+sends essential notifications to patients and healthcare providers. The blockchain was used
+to securely log transactions (such as data reads and doctor’s commands). However, patient
+data was stored on a local database.
+They have proved that blockchain could be used
+to resolve security concerns about the transfer and logging of data transactions in an IoT
+healthcare system. The limitation rests in perfecting the time of the transmission of the
+aggregated data sent by the gateway to the blockchain nodes.
+The authors in [13] proposed a decentralized peer-to-peer remote health monitoring sys-
+tem. The proposed architecture uses Tor hidden services for off-chain data delivery between
+patients and doctors. The authors in [14] proposed HealthFog, a Fog-based healthcare sys-
+tem that integrates Edge computing and IoT. Their work was motivated by latency-sensitive
+healthcare applications, especially deep learning-based algorithms. The proposed system was
+validated for a health disease use case.
+In [15], an AI-driven e-health solution was proposed. The solution integrates IoT with
+cloud computing. The key difference of this solution is the distribution of the AI intelligence
+across the three architecture layers, namely: the Device layer, Fog layer, and Cloud layer.
+Moreover, the authors proposed hardware implementation of the SVM, ANN, and CNN
+algorithms using digital circuits. The authors in [16] proposed a framework for accelerating
+the response to remote patients requiring the execution of smart eHealth services. Their
+proposed framework supports distributed offloading to fog servers and multicore processors’
+capacity to accelerate its execution.
+The authors in [17] proposed a blockchain-based lightweight authentication and autho-
+rization scheme for IoT devices. The proposed scheme uses distributed ledger technology
+IOTA to design a lightweight and scalable mechanism for identity management and access
+control of IoT devices. However, this solution did not support smart contacts. The authors
+in [18] proposed a blockchain-based architecture that enables data owners to define their
+desired access policies over their privacy-sensitive healthcare data. The architecture used
+7
+
+two separate chains; one for storing data transactions and one for storing access policies.
+Several lightweight blockchain architectures have been proposed in the literature [19, 20,
+21, 22, 23, 24]. For example, the ECLB protocol in [19] saves the full blockchain on edge
+nodes, while the IoT nodes store what the authors call the fragmented ledger structure,
+which contains the block headers and some of the transactions in each block that are needed
+by the lightweight node. A multi-layer blockchain model is proposed in [20]. The blockchain
+network is divided into three layers. At the first layer, ordinary IoT nodes are divided into
+clusters. At the second layer, IoT cluster heads (CHs) store the local (i.e., cluster) copy of
+the BC. The cellular base stations (BSs) store the full global BC at the third layer. Nodes
+in Layers 2 and 3 collaborate to create new blocks and execute the consensus algorithm. On
+the other hand, some IoT nodes at layer one can be peers that maintain a copy of the local
+BC and act as transaction endorsers or committers, while CHs and BSs act as Hyperledger
+orderers who order transactions and create blocks.
+From the study of the existing work, we note that several works address only the per-
+formance and energy aspect of their proposed RPMs by adding the fog layer that manages
+the computing and data processing tasks [7, 8, 1, 9, 10, 14, 15, 16]. However, these schemes
+did not address the security aspects, and therefore, they are vulnerable to attacks. Other
+schemes such as [11, 12, 13, 17, 18], addressed the security aspect by adopting the blockchain
+technology, however, they used classical blockchain platforms and architectures that cannot
+fit the resource-constrained IoT devices. In this paper, we leverage the fog computing layer
+not only to improve the performance of the RPM system but also to lighten the load of
+blockchain technology. More precisely, the fog layer permits the proposal of a lightweight
+blockchain architecture that provides security services adaptable to resources constrained
+IoT devices. Moreover, the proposed consensus mechanism frees the architecture from the
+burden of classical consensus algorithms such as PoW or PoS [25, 26].
+3. The Proposed RPM System Overview
+This section gives an overview of the proposed RPM system. It highlights its three-layer
+architecture and the different communication interaction between the components.
+3.1. RPM System Architecture
+The proposed remote patient monitoring system is a three-layer architecture as shown
+in Figure 1, and which are:
+• The IoT devices layer: This layer, composed of biomedical sensor nodes, wearable
+sensor nodes, and IoT medical devices, is responsible for collecting the vital signs of the
+monitored patient. These sensor readouts are collected continuously; however, their
+transmission to the gateway node located at the fog layer can be done periodically. The
+transmission period depends on the nature of the vital sign and generally is determined
+by the patient supervising doctors.
+• The Fog Computing layer: This layer is responsible for the lightweight processing
+of vital signs received from the IoT layer. For example, suppose the monitored vital
+8
+
+Cloud Layer
+DB
+DB
+DB
+Gateway
+Gateway
+Gateway
+Gateway
+Edge/ Fog Layer
+IoT Layer
+Global Blockchain
+Big data Analytics
+Patient
+Doctor
+IoMT devices
+GetVitalSign
+Write/Order
+Local Blockchain
+GiveAccess/
+Audit Data
+GetVitalSign/ 
+GetNotification
+cloud 
+server 
+cloud 
+server 
+cloud 
+server 
+Blockchain 
+update
+Figure 1: The three-layer architecture of the proposed remote patient monitoring system
+sign exceeds a specific threshold. In that case, an alert message will be triggered and
+sent to the patient and the supervising doctor to make the right decision. Moreover,
+the fog layer first decides which data needs to be recorded in the blockchain network
+and then interacts with this latter. The fog layer will also aggregate continuous sensed
+data before sending it to the cloud server for permanent storage and data analytics.
+Additionally, the fog layer contains IoT gateways that include the local blockchain
+network, which is a subset of the global blockchain network (please refer to Section 5 for
+full description). In the proposed system, the fog computing module consists of many
+geographical intelligent gateways, that is, forming the fog. Each gateway supports
+different protocols for communication and serves as a point of contact between the
+sensor network and the cloud. It collects data from different sub-networks, translates
+protocols, and offers other higher-level services, including filters, data aggregation,
+analysis, and so on. The fog computing layer extends cloud computing to the edge of
+the network and its facilities. From cloud to end users/devices, the fog recognizes real-
+time interaction, dense geographical distribution, heterogeneity, accessibility support,
+pre-processing interoperability along with cloud interaction [27]. This enables latency
+to be decreased, particularly for real-time applications such as in-house IoT monitoring
+of patients. The fog reduces contact with the cloud, particularly in the event of a loss
+of cloud connectivity, where the data is stored locally on these gateways, and patients’
+data is sent to the cloud when the connection is restored.
+• The Cloud Computing layer: This layer is responsible for permanent data storage
+and data analytics. Complex AI and deep learning algorithms can be implemented
+at this layer for data classification, disease detection and prediction, and treatment
+9
+
++OAEplan decision.
+Moreover, in the proposed architecture, cloud servers play the role
+of full blockchain nodes that store the full copy of the blockchain and participate in
+transaction validation, block generation, and consensus. The blockchain records pa-
+tient data and actions of patients and caregivers and permits the patients to decide to
+whom they give access to their data. Moreover, blockchain technology contains pieces
+of code called smart contracts that can be automatically triggered when an event is
+achieved. These smart contracts are a powerful tool for a remote patient monitoring
+system as they can trigger an alarm and notify the doctor in an abnormal situation
+(for example, when the vital sign value exceeds a specific threshold). In addition, the
+blockchain is used to ensure patient data privacy and the system’s security. First,
+thanks to blockchain technology, the patient will be given an anonymous identity.
+This permits hiding the real patient’s identity; therefore, doctors can treat his/her
+data privately. Moreover, in our system, we propose to use a private blockchain. This
+type of blockchain has the advantage of restricting access to users’ data to only au-
+thorized persons (such as patients, doctors, and caregivers). Furthermore, blockchain
+architecture permits a patient-centric data management architecture. More precisely,
+the patient will decide to whom he/she shares data access (please refer to Section 4
+for more details).
+3.2. Communication Models
+The fog layer enables us to control access to IoT devices for medical applications. Each
+fog node manages and operates a group of medical IoT devices. This layer also interacts
+with a network of fog nodes allowed by blockchain, which function together on the Internet.
+All the related smart medical devices are connected with the closest blockchain-enabled
+fog node, e.g., in an in-house monitoring scenario. These blockchain-enabled fog nodes are
+communicated by IoT nodes and system users for authentication, authorization, and safe
+communication synchronization. An intelligent contract with a collection of rules can also
+be established on top of the fog nodes allowed by blockchain. Furthermore, the consensus
+algorithm is performed to validate the transactions and blocks for those transactions after
+they are created.
+Transaction blocks can be exchanged between cloud servers and the
+blockchain-enabled fog nodes or between the fog nodes to support robust authentication,
+permission, and distributed secure communication. The proposed solution mainly includes
+four forms of communication:
+(1) Medical caregiver-to-fog communication: Where the end user (e.g., a healthcare
+provider) is willing to use a particular IoT system, he first sends a request for au-
+thentication with a query authentication function specifying the sensor details to the
+blockchain-enabled fog node. The fog node with the blockchain feature will search for
+that medical attendant in the available list of approved sensor equipment. A reject
+message will be given when the user is not allowed to access the requested data. Oth-
+erwise, if the user is approved, the blockchain-enabled fog node issues an access token
+containing Unique Identification (UID) information, length, time of access, blockchain
+address for the data, user blockchain address, and blockchain address of the fog node
+10
+
+that stores the requested data. Notice that every fog node, sensor, and the user has a
+unique blockchain address.
+(2) Medical sensor-to-fog communication: The sensor-to-fog correspondence has two
+principle objectives in our framework.
+IoT medical services system mainly aims to
+validate and authorize the clinical sensors. The following goal is to insert a blockchain-
+enabled fog connected to sensor devices. It helps new sensors to enlist with the mist and
+ensures that all sensors are recognizable by the blockchain network. In our context, each
+IoT medical care system has at least one blockchain-enabled fog node close to the entire
+blockchain network and is used for the enlistment, confirmation, and authorization of
+IoT medical care gadgets with the same framework. Initially, the gadgets will enroll
+with their associated blockchain fog node. As an exchange and blocks are made for
+them, data concerning these gadgets are placed in the blockchain. These blocks would
+then be transported between the wide range of different blockchain fog nodes. Should a
+system with a collecting place need confirmation and consent, the associated blockchain-
+enabled fog node should be given its certifications. The blockchain approves the provided
+accreditation, and if there are significant requirements, the IoT gadgets for medical care
+are effectively checked and authorized. If the certification is not valid, the gadget is
+refused and will not obtain permission to access the blockchain data.
+(3) Fog-to-fog communication: The main goal is to synchronize the information associ-
+ated with IoT medical service confirmation and approval across all blockchain-enabled
+fog nodes [28]. Several biological or physiological parameters are obtained by medical
+sensors transmitted by patients. Medical IoT programs should be reliable and diligent
+in supporting patients moving to a hospital or home. Typically, the mobility support
+of the medical sensors from the upper layer (i.e., fog layer) should be given so that zero
+reconfiguration in the sensor layer is essential. The strategic location and distribution
+of smart gates in the fog layer can be used to provide smooth mobility for medical
+sensors and relieve processing loads. Fog-to-fog contact helps patients wander around
+the hospital wards, ensuring their health monitoring is not disrupted. The patient-free
+movement provides a high level of medical services using a portable patient monitoring
+system. Support of mobility for healthcare IoT systems is one of the most critical prob-
+lems [29]. The improvements to patients’ quality of life in such programs are essential
+[27]. It is important to encourage patients to walk into the hospital/medical facilities
+knowing that monitoring their well-being is not disrupted. It is necessary to establish
+self-configuration or transfer mechanisms to ensure safe and successful data transfer be-
+tween different Medical Sensor Networks (MSNs) [30] to achieve ongoing monitoring of
+patients considering mobility support. For example, when a patient is moving across the
+clinics, a data transfer mechanism is described as the process of modifying or updating
+the registration of mobile sensors on its MSN base. Data handover solutions should
+allow ubiquity when they need to function independently without human interference.
+(4) Medical sensor-to-medical sensor communication: When two clinical devices are
+effectively tested and approved (both have a position with a similar system or another
+one), they may create a safe link to each other and convey information.
+In a case,
+for example, where a patient is released from a clinic but still needs to be constantly
+11
+
+monitored. The doctors bind the patient’s body before he/she leaves the medical cen-
+ter to health tracking devices, including blood pressure monitors, pulse sensors, blood
+glucose monitoring sensors, etc. These devices sense the patient’s blood pressure, heart
+rate, and glucose level and transmit them through a safe channel to the health workers.
+These devices can also interact with the patient’s intelligent home devices. For exam-
+ple, if the patient’s condition becomes severe or a fall is detected, an immediate alarm
+may automatically be activated. The hospital-related devices must interact to check the
+availability of hospital beds in a smart city to ensure a correct count. The proposed
+framework provides medical devices with access control in the IoT healthcare system.
+Under this mechanism, devices can only communicate with recorded and successfully
+authenticated and certified devices with blockchain-enabled fog nodes. A device not reg-
+istered in the blockchain cannot authenticate itself or communicate with other devices
+within the same healthcare ecosystem or external ones. The contact between malicious
+devices and legitimate devices would also be alleviated.
+In what follows, we detail the proposed lightweight blockchain model and the Fog layer
+functions and properties.
+4. The Blockchain Module Description
+Blockchain technology provides a decentralized, transparent, authenticated platform that
+applies a consensus-driven approach to facilitate the interactions of multiple entities through
+the use of a shared ledger. Beyond the financial sector, where much of the initial develop-
+ment is taking place, blockchain has the potential to revolutionize the healthcare system.
+By providing doctors, patients, researchers, and other healthcare professionals with a mech-
+anism for the controlled exchange of sensitive, permissioned data, blockchain technology can
+improve data sharing and transparency between clinical and research data systems. Any
+healthcare organization participating in a blockchain consortium would be able to share
+medical information, regardless of their native electronic health record system. Blockchain
+provides significant opportunities for healthcare organizations to deliver more efficacious
+treatments and diagnoses through increased provider data sharing and potentially safer and
+more effective remote patient monitoring through advanced technologies such as AI.
+4.1. Blockchain Architecture of Proposed RPM System
+We propose a lightweight blockchain architecture to manage the data storage and re-
+trieval operations in the remote patient monitoring system. In our system, we implement a
+lightweight blockchain architecture that aims at reducing the delay in accessing the cloud
+by the end users while maintaining the security and immutability of data at all nodes. The
+blockchain will store all healthcare-related data, such as the IoT sensor readings, lab test
+results, physicians’ decisions, commands, etc.
+In addition, the blockchain will comprise
+transactions that contain management and security-related data, such as nodes’ and users’
+registrations, access requests, smart contract results, etc.
+In the proposed architecture, cloud servers play the role of full blockchain nodes that store
+the full copy of the blockchain and participate in transaction validation, block generation,
+12
+
+and consensus. On the other hand, IoT gateways play the role of light blockchain nodes
+that store part of the blockchain. In our system, each gateway will be connected to a certain
+number of IoT networks. For example, an IoT gateway at a patient’s home will connect to
+a single IoT network that contains the IoT devices that are monitoring the patient. On the
+other hand, an IoT gateway at a hospital could connect several IoT networks, such as IoT
+devices, in several patients’ rooms. Here, the IoT devices in a certain room or Lab form a
+separate IoT subnetwork since the data produced by these devices will be linked together
+(for example, data related to a specific patient, doctor, lab, etc.).
+The IoT gateways and sensors that exist in the same IoT ecosystem (for example, home,
+hospital, health institution, etc.) form a cluster that store and manage a local blockchain.
+Each local blockchain is created as part of the full blockchain that is related to the cor-
+responding ecosystem. For example, in a certain patient’s home, a set of IoT devices are
+connected to an IoT gateway. The devices and gateway form a cluster that store and man-
+age a local blockchain that contains the blockchain blocks related to that home only. In a
+hospital, several gateways and sets of IoT devices will form a cluster that store and manage
+the blockchain of the hospital.
+In each cluster, the sensor nodes store only the blocks headers of the full blockchain,
+while the gateways store the block headers of the full blockchain in addition to the full blocks
+of the local blockchain (as illustrated in Figure 2). In addition, to avoid overwhelming the
+gateways with excessive storage as the blockchain grows, each transaction will have an expiry
+time after which it becomes obsolete (for example, when the information in the transaction
+becomes old and is no more relevant). Each gateway saves a data structure that contains,
+for each transaction, the ID of the block in which the transaction is stored (BlockID) and
+the transaction expiry date (Tex). The gateway continuously updates the data structure
+when a transaction expires. In addition, the gateway searches the data structure to detect
+any block in which all transactions have expired and deletes it. Using this approach allows
+the gateway to remove old blocks and create room in its storage for new blocks in the local
+blockchain.
+In the proposed system, IoT nodes continuously generate data and send them to the
+IoT gateway. In addition, healthcare providers (doctors, nurses, scientists, etc.) send their
+data (such as prescriptions, sensors’ configurations, lab test results, commands to activate
+actuators, data analytic results, etc.) to the nearest IoT gateway in their institution’s IoT
+cluster. The IoT gateway stores the data it receives in a temporary cache. Each small period
+(for example, every 100 ms), the IoT gateway aggregates and groups the received data into a
+blockchain block and sends it to the cloud server. Note that each block can contain multiple
+transactions. For example, the readings of a certain sensor can be aggregated into a single
+transaction. Similarly, if the doctor is sending configuration commands to the IoT sensor,
+the configuration settings of each sensor can be grouped into a transaction. Each transaction
+will be signed by the owner that created the transaction.
+4.2. Consensus Protocol
+We consider a network of cloud servers that are used by various healthcare providers
+to manage the system. As mentioned, the cloud servers act as full blockchain nodes that
+13
+
+Figure 2: The architecture of the proposed blockchain model: the cloud servers store the full blockchain,
+the IoT gateways save the local blockchain, while the IoT nodes store the block headers.
+store all the blockchain blocks. In addition, the cloud servers participate in the blockchain
+consensus protocol. Each cloud server has a unique blockchain ID. The cloud servers create
+the blockchain blocks successively based on their IDs. In other words, the server with the
+smallest ID creates the first block, followed by the server that has the second smallest ID,
+and so on. When the server that has the biggest ID creates a block, the turn goes back to
+the first server. Note that the block generation time at the IoT gateway should be adjusted
+to allow all the cloud servers to generate their blocks in order to avoid block accumulation
+at the cloud servers.
+When its turn to create the new block arrives, a cloud server CS 1 broadcasts the block
+that it received from the gateway to all the cloud servers. Each cloud server CS i verifies that
+all transactions in the block are legitimate by validating the signature of each transaction.
+Next, CS i replies with a CONFIRM message to CS 1. The confirm message contains CS i’s
+signature of the new block. However, If CS i discovers that one or more transactions in
+the block are not valid, it replies with an ERROR message. In its turn, CS 1 waits until it
+receives at least (N /2+1) CONFIRM messages before it adds the block to the blockchain
+and broadcasts its ID in a “Block Add” message to all cloud servers. Here, N is the number
+of the cloud servers.
+This mechanism allows a cloud server to add the new block after
+the majority of cloud servers confirm its validity. The ”Block Add” message contains the
+signatures that CS 1 received in the CONFIRM messages. When a cloud server CS j receives
+a ”Block Add” message, it checks the attached signatures to ensure that more than (N ÷ 2)
+cloud servers have validated and confirmed the new block, before adding it to its blockchain.
+After receiving the “Block Add” message, each cloud server adds the new block to its
+blockchain and broadcasts it to its clusters. Note that each cloud server can serve multiple
+14
+
+Block Header
+Headers Blockchain
+TX 1
+Block Body!
+TX 6
+Block
+Block
+IoT Sensors
+Full
+Blockchain
+Cloud
+IoT Gateway
+Server
+Local Blockchain
++
+Block 0
+Block 1
+Block 2
+Block k-1
+Block k
+Header
+Header
+Header
+Header
+Header
+Body
+Body
+Local BlocksFigure 3: A sample scenario of the proposed consensus algorithm.
+institutions and organizations, with each institution/organization having its own IoT cluster.
+Each gateway in a cluster examines the new block to determine if it contains transactions
+that were generated by one of the IoT networks in the cluster. If yes, the gateway stores
+the block in its local blockchain and sends it to the IoT devices that are connected to it.
+Each IoT device validates the block (by hashing it and comparing the result to the hash
+in the block header) and then stores the block header in the headers’ blockchain. Next,
+the IoT device caches the block body for a small period of time before deleting it. On the
+other hand, if the new block does not contain transactions that were generated by an IoT
+network in the cluster, the gateway validates the block, sends it to the IoT devices that are
+connected to it, extracts the block header, and adds it to the headers’ blockchain, and then
+deletes the block. Each IoT device that receives the block performs the same operations
+as the gateway. This allows the gateway and IoT devices to maintain the headers of all
+blocks in the blockchain and use these headers to validate any block from outside their local
+blockchain that they obtain from the cloud servers in the future. The proposed consensus
+protocol is illustrated in Figure 3. In the figure, gateways G1 and G2 are connected to cloud
+server CS1, while gateway G3 is connected to cloud server CS2. At a certain time, G2 creates
+a new block B1 and sends it to CS1. When its turn to generate a new block arrives, CS1
+broadcasts B1 to the cloud servers. Each cloud server confirms B1 by sending a CONFIRM
+packet to CS1. Next, CS1 sends a “Block Add” packet to the cloud servers, and each cloud
+server sends the new block to its gateways. G1 and G2 receive B1 from CS1 and add it to
+15
+
+Block B1
+Block B1
+Block Bi
+Block B1
+Send Bi
+Creation
+Broadcast :
+Commit
+ppy
+to Gateways
+Gateway Gi
+Gateway G2
+B1
+Gateway G3
+Cloud Server CS
+wait for turn
+Cloud Server CS2
+Cloud Server CS3
+i
+Cloud Server CS.
+New Block Packet
+Add B ock Packet
+* : Add Block to Local Blockchain
+Block Broadcast Packet
+^ : Add Block Header to Local Blockchain
+Block ACK Packettheir copies of the local blockchain (since B1 was generated by a gateway in CS1’s cluster),
+while G3 receives B1 from CS2 and adds its header only to the local blockchain.
+4.3. Smart Contracts and Data Management
+When an IoT device or a user requires data from the blockchain, it sends a request to the
+IoT gateway. The latter searches for the data in its local chain. If it finds it, the gateway
+authenticates the sender and verifies that it has access to the requested data. If yes, the
+gateway replies directly to the sender with the block that contains the data and the token
+that enables the sender to access the data (more about this soon). If the gateway finds
+that the required data doesn’t exist in the local blockchain, it forwards the request to the
+cloud server. The latter performs the same operation, i.e., it authenticates the requesting
+node and verifies that it has access to the requested data. If yes, the cloud server sends the
+block that contains the data and the access token to the gateway, which forwards them to
+the sender. When the latter receives the block, it validates it using the headers blockchain
+before it retrieves the required transactions from the block and decrypts it using the access
+token.
+Note that in our system, all transactions that can be accessed together are assigned an
+access token by the creator and saved into a smart contract. When the creator wants to
+grant access to the transaction to a certain node/user, the creator executes a smart contract
+function that adds the ID of the node/user to the access list of these transactions that is
+saved in the smart contract. When the node/user wants to access the transactions, it should
+authenticate itself and obtain the access token as described before. If the transactions belong
+to the local chain, the smart contract is executed by the gateway within the local chain.
+Else, the smart contract is executed by the cloud server within the full chain. The various
+subsystems and interactions in the proposed RPM platform are presented in Figure 4.
+5. Specifications of Gateways at the Fog Layer
+The fog layer is made up of IoT gateways which function primarily as a hub between
+the cloud and IoT levels [31].
+With an in-depth study of the role of the gateway in a
+smart home/hospital, where the location and mobility of things and users are confined
+to hospital premises or buildings, it can be recognized that the stationary nature of the
+gateways empowers them with the property of being non-resource constrained in terms of
+power consumption, processing power, and communication. These advantages can be used
+by allowing gateways with ample intelligence, computing power, and structured networks.
+An inter-device communication is the key task of a gateway and supports numerous
+wireless protocols.
+We broaden the function of such gateways into fog enablers by (1)
+building a distributed gateway network and (2) implementing features such as the repository
+(i.e., local data processing and storage using blockchain) to temporarily preserve data for
+analysis by sensors and users. These are important to provide local pre-processing of sensor
+information and, therefore, to be an intelligent gateway for medical services. In a smart
+gateway, the main functions are:
+16
+
+Figure 4: Interaction Model of the proposed blockchain-based remote patient monitoring system.
+5.1. Local data processing and storage
+Local data processing is a key aspect of fog computing and is performed locally so
+that intelligence is accessible at the doors.
+Based on the device architecture, fog/edge
+layers must continuously handle a large amount of information and respond to different
+conditions in a short time. In the remote patient management system, this becomes more
+important by allowing the system to respond to medical emergencies as quickly as possible.
+Gateways should store inbound information in local storage to ensure that the remote patient
+monitoring system can quickly recuperate patient medical data. In the proposed system, we
+make use of the local blockchain to achieve this objective in a secure manner. The patient
+data can be stored as blockchain transactions in an encrypted or compressed form depending
+on their context and security requirements. The gateway stores all data related to the local
+cluster in the local blockchain. In addition, when the gateway receives a blockchain block
+that contains data related to other clusters, it caches the block for a small period of time to
+allow users in the cluster who require data from the block to access it in a fast manner while
+the data is hot. Moreover, since the network bandwidth is limited between the gateway and
+the cloud, the locally cached blocks can be used to maintain a continuous data flow in the
+event of a weak or unstable connection.
+5.2. Data filtering
+Data from various medical sensors must be obtained before further processing, e.g., data
+analysis, on the fog layer. The major sources of knowledge for the assessment of the health
+status of a patient in the remote patient monitoring system [32, 33] are bio-signals, for exam-
+ple, Electroencephalography (EEG), Electrocardiogram (ECG or EKG), and Electromyog-
+raphy (EMG). They typically have complex types that have small amplitudes and varying
+17
+
+Remote Patient Monitoring Healthcare System
+Smart Contract
+Call
+Smart
+Identity
+Storage
+Communication
+Consensus
+Smart Contract
+Contract
+Management
+Management
+Protocol
+Reply
+Interface
+Patient
+Data Access
+Data
+Local
+All Block
+All Blocks
+Data
+Blocks
+Validation
+Headers
+Enrollment
+loT Gateway
+Cloud Server
+Certificate
+Smart Contract
+Healthcare Blockchain Platform
+Nurse
+Result
+Healthcare
+Registration
+Certificate
+Data
+Sensor1
+Sensor2
+Sensor3
+Sensorn
+Sensor4
+Healthcare Sensors
+Doctorfrequencies. It is important to remember that noise is often introduced to the signals during
+a patient’s body sensing in a way that distorts the accuracy of the signals. These noises
+are caused by different sources, including electromagnetic interference from other electrical
+devices, shifts in current in the electricity grid, and inappropriate attachment of sensors
+to the body of users. At the fog level, due to the proximity of the sensors, the gateway
+addresses this issue. The fog layer is digitized via different contact protocols by sensors
+(e.g., 6LoWPAN, Zigbee, etc.). While sensors are able to perform lightweight filtering to
+eliminate certain noises during the data collection process, the fog layer offers more complex
+and robust data filtering.
+5.3. Data analysis
+With local data analysis in the fog layer, the sensitivity of the device can be corrected.
+It helps the device to anticipate and diagnose situations of emergency. The developed deep
+learning module for detecting irregular cardiac conditions is implemented in the fog layer in
+our proposed RPM medical system. The deep learning module can categorize signals and
+detect abnormal conditions on the basis of the sensed ECG signal. As a result, the device
+responds more accurately, rapidly, and in real time to emergency situations. In addition,
+local input and locally sensed data analysis change the quality and reliability of the device
+in the event of the unavailability of the Internet link. Internet disconnection may occur
+regularly for the long-term monitoring of patients with chronic diseases. Fog computing, in
+this case, provides local maintenance of the system’s features. Thus, the sensed data and
+processing results can be kept locally on the fog layer and later synchronized to the cloud via
+the blockchain. Data analysis in the fog often helps the device minimize severe parameter
+processing latencies.
+5.4. Improved latency
+Agile responses and quick decision-making for acute diseases and emergencies, where
+transmission time and data processing are to be reduced, are important for a continuous
+remote control system. When raw medical data is transferred from medical sensor nodes to
+the cloud, cloud computing can trigger response latencies indefinitely if the network condition
+is not predictable. This becomes serious when streaming-based data processing, such as that
+EEG or ECG signals that are obtained from patients, is needed. Hence, deploying high-
+priority data analytics in distributed gateways in the fog later and making time-sensitive
+and critical decisions inside the local network make the remote patient monitoring system
+more predictable and robust. The processed data can then be transmitted for storage and
+further processing to the cloud.
+5.5. Sensor nodes energy efficiency
+There are various drawbacks to the processing of data at sensor nodes, as medical sensors
+are resource-restricted devices. Complicated tasks can, in certain cases, be performed suc-
+cessfully at sensor nodes but at significant energy costs. The transfer of heavy-weight tasks
+from sensors to intelligent gateways in the fog layer can be an effective solution for solving
+the above-mentioned problem, in particular when sensors do not have sufficient resources.
+18
+
+Much energy can be saved with the aid of fog computing by outsourcing tasks from medical
+sensors to intelligent gateways.
+6. Performance Evaluation
+In this section, we present the performance evaluation of the proposed RPM system. We
+have mainly evaluated the performance of the proposed blockchain module and demonstrated
+the efficiency of Fog computing in dealing with critical healthcare applications.
+6.1. Blockchain Implementation and Performance Evaluation
+The proposed blockchain model was implemented via the Hyperledger platform. Hy-
+perledger is an open-source development platform for blockchain applications. It has been
+widely used as an implementation platform by the research community and is considered
+a benchmark tool to evaluate the performance of the proposed approach against state-of-
+the-art approaches. For smart contracts, the Hyperledger tool provides easy-to-configure
+and user APIs, thus making validation easy for our research work. Furthermore, the REST-
+ful API is utilized to provide the functionality of interoperability and expose the back-end
+blockchain services to the client application through which the patients or other medical
+personnel interact with the system. The smart contacts are designed and aggregated in the
+form of .bna files known as business network archive. Hyperledger Composer [34] is used
+to implement and design the proposed medical blockchain, which aims to enhance system
+operations in terms of throughput and latency. Hyperledger Composer is an open-source
+tool used to design blockchain applications. The .bna in the designed platform consists
+of a model, query definition, transaction, and access control rules. The model file is the
+combination of participants, assets, and transactions. The participants are the user of the
+system who can interact with the system to commit transactions. Similarly, the assets are
+the medical services that are used by the system users (participants), which are stored in
+the blockchain. Likewise, transactions are operations that are used to communicate with
+assets. Moreover, transactions are also used to amend the values of assets and participants.
+Similarly, the access control rules are also defined to yield authentication and authorization
+to the users of the system. We also used the world state database to store the blockchain
+data. We specified the queries that are required to determine the interaction between the
+blockchain and the world state database. The queries are also used to fetch the user-based
+customized data from the database.
+Table 2 encapsulates the business network archive file with transactions, assets, and
+participants. The users are patients, doctors, and nurses. Similarly, the assets comprise
+patients’ medical records, sensors, vital sign readings, and other healthcare records. Lastly,
+transactions include getVitalSignReadings, AddHealthcareSensor, and DetectStatus.
+The business network archive is then used to construct a Representational state trans-
+fer (REST) Application Program Interface (API) in order to provide communication be-
+tween the client application and the back-end database. The RESTful API provides cross-
+accessibility, where the user of the system can access it from any platform with authentic
+credentials. Table 3, presents the RESTful API for the proposed medical blockchain, which
+19
+
+Table 2: Smart Contract Modeling for Proposed RPM System
+Type
+Components
+Description
+Asset
+Healthcare˙Sensor
+Healthcare sensors, such as ECG, or
+EMG etc.
+Vital Sign˙Sensing˙Data
+The vital signs of patients acquired
+from healthcare sensor.
+HealthRecord
+The patient medical information,
+such as current health condition, de-
+ployed sensors, etc.
+Participant
+Doctor
+System user.
+Patient
+System user.
+Nurse
+System user.
+Transaction
+getVitalSignReadings
+Get vital sign reading from health-
+care sensors.
+Add˙Healthcare˙Sensor
+Addition of new healthcare sensor in
+a medical blockchain platform.
+Modify˙Sensor
+Modify sensor composition.
+Detect˙Status
+Detect the patient vital sign status.
+is based on HTTP protocol. The generated RESTful API is used to expose the medical plat-
+form services to the client application. The services are related to patients, nurses, doctors,
+EMR, and other medical information. Figure 5 demonstrates how the major components of
+the proposed RPM system have interacted during the simulation study.
+Table 3: RESTful API for proposed Medical Blockchain
+Action
+Verb
+Media Type
+URI
+Patient Dashboard
+ALL
+Application/json
+/api/Patient
+Doctor Dashboard
+ALL
+Application/json
+/api/Doctor
+Nurse Dashboard
+ALL
+Application/json
+/api/Nurse
+Healthcare Sensor Dashboard
+ALL
+Application/json
+/api/Sensor
+Vital˙Sign
+Application/json
+/api/VitalSignReading
+EMR Dashboard
+ALL
+Application/json
+/api/PatientRecord
+Share patient record with healthcare personnel
+POST
+Application/json
+/api/ShareRecord
+Blockchain Network Text
+GET
+Application/json
+/api/system/ping
+Issue identity to system user
+POST
+Application/json
+/api/SystemIdentities/issue
+Get Identities
+GET
+Application/json
+/api/System/identities
+Retrieve historian records
+GET
+Application/json
+/api/System/historian
+Within our blockchain implementation, each piece of medical record has one user (owner)
+who can share the data they own with other users (doctors) at varying levels of access. Data
+sharing between users is modeled by a system where users can share data with other users
+in different groups, as well as receive data requests from other users at any access level.
+If a user responds to a request by granting data access, an access token is provided to the
+receiver in a way that allows that receiver to access the data at the specified access level only.
+Our system ensures that sensitive information is never exposed on the blockchain, including
+20
+
+Figure 5: Sequence of interactions conducted during simulation
+both private and document keys, which is necessary in order to maintain the privacy and
+security of user-controlled data.
+We evaluate the performance of the proposed blockchain model using Hyperledger Caliper
+[35]. For experimental analysis, we carried out several experiments in terms of the execution
+time when adding a new healthcare device and executing a healthcare data query. We also
+measure the average time of the proposed consensus algorithm. The execution time is the
+round-trip time which includes the total time of sending the request by the client and getting
+the response from the network. In order to evaluate the execution time, we utilized the Post-
+man tool, which is used to explore and test the RESTful APIs by simulating a customized
+user load within the network. In this study, we created three groups of devices: 150, 300,
+and 500, in order to investigate the execution time of registering a device in the proposed
+blockchain model. Furthermore, the execution time is analyzed using different statistical
+21
+
+Doctor/Nurse
+Gateway/Fog Node
+REST Server
+Patient
+(Sensor)
+(GUI)
+(Local Chain)
+(Global Chain)
+1
+Device Registration
+Device Registation
+smart
+contract
+Device ldentity and Certificate
+call
+Vital Sign (ECG Signal)
+create
+block
+New Block
+block
+mining
+New Block Broadcast
+save block
+or block
+New Block Broadcast
+header
+save block
+or block
+header
+Data Request
+check
+data
+location
+alt
+certify
+sender's
+[data in local chain]
+Secure Health Data
+certificate
+[data in global chain]
+Data Request
+certify
+ sender's
+certificate
+Secure Health Datameasures, such as the minimum, maximum, and average times. As shown in Figure 6, in
+the case of 150 users, the average, minimum, and maximum execution time to register the
+healthcare device is recorded as 2335 ms, 2257 ms, and 2795 ms, respectively. Likewise,
+the minimum, maximum, and average execution times for 300 healthcare device-group is
+are 1785 ms, 3204 ms, and 2454 ms, respectively. Finally, for 500 devices the minimum
+execution time is recorded as 2810 ms, whereas the maximum and average execution time
+is 3524 ms and 3015 ms respectively (Figure 6).
+Figure 6: Healthcare device registration execution time
+The execution time of the proposed system is also evaluated in the case of retrieving
+healthcare data from the blockchain network.
+Every healthcare device in the proposed
+platform has the HTTP client functionality which is used to send requests for vital sign
+sensing data through the IoT gateway. The request is initially processed by the IoT gateway.
+If the requested data is found in the local chain, the IoT gateway validates the device
+certificate via the local smart contract and then replies to the device with the encrypted
+data. Else, the IoT gateway forwards the request to the REST server, which performs a
+similar process. The execution time of reading the vital sign data is illustrated in Figure 7.
+The same set of device groups has been considered for the experimental evaluation, i.e., 150,
+300, and 500 devices. It is observed from the graph that the increase in the device scale in
+the proposed healthcare system will also create an impact on the execution time. However,
+the overall execution time of the network remains stable until there is high congestion in the
+network. The average execution time of vital sign sensing data in the case of 150, 300, and
+22
+
+4000
+■150 Devices
+300Devices
+500 Devices
+3500
+3000
+Execution Time (ms)
+2500
+2000
+1500
+1000
+500
+0
+Minimum
+Average
+Maximum500 devices are 2552 ms, 2525 ms, and 2775 ms, respectively, which are comparable to the
+execution times of registering a device that is shown in Figure 6.
+Figure 7: Vital signs reading execution time
+In order to evaluate the effectiveness of the proposed consensus method, we tested several
+scenarios in which we deployed five REST servers and five IoT gateways. The IoT devices
+were distributed evenly among the gateways, and each gateway was connected to a REST
+server. The servers saved all the blocks that were confirmed by the consensus protocol,
+while the IoT gateways saved the blocks of the devices that connected to them only. In
+these scenarios, we measure the consensus time of each created block, then we calculate the
+minimum, maximum, and average values for all the created blocks. The results are shown
+in Figure 8. We notice that the consensus time generally increases as the number of devices
+increases, which is logical since, with more devices, the total number of transactions increase,
+which adds more time to validate the new blocks. However, the increase in the consensus
+time is only 12.5 ms (on average) as the number of devices increases from 150 to 500, which
+proves the efficiency of the proposed consensus approach. In addition, the average consensus
+time of the system is 140 ms. In case of Ethereum and Bitcoin, it requires 10 to 19 seconds
+and 10 minutes to an hour respectively to mine a new block. Hence, the proposed consensus
+algorithm outperforms those of other blockchain platforms in terms of consensus time.
+23
+
+4000
+150Devices
+300Devices
+500Devices
+3500
+3000
+Execution Time (ms)
+2500
+2000
+1500
+1000
+500
+0
+Minimum
+Average
+MaximumFigure 8: Block consensus time
+6.2. Efficiency of the Fog computing infrastructure
+Figure 9 depicts the distributed data flow model for our proposed IoT-driven critical
+healthcare applications. According to this model, data signals generated by the IoT devices
+are pushed into the client module, an initial application interface for interacting with the IoT
+devices and actuators and receiving the user’s information, such as name, location, address,
+sex, and age of the patient.
+After pre-processing and filtering the data that is coming
+from the IoT devices, the client module forwards the data to the Data Processing module
+for further processing. Here, AI-enabled modules can execute data analytics processes for
+testing purposes. Based on the outcome of the data processing, a command is issued by
+the Data Processing module for the client module so that it can trigger physical emergency
+actions through the actuators. Next, the Data Processing module dispatches the processed
+data to the aggregator module, which simultaneously interacts with the blockchain module
+at the IoT gateway and cloud server to add the data to the blockchain and ensure data
+integrity and location-independent data access. The blockchain module interacts with the
+storage module in case the data is to be stored off-chain. Finally, The Data Processing
+module at the cloud server interacts with the blockchain module to consistently produce
+the results that are requested by the application users. Since the client module directly
+interacts with the IoT devices, it is preferable to be deployed at the IoT gateways (e.g.,
+ECG machines). For the deployment of other modules, there exist different approaches in
+the literature. For instance, cloud computation has been exploited in [36] [37] to execute the
+data analytics, aggregator, blockchain, storage, and training module. On the other hand,
+the proposed RPM system adopts Fog computing for executing these modules and utilizes
+the cloud to host the blockchain, storage, and processing modules.
+24
+
+200
+150 Devices
+300Devices500Devices
+175
+150
+(ms)
+Time (
+125
+Consensus
+100
+75
+50
+25
+0
+Minimum
+Average
+MaximumFigure 9: Data flow model for the proposed RPM system
+In this phase of performance evaluation, we demonstrate how the augmentation of Fog
+computing in remote patient monitoring improves the service latency and the energy usage
+in comparison to harnessing cloud-based resources. The experiments are conducted in an
+iFogSim [38] simulated Fog-Cloud computing environment. The computing resources within
+the simulation environment are organized in a hierarchical order, as shown in Figure 10. At
+the lower level of the simulation environment, twenty-four ECG machines (EMs) equipped
+with ECG sensors and emergency alert systems are placed. Based on the simulation design,
+an EM can connect with any of the four Fog local servers (FLSs) at the upper level. All
+FLSs are also set connected with a Fog regional server (FRS) that helps the lower-level
+computing devices to maintain seamless communication with the Cloud datacenter. Table
+4 presents the details of the simulation parameters used in the experiments. The numerical
+values have been extracted from real-world references as specified in [39] [40]. Additionally,
+Table 5 illustrates the configuration of different application modules for the simulations,
+Figure 10: Architecture of the simulated Fog-Cloud computing environment
+25
+
+Cloud
+Server
+FRS#1
+FLS#1
+FLS#6
+EM#1
+EM#2
+EM#3
+EM#4
+EM#21
+EM#22
+EM#23
+EM#24
+883
+89loT layer
+Fog layer
+Cloud layer
+Data
+600
+Global
+Signal
+Processed
+Aggregator
+Records
+Blockchain
+Data
+Module
+ECG
+Raw Data
+Module
+Sensor
+Data
+Client
+Processing
+Data
+Retrieve
+Response
+Module
+Records
+Query
+Module
+Command
+ Save
+Local
+Analytic
+Alert
+Storage
+Blockchain
+Training
+Module
+Block Metadata
+Module
+Emergency
+Module
+IndicatorTable 4: Parameters of simulated environment
+Device configuration
+Name
+Processing
+speed
+Downlink
+bandwidth
+Uplink
+bandwidth
+Memory
+capacity
+Busy
+power
+Idle
+power
+(in MIPS)
+(in MB)
+(in MB)
+(in GB)
+(in MWh)
+(in MWh)
+EM
+1000
+10
+5
+8
+1.1
+0.2
+FLS
+7000
+8
+3
+12
+1.3
+0.4
+FRS
+15000
+6
+2
+16
+1.6
+0.8
+Cloud
+40000
+3
+4
+32
+3.2
+1.4
+Sensing frequency of ECG sensors
+5 signals per second
+Simulation time
+500 seconds
+Table 5: Module configuration
+Name
+Program size
+Packet size
+RAM usage
+(in MB)
+(in KB)
+(in GB)
+Client module
+2000
+500
+1
+Data analytic module
+4000
+1500
+6
+Aggregator module
+1500
+1800
+2
+Blockchain module (periodic)
+1000
+2000
+4
+Storage module
+1000
+2000
+2
+Analytic training module
+8000
+2000
+12
+Figure 11: Performance in reducing sense-process-actuation delay
+which have been approximated based on the profiled run-time, resource utilization, and
+data communication delay of the proposed solutions in heterogeneous computing devices
+and networking context.
+The results of the simulation experiments conducted in the aforementioned computing
+26
+
+350
+ms)
+300 -
+Sense-Process-Actuation delay (in
+250
+200
+150
+100.
+S
+Proposed RPMS
+Cloud-based RPMSFigure 12: Performance in reducing energy consumption
+Figure 13: Performance in blockchain transaction retrieval
+setup demonstrate that our proposed Fog computing-based RPMS outperforms the Cloud
+computing-based RPMS both in terms of reducing sense-process-actuation delay (calculated
+using iFogSim AppLoop model on ECG sensors → client module → data analytic module →
+client module → emergency alert system data flow) and energy usage. Figure 11 indicates
+that the augmentation of Fog computing can improve the responsiveness of RPMS by 40%
+in initiating alert messages during emergency situations compared to its cloud counterpart.
+Such performance improvement happens mainly for executing the data analytics module
+closer to the sources, that consequently decreases the data transfer delay to remote cloud
+servers. Moreover, the computing devices in the Fog paradigm consume a reduced amount
+of energy than a cloud server because of their capacity constraints. Statistically, this feature
+27
+
+0.40
+0.35 -
+0.30-
+0.25
+0.20
+0.15
+0.10.
+0.05 -
+0.00
+Proposed RPMS
+Cloud-based RPMS25
+ProposedRPMS
+ Transaction Retrieval
+WCloud-basedRPMs
+20
+ime (ms)
+15
+3lockchain
+B
+0
+Transaction size = 500 KB
+Transactionsize=2000KBalso has an influence in lowering the idle energy consumption of Fog computing devices.
+Therefore, when the time-based energy consumption model (as programmed in the iFogSim
+simulator) is applied, the Fog computing-based RPMS promises to deliver its services by
+consuming around 36% less energy than its Cloud-based implementation (as shown in Figure
+12).
+On the other hand, due to executing the blockchain module at the fog devices, the delay
+required to retrieve a random blockchain transaction decreases as compared to the cloud-
+based RPMS, as shown in Figure 13. The figure illustrates that when the transaction size
+is equal to 500 KB, the proposed system requires an average of 7.16 ms to retrieve the
+transaction from the blockchain, while cloud-based RPMS needs 16.54 ms. On the other
+hand, for a 2000 KB transaction, the proposed RPMS produces a delay equal to 8.9 ms,
+while the cloud-based RPMS needs 19.34 ms.
+Hence, the proposed RPMS reduces the
+transaction retrieval delay by an average of 55.1%. This is mainly due to the cases in which
+the transaction is fetched from the local chain, which require much less end-to-end delay
+than retrieving the transaction from the global chain, due to the deployment of fog nodes
+at locations that are much nearer to the sensor nodes than the cloud servers.
+7. Security Analysis
+Having a robust architecture encryption scheme as part of a blockchain-based data-
+sharing system is particularly critical from a security perspective because most blockchain
+implementations replicate the entire transaction ledger onto each node, therefore, multiply-
+ing the potential attack surface by the number of nodes in the network. In the following,
+we discuss the security analysis which we performed on the proposed patient monitoring
+system.
+• Key attack: Elliptic curve encryption method is employed from a key pair, and an
+attacker can’t calculate the private key to address the elliptic curve logarithm problem;
+hence the security of the proposed model is ensured. Moreover, for each session, a
+temporary private key is generated for interaction among the nodes. In such a way, if
+a private key gets compromised in terms of leakage, then this will not have an impact
+on the session, as the attacker would not be able to calculate a session key for a session
+that is currently going on among the nodes; and (b) the leaked private key is of no use
+until the session is completed.
+• Replay attack: The proposed model uses an individual temporary private key that
+is different for each session agreement among the interacting nodes. It is improbable
+that a replay attack becomes successful since private keys hold a bounded lifetime.
+• Impersonation attack: This attack is executed only if the attacker has successfully
+obtained the private key. The proposed model employs an individual private key and
+elliptic curve encryption. Therefore, this attack cannot be executed.
+28
+
+• Sybil attack: there are different methods to remove the impact of Sybil attack on
+the proposed model, such as increasing the price to form a new identity. This method
+restricts attackers from obtaining fake identities, using a two-factor authentication
+mechanism and accumulating the MAC and IP addresses of the participants, which
+permits the detection of those participants who have varying identities.
+• False data injection attack: Prior to validating the records, the consensus algorithm
+is executed by the blockchain nodes. On arrival of the positive consensus, a node can
+confirm the legitimacy of the received record.
+• Tampering attack: For encryption and signing the transaction, a public key crypto-
+system is employed. This indicates that the tampering node cannot amend the transac-
+tion as it does not hold the private key of the signing node. Furthermore, the proposed
+model can handle the key attacks; therefore, the adversaries cannot exploit the private
+keys.
+• Modification attack: As explained above, this attack is impossible because the
+adversaries cannot exploit the private keys.
+• Hiding blocks attack: A record in the proposed vital sign monitoring platform
+holds a unique sequence number. It is a must for a blockchain node to provide its
+saved records if requested. If a node in the network does not offer its records, it is
+detached from the network and disallowed to interact with other nodes.
+• Man-in-the-middle attack: A mutual authentication is performed between the
+nodes in the proposed model, which employs private keys for each session agreement,
+therefore, man-in-the-middle attacks are prevented.
+• Compromisation attack: If an attacker compromises a cloud server and attempts
+to sabotage the consensus operation by sending a ”Block Add” message that contains
+an invalid block, the legitimate cloud servers will detect the attack from the invalid
+signatures in the ”Block Add” message, since the attacker will not be able to generate
+the valid signatures of the other cloud servers. If the attacker drops the block that
+it receives from the IoT gateway, the latter reports the attack to the IoT ecosystem
+administrator when it detects that its block was not added to the blockchain in due
+time. Finally, if the attacker sends a wrong reply message when it receives a new block
+from another cloud server, the attack will not have an effect as long as the number of
+legitimate cloud servers is greater than N /2.
+8. Conclusion and Future Work
+In this work, we have presented a three-layer remote patient monitoring system that
+leverages blockchain technology for better security and Fog technology for providing low-
+latency services to IoT devices and healthcare users. The most important functions that
+encompass the system components are described and evaluated. In addition, a new consensus
+29
+
+protocol that is tailored to the RPM environment is discussed and analyzed. Moreover, the
+blockchain module was implemented and tested using Hyperledger Fabric Framework, and it
+achieved low execution and consensus delays. Moreover, the augmentation of Fog computing
+can improve the responsiveness of the remote patient monitoring system by 40%.
+Several future works are being studied to enhance the proposed system. For example,
+we are planning to perform the simulations using real healthcare datasets (such as that in
+[41]). In addition, we intend to add a prediction module at the cloud layer that can predict
+a heart disease problem before its occurrence. The module would analyze the patient’s data
+from the global blockchain over an extended period to enhance prediction accuracy. Another
+enhancement would be the integration of the proposed blockchain system with a body area
+network (BAN) framework that is used to collect patient medical data in an efficient manner.
+Such integration should be carefully designed in order to secure the BAN operations without
+adding significant overhead in terms of computation and energy consumption on the BAN
+nodes. A similar system was proposed in [42]. Hence, we aim to study the literature in order
+to adjust the proposed blockchain system to make it suitable for a BAN environment.
+Another important future work is to enhance the proposed fog layer by augmenting it
+with modern technological tools that will improve its performance. For example, federated
+learning can be used by fog nodes to filter and analyze the readings of IoT devices in order
+to provide more accurate results to healthcare providers. Another important aspect is to
+design the scheduling of IoT data on the fog layer using the blockchain. For this aspect, we
+intend to adopt a previous strategy that we proposed in [43] to guarantee that a fog node
+treats data from IoT nodes fairly and provides equal opportunities for IoT nodes to save
+their data in the blockchain.
+Finally, we will study the scalability of the proposed system and its ability to support a
+large number of IoT ecosystems. For this purpose, we will design a hierarchical clustering
+framework that distributes cloud servers, fog nodes, and IoT devices into clusters based on
+their geographic locations and the deployed healthcare application. Using clustering will
+allow us to reduce the delay overhead when the application contains a huge number of
+blockchain nodes. In such a system, it is possible to execute a blockchain query in parallel
+by distributing it over the cluster heads, which would result in a reduced end-to-end delay
+between the patient and the healthcare provider.
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+

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+Entangled States are Harder to Transfer than Product States
+Tony J. G. Apollaro
+,1, ∗ Salvatore Lorenzo
+,2 Francesco
+Plastina
+,3, 4 Mirko Consiglio
+,1 and Karol ˙Zyczkowski
+5, 6
+1Department of Physics, University of Malta, Msida MSD 2080, Malta
+2Universit`a degli Studi di Palermo, Dipartimento di Fisica e Chimica - Emilio Segr`e, via Archirafi 36, I-90123 Palermo, Italy
+3Dipartimento di Fisica, Universit`a della Calabria, 87036 Arcavacata di Rende (CS), Italy
+4INFN, gruppo collegato di Cosenza, 87036 Arcavacata di Rende (CS), Italy
+5Institute of Theoretical Physics, Jagiellonian University, ul. �Lojasiewicza 11, 30–348 Krak´ow, Poland
+6Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotnik´ow 32/46, 02-668 Warszawa, Poland
+The distribution of entangled states is a key task of utmost importance for many quantum infor-
+mation processing protocols. A commonly adopted setup for distributing quantum states envisages
+the creation of the state in one location, which is then sent to (possibly different) distant receivers
+through some quantum channels. While it is undoubted and, perhaps, intuitively expected that the
+distribution of entangled quantum states is less efficient than that of product states, a thorough
+quantification of this inefficiency (namely, of the difference between the quantum-state transfer fi-
+delity for entangled and factorized states) has not been performed. To this end, in this work, we
+consider n-independent amplitude-damping channels, acting in parallel, i.e., each, locally, on one
+part of an n-qubit state. We derive exact analytical results for the fidelity decrease, with respect to
+the case of product states, in the presence of entanglement in the initial state, for up to four qubits.
+Interestingly, we find that genuine multipartite entanglement has a more detrimental effect on the
+fidelity than two-qubit entanglement. Our results hint at the fact that, for larger n-qubit states,
+the difference in the average fidelity between product and entangled states increases with increasing
+single-qubit fidelity, thus making the latter a less trustworthy figure of merit.
+I.
+INTRODUCTION
+Distributing entangled states among several distant recipients is a task of paramount importance in a variety of
+quantum-information processing protocols, ranging from n-party quantum key distribution [1] to distributed quantum
+computing [2]. In many of these protocols, an n-partite entangled state is created at location S (the sender’s location),
+and its parts are distributed among m ≤ n receivers, generally at different locations (which we dub R, the receivers’
+location).
+The special case of distributing a bipartite entangled state was already considered in the seminal paper by Bose on
+quantum-state transfer (QST), where the transfer protocol is employed to send (the state of) one party of a two-qubit
+Bell state to the opposite edge of a spin chain [3]. After this first instance, a considerable amount of research, both
+theoretical and experimental, has been performed in order to improve the transfer performance and optimize the Bell
+state distribution protocol [4–10]. Moreover, with the increasing exploration (and exploitation) of the fascinating
+realm of quantum correlations by quantum technological applications, the distribution of n-partite entangled states,
+with n > 2, has become a very active research topic [11–15].
+At variance with the entanglement of two-qubits, which is the only system whose entanglement properties have been
+fully characterized both for pure and mixed states, for n > 2 there are only a handful of closed, analytical results for
+the quantification of entanglement [16], making the task of evaluating the efficiency of an entanglement distribution
+protocol very difficult to assess.
+Here, we address the distribution of an n-partite entangled state utilizing the fidelity between the sender’s and the
+receivers’ state as a figure of merit for the quality of the protocol. Although the fidelity is not a bona fide tool to
+characterize quantum resources [17, 18], it is nevertheless widely employed in constructing entanglement witnesses
+following the idea that states close to an entangled state must be entangled as well [19]. Hence, building on recent
+results [20, 21] reporting the fidelity of an n-qubits QST (n-QST) protocol for arbitrary quantum channels, we
+investigate the effect of the presence of entanglement on the n-QST fidelity, which is evaluated when each qubit is
+subject to an independent U(1)-symmetric quantum channel, e.g., an amplitude-damping channel. In particular, we
+find that the presence of entanglement in the sender state is detrimental to the efficiency of the n-QST protocol. It
+is not surprising that independent quantum channels acting on n qubits tend to destroy their quantum correlations,
+thus lowering the transmission fidelity. We are able to provide a quantification of the fidelity reduction as a function
+∗ tony.apollaro@um.edu.mt
+arXiv:2301.04443v1  [quant-ph]  11 Jan 2023
+
+2
+of different entanglement monotones . In particular, we show that genuine multipartite entanglement, as quantified,
+e.g., by the three-tangle, has a more pronounced effect on lowering the n-QST fidelity than bipartite entanglement
+between two qubits, as quantified by the concurrence [22].
+The paper is organised as follows: in Section II, we introduce our model and provide a brief recap of the n-QST
+fidelity; in Section III, we apply the developed formalism to the case of n = 2, 3, 4 qubits; finally, in Section IV, we
+draw our conclusions.
+II.
+N -QST FIDELITY FOR INDEPENDENT AMPLITUDE-DAMPING CHANNELS
+Let us consider an n-qubit quantum-state transfer protocol as depicted in Figure 1. A sender, located at position
+S, prepares an n-qubit arbitrary state and wants to transfer each party to different receivers to which the sender is
+connected by different quantum channels. Without a loss of generality, let us assume the sender state to be a pure
+state, ρS = |Ψ⟩⟨Ψ|n. The state at the receivers’ location reads
+ρR(t) = [Φ1 ⊗ Φ2 ⊗ · · · ⊗ Φn] (t) (ρS) .
+(1)
+The fidelity between the sender and the receivers’ state is given by the Uhlmann–Jozsa fidelity [23]
+F (|Ψ⟩ , ρ(t)) = ⟨Ψ| ρ |Ψ⟩ .
+(2)
+Expressing an arbitrary input state in the computational basis
+|Ψ⟩ =
+2n
+�
+i=1
+ai |i⟩ ,
+(3)
+the elements of the receivers’ density matrix read (sum over repeated indexes is assumed)
+ρR
+ij = Anm
+ij ρS
+nm
+(4)
+yielding the fidelity
+F (|Ψ⟩ , ρ) =
+d−1
+�
+ijnm=0
+a∗
+i ajana∗
+mAnm
+ij ,
+(5)
+where all of the amplitudes a refer to the initial state of the sender.
+For the case represented in Figure 1, the total map is given by the tensor products of n independent maps as in
+Equation (1). Hence, Equation (4) can be cast in the following form: [24]
+ρR
+i1i2...in;j1j2...jn = Ap1p2...pn;q1q2...qn
+i1i2...in;j1j2...jn ρS
+p1p2...pn;q1q2...qn,
+(6)
+where i, j, p, q = 0, 1 and the corresponding subscript refers to the i’s qubit, with
+Ap1p2...pn;q1q2...qn
+i1i2...in;j1j2...jn
+= Ap1;q1
+i1;j1 Ap2;q2
+i2;j2 . . . Apn;qn
+in;jn .
+(7)
+Each A in Equation (7) comes from a single qubit map connecting the sender qubit si and the receiver qubit ri,
+which, for an U(1)-symmetric channel, can be expressed as
+�
+�
+�
+ρ00
+ρ01
+ρ10
+ρ11
+�
+�
+�
+ri
+=
+�
+�
+�
+�
+1
+0
+0
+1 −
+��f ri
+si
+��2
+0 f ri
+si
+0
+0
+0
+0
+�
+f ri
+si
+�∗
+0
+0
+0
+0
+��f ri
+si
+��2
+�
+�
+�
+�
+�
+�
+�
+ρ00
+ρ01
+ρ10
+ρ11
+�
+�
+�
+si
+,
+(8)
+where f ri
+si is the transition amplitude for the excitation initially on si to reach ri. A widely used U(1)-symmetric
+quantum channel is given by the so called XXZ spin- 1
+2 Hamiltonian,
+H =
+�
+i,j
+Jij
+�
+σx
+i σx
+j + σy
+i σy
+j
+�
++ ∆ijσz
+i σz
+j + hiσz
+i
+(9)
+
+3
+FIG. 1. A quantum router. A dispatch center, encircled in red, creates an n-qubit entangled state (red spheres) with the aim
+to send each party to a different receiver (green spheres) along independent quantum channels (blue spheres).
+where σα
+i (α = x, y, x) are Pauli matrices and i, j are the position indexes on an arbitrary d-dimensional lattice.
+Assuming that each quantum channel is fully polarized, for the sender state , the map Φi reduces to an amplitude-
+damping channel [25]. In particular, for f ri
+si = 1, the map Φi entails a SWAP operation. Therefore, our formalism
+also describes entanglement swapping protocols via imperfect operations [26]. Finally, to express Equation (6) in the
+form of Equation (4), it is sufficient to express the bit strings in decimal notation.
+By making use of Equations (4), (7), and (8), it is straightforward to evaluate the average fidelity [13] of an arbitrary
+quantum state |Ψ⟩
+⟨F⟩ = 1
+Ω
+�
+Ω
+dΩ F (|Ψ⟩ , ρ(t)) ,
+(10)
+with Ω denoting the space of pure states and, with an abuse of notation, its volume and the measure of it .
+An average with respect to an n qubit system will be denoted as ⟨Fn⟩. In the case of n independent channels,
+making use of the transition amplitude f introduced in Equation (8), one arrives at the expression,
+⟨Fn⟩ =
+1
+2n + 1 +
+1
+2n (2n + 1) |1 + f|2n .
+(11)
+Notice that the average fidelity ⟨Fn⟩ ≤ �n
+i=1 ⟨F1⟩, with equality holding only for f = 0, 1. While the left-hand side of
+the latter inequality gives the average over all possible pure input states, its right-hand side, on the other hand, gives
+the average restricted to fully factorized states only, i.e., to product states of the form |Ψ⟩n = �n
+i=1 |ψ⟩i, thus not
+including the entangled states. Hence, we conclude that, when n ≥ 2, in the set of all pure input states, entangled
+states have a lower n-QST fidelity than the product state. In the next sections, we will provide a quantitative analysis
+for this intuitive observation.
+III.
+N -QST FIDELITY AS A FUNCTION OF ENTANGLEMENT
+This Section contains our main result, namely, that the presence of entanglement reduces the transfer fidelity.
+Below, we illustrate this idea separately for two, three, and four qubit transmissions. In particular, we will show that,
+in the presence of entanglement in the states to be sent, a reduction factor exists, which we dub Rn, such that the
+average fidelity for the QST of n-qubits can be generically written
+⟨Fn⟩ = ⟨F1⟩n − En Rn.
+
+4
+Here, ⟨F1⟩n gives the average fidelity for the transfer of factorized states (indeed, intuitively, qubits can be transferred
+one by one, in this case), so that the difference ⟨Fn⟩ − ⟨F1⟩n is entirely due to the fact that entangled states are
+possibly transferred. The coefficient En is an entanglement quantifier that changes with n. It is given by twice the
+square of concurrence for n = 2, while for n = 3 it is proportional to a linear combination of the invariant polynomials
+identifying the different classes of entangled states. Finally, the factor Rn (defined below for the various cases) gives
+the weight of entanglement-induced fidelity decrease, and it also enters the fidelity averaged over specific entanglement
+classes of states (for n = 3, 4).
+A.
+Two Qubits
+Adopting the (Schmidt-)parametrization of two-qubits pure states in terms of their entanglement [27], we write
+|Ψ(s)⟩ =
+�
+1 + s
+2
+|00⟩ +
+�
+1 − s
+2
+|11⟩
+(12)
+where the parameter s ∈ [−1, 1] is related to the concurrence [22] via C =
+√
+1 − s2, and every two-qubit pure state
+can be obtained from Equation (12) via local, unitary operations |Φ(s)⟩ = U1U2 |Ψ(s)⟩, with Ui ∈ SU(2) acting on
+qubit i = 1, 2. Below, we obtain the average fidelity for a two-qubit QST protocol with independent channels as a
+function of the amount of entanglement of the sender state, which is invariant under local unitaries, and which we
+denote as ⟨·⟩U ⊗2
+This average fidelity (which, to say it shortly, is averaged at fixed values of entanglement) reads
+⟨F2⟩U ⊗2 = 1
+36
+�
+3 + |f|2 + 2 |f| cos φ
+�2
+− 1
+18
+�
+|f|2 + 2 |f| cos φ
+� �
+3 − |f|2 + 2 |f| cos φ
+�
+C2
+(13)
+where we expressed the complex transition amplitude f as f = |f| eiφ.
+Following the procedure outlined by Bose [3], in order to maximise Equation (13), one sets cos φ = 1, which,
+physically, can be obtained, e.g., by a uniform magnetic field applied over the spin chain. Hence, the average two-
+qubit fidelity F2 can be cast in the form
+⟨F2⟩U ⊗2 = 1
+36
+�
+3 + |f|2 + 2 |f|
+�2
+− 1
+18
+�
+|f|2 + 2 |f|
+� �
+3 −
+�
+|f|2 + 2 |f|
+��
+C2.
+(14)
+From Equation (14), since 0 ≤ |f| ≤ 1, one can readily appreciate that the more concurrence the sender’s pure state
+contains, the lower the fidelity with the received state. Equation (14) can also be rewritten as a function of the
+single-qubit QST average fidelity
+⟨F1⟩ = 1
+6
+�
+3 + 2 |f| + |f|2�
+,
+(15)
+to read
+⟨F2⟩U ⊗2 = ⟨F1⟩2 − 2
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩) C2 = ⟨F1⟩2 − 2R2C2.
+(16)
+Again, as 1
+2 ≤ ⟨F1⟩ ≤ 1, the average fidelity ⟨F2⟩ decreases with the amount of concurrence of the sender state.
+From Equation (16), we see that the average 2-QST fidelity is reduced in the presence of the squared concurrence
+by a factor of
+R2 =
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩) ,
+(17)
+which is reported in Figure 2 (left panel), together with the two-qubit average fidelity ⟨F2⟩, displayed for different
+values of the squared concurrence as a function of the one-qubit fidelity ⟨F1⟩ (right panel).
+B.
+Three Qubits
+Having obtained a quantitative expression giving the reduction in the transmission fidelity due to the presence
+of entanglement for two qubits, we move to the more intricate three qubit case in order to try and obtain similar
+relations.
+
+5
+0.6
+0.7
+0.8
+0.9
+1.0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.6
+0.7
+0.8
+0.9
+1.0
+0.4
+0.6
+0.8
+1.0
+FIG. 2. (left) Reduction factor (17) for entangled states of the 2-QST average fidelity ⟨F2⟩ (16) as a function of the 1-QST
+average fidelity ⟨F1⟩. The dotted, vertical line reports the maximum of R2 attained at ⟨F⟩1 = 0.75. (left)
+1.
+Three-qubit pure-state entanglement
+Let us now consider a system of three qubits A, B, and C. A three-qubit pure state can be written in canonical
+form as [28]
+|Ψ⟩ABC = λ0 |000⟩ + λ1eiφ |100⟩ + λ2 |101⟩ + λ3 |110⟩ + λ4 |111⟩ ,
+(18)
+where λi ≥ 0, 0 ≤ φ ≤ π, and the normalisation condition reads �
+i λ2
+i = 1.
+In terms of the coefficients of the state in Equation (18), one can introduce five invariant polynomials, allowing to
+identify different entanglement classes [29]:
+J1 =
+��λ1λ4eiφ − λ2λ3
+��2 , J2 = λ2
+0λ2
+2 , J3 = λ2
+0λ2
+3
+(19a)
+J4 = λ2
+0λ2
+4 , J5 = λ2
+0
+�
+J1 + λ2
+2λ2
+3 − λ2
+1λ2
+4
+�
+.
+(19b)
+The relation between invariant polynomials and entanglement measures is given by
+C2
+jk = 4Ji,
+(20)
+for i ̸= j ̸= k = 1, 2, 3, and where now, (1, 2, 3) = (A, B, C) holds on the LHS of Equation (20) . At variance with the
+two-qubit case, no single entanglement measure can capture genuine three-partite entanglement as three qubits can
+be entangled in two inequivalent ways [30].
+One type of entanglement is quantified by the three-tangle [31],
+τ 2
+3 = 4J4 ,
+(21)
+while an inequivalent type of genuine multipartite entanglement (GME) is quantified by the so called GME concur-
+rence, CGME [32], defined, in terms of invariant polynomials, for a three-qubit pure state as:
+CGME = 4 (min {J2 + J3, J1 + J3, J1 + J2} + J4) .
+(22)
+Hence, three qubit states can be sorted in the following entanglement classes [30]:
+• Product states. All Ji = 0, resulting into A − B − C (class 1). All entanglement measures vanish.
+• Biseparable states. All Ji = 0 except i) J1 for A − BC, ii) J2 for B − AC, and iii) J3 for C − AB (class 2a).
+Only the concurrence for one single pair of qubits is different from zero.
+• W-states. CGME > 0 and τ3 = 0
+1. J4 = 0 and J1J2 + J1J3 + J2J3 = √J1J2J3 = J5
+2 (class 3a).
+2. J4 = 0 and √J1J2J3 = J5
+2 (class 4a).
+
+6
+• GHZ-states. CGME > 0 and τ3 > 0, with 5 possible cases:
+1. All Ji = 0 except J4 (class 2b).
+2. J1 = J2 = J5 = 0, or J1 = J3 = J5 = 0 or J2 = J3 = J5 = 0 (class 3b).
+3. J2 = J5 = 0 or J3 = J5 = 0 (class 4b).
+4. J1J4 + J1J2 + J1J3 = √J1J2J3 = J5
+2 (class 4c).
+5. √J1J2J3 = |J5|
+2
+and (J4 + J5)2 − 4 (J1 + J4) (J2 + J4) (J3 + J4) = 0 (class 4d) ,
+where, with the notation X − Y , we indicate that subsystems X and Y do not share any type of entanglement.
+Notably, for 3-qubit pure states, a monogamy relation exists between the amount of entanglement that can be
+shared among the parties [31]
+C2
+A|BC = C2
+AB + C2
+AC + τ 2
+3 .
+(23)
+2.
+Fidelity of 3-QST
+Here, we derive the fidelity of the QST of a tree-qubit pure state. First, we average the state in Equation (18) over
+single-qubit, local operations U, and use the notation ⟨·⟩U ⊗3, in order to indicate the average fidelity at given values
+of (and, thus, as a function of) the entanglement quantifiers. Subsequently, utilizing the averages of the invariant
+polynomials obtained in Ref. [29], we derive the average fidelity of each three-qubit class and use the notation ⟨·⟩.
+The 3-QST fidelity, expressed in terms of the single-qubit average, reads
+⟨F(|Ψ⟩ , ρ0)⟩U ⊗3 = ⟨F1⟩3 − 8 ⟨F1⟩
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩)
+�
+J1 + J2 + J3 + 3
+2J4
+�
+.
+(24)
+Since 1
+2 ≤ ⟨F1⟩ ≤ 1, and 0 ≤ Ji ≤ 1
+4, the second term on the right hand side of the latter equation is always
+negative, so that one sees at once that entanglement reduces the 3-QST fidelity.
+Introducing the reduction factor R3,
+R3 = ⟨F1⟩
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩) ,
+(25)
+we can write
+⟨F(|Ψ⟩ , ρ0)⟩U ⊗3 = ⟨F1⟩3 − 8 R3
+�
+J1 + J2 + J3 + 3
+2J4
+�
+.
+(26)
+Now, using the averages of the invariant polynomials obtained in Ref. [29], ⟨J4⟩ =
+1
+12 and ⟨Jk⟩ =
+1
+24 (k = 1, 2, 3),
+the average fidelity ⟨F3⟩ (with average taken over the full three qubit Hilbert space) is given by
+⟨F3⟩ = ⟨F1⟩3 − 2R3
+(27)
+Here, we see that, at fixed single-particle average fidelity, entanglement is responsible for a decrease in the 3-QST
+average fidelity by twice the reduction factor R3.
+In particular, the fidelities for the canonical states belonging to the different entanglement classes read
+• class 1 (product state)
+⟨Fc1⟩U ⊗3 = ⟨F1⟩3
+(28)
+⟨Fc1⟩ = ⟨F1⟩3
+(29)
+• class 2a (biseparable states)
+⟨Fc2a⟩U ⊗3 = ⟨F1⟩3 − 2 ⟨F1⟩
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩) C2
+jk
+(30)
+⟨Fc2a⟩ = ⟨F1⟩3 − R3
+3
+(31)
+where i ̸= j ̸= k = 1, 2, 3;
+
+7
+• class 2b (GHZ-states)
+⟨Fc2b⟩U ⊗3 = ⟨F1⟩3 − 3 ⟨F1⟩
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩) τ 2
+3
+(32)
+⟨Fc2b⟩ = ⟨F1⟩3 − R3
+(33)
+• class 3a (J4 = 0 and J1J2 + J1J3 + J2J3 = √J1J2J3 = J5
+2 )
+⟨Fc3a⟩U ⊗3 = ⟨F1⟩3 − 2 ⟨F1⟩
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩)
+�
+C2
+BC + C2
+AC + C2
+AB
+�
+⟨Fc3a⟩ = ⟨F1⟩3 − R3
+(34)
+• Class 3b (J1 = J2 = J5 = 0 or J1 = J3 = J5 = 0 or J2 = J3 = J5 = 0)
+⟨Fc3b⟩U ⊗3 = ⟨F1⟩3 − ⟨F1⟩
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩)
+�
+2C2
+BC + 3τ 2
+3
+�
+⟨Fc3b⟩ = ⟨F1⟩3 − 4
+3R3
+(35)
+• Class 4a J4 = 0 and √J1J2J3 = J5
+2
+⟨Fc4a⟩U ⊗3 = ⟨F1⟩3 − 2 ⟨F1⟩
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩)
+�
+C2
+BC + C2
+AC + C2
+AB
+�
+⟨Fc4a⟩ = ⟨F1⟩3 − R3
+(36)
+• Class 4b (J2 = J5 = 0 or J3 = J5 = 0)
+⟨Fc4b⟩U ⊗3 = ⟨F1⟩3 − ⟨F1⟩
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩)
+�
+2
+�
+C2
+BC + C2
+AC
+�
++ 3τ 2
+3
+�
+⟨Fc4b⟩ = ⟨F1⟩3 − 5
+3R3
+(37)
+• Class 4c (J1J4 + J1J2 + J1J3 + J2J3 = √J1J2J3 = J5
+2 )
+⟨Fc4c⟩U ⊗3 = ⟨F1⟩3 − ⟨F1⟩
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩)
+�
+2
+�
+C2
+BC + C2
+AC + C2
+AB
+�
++ 3τ 2
+3
+�
+⟨Fc4c⟩ = ⟨F1⟩3 − 2R3
+(38)
+• Class 4d (√J1J2J3 = |J5|
+2
+and (J4 + J5)2 − 4 (J1 + J4) (J2 + J4) (J3 + J4) = 0)
+⟨Fc4c⟩U ⊗3 = ⟨F1⟩3 − ⟨F1⟩
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩)
+�
+2
+�
+C2
+BC + C2
+AC + C2
+AB
+�
++ 3τ 2
+3
+�
+⟨Fc4d⟩ = ⟨F1⟩3 − 2R3
+(39)
+Comparing Equation (30) with Equation (32), it turns out that, for an equivalent amount of the entanglement
+monotone C2
+jk and τ 2
+3 , at fixed ⟨F1⟩ (or, equivalently, at a fixed transition amplitude f), the fidelity of the canonical
+state in class 2a is greater than that in class 2b. This is in line with our intuition that the more entangled a state
+is, the harder it is to achieve high fidelity in our parallel QST protocol, as shown in Figure 3 (right panel), where we
+plot the average fidelity for different three qubit classes. In the left panel of Figure 3 we report the reduction factor
+R3 of Equation (25) as a function of the single-particle average fidelity ⟨F1⟩.
+From the above equations of the average fidelity of the three-qubit classes, we see that the average fidelity is
+decreased, with respect to the product state class, whenever there is two-qubit concurrence or genuine multipartite
+entanglement, both as CGME and as τ3. Moreover, per equal amount of squared two-qubit concurrence C2 and genuine
+multipartite entanglement CGME, the reducing factor is respectively 2 and 3 times R3. As a consequence, we state
+that, at fixed amount of entanglement, GME states are harder to transfer than biseparable states.
+
+8
+0.6
+0.7
+0.8
+0.9
+1.0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.6
+0.7
+0.8
+0.9
+1.0
+0.2
+0.4
+0.6
+0.8
+1.0
+FIG. 3. (left) Reduction factor for the average fidelity in the presence of entanglement as in Equation (25). (right) Average
+fidelity for the three-qubit classes. The dotted, vertical line is at ⟨F⟩1 = 1
+2
+�
+1 +
+1
+√
+3
+�
+≃ 0.789.
+C.
+Four Qubits
+While for two and three qubits, the entanglement of pure states has been fully characterized, for four (or more)
+qubits there are infinitely many inequivalent entanglement classes [30, 33] under SLOCC operations (stochastic local
+operations and classical communication).
+Here, we consider the fidelity of specific four-qubit states averaged over random local unitaries on each qubit.
+Whereas this does not account for all entangled states within a given class, as the group of stocastic local operations
+includes deterministic local operations, SU(2) ⊆ SL(2, C) (with equality holding for pure states), the results nev-
+ertheless hint at the fact that the average fidelity decreases with the entanglement of the sender state and that the
+reduction factor depends on the type of entanglement contained in the state.
+We will consider the three irreducibly balanced states [34]: the 4-qubits GHZ-state, the cluster, and X4 :
+|GHZ4⟩ =
+1
+√
+2 (|0000⟩ + |1111⟩)
+(40a)
+|Cl4⟩ = 1
+2 (|0000⟩ + |0111⟩ + |1011⟩ + |1100⟩)
+(40b)
+|X4⟩ =
+1
+√
+6
+�√
+2 |1111⟩ + |0001⟩ + |0010⟩ + |0100⟩ + |1000⟩
+�
+(40c)
+and two additional entangled states: the product of two-Bell states and the 4-qubit W-state:
+|B2⟩ = |Φ⟩12 ⊗ |Φ⟩34
+(41a)
+|W4⟩ = 1
+2 (|0001⟩ + |0010⟩ + |0100⟩ + |1000⟩) .
+(41b)
+The average fidelities of the states reported in Equations (40) and (41), expressed in terms of 1-QST average fidelity,
+read
+⟨FGHZ4⟩U ⊗4 = ⟨FB2⟩U ⊗4 = ⟨F1⟩4 − 2 ⟨F1⟩
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩)
+�
+1 − 3 ⟨F⟩ + 4 ⟨F⟩2�
+(42a)
+⟨FCl4⟩U ⊗4 = ⟨FX4⟩U ⊗4 = ⟨F1⟩4 − 4 ⟨F1⟩2
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩)
+(42b)
+⟨FW4⟩U ⊗4 = ⟨F1⟩4 − 3 ⟨F1⟩2
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩) .
+(42c)
+Notice that the reduction factor for the states |Cl4⟩ , |X4⟩ , |W4⟩ is the same, although with different weights, and
+
+9
+0.6
+0.7
+0.8
+0.9
+1.0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.6
+0.7
+0.8
+0.9
+1.0
+0.2
+0.4
+0.6
+0.8
+1.0
+FIG. 4. (left) Reduction factor for the average fidelity in the presence of entanglement as in Equations (43). (right) Average
+fidelity for the entangled classes as reported in Equations (42). The blue and red dotted, vertical lines are, respectively, at
+⟨F⟩1 = 0.82 and ⟨F⟩1 = 0.85.
+differs from the reduction factor for the states |GHZ4⟩ , |B2⟩, reading, respectively,
+R4a = ⟨F1⟩
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩)
+�
+1 − 3 ⟨F⟩ + 4 ⟨F⟩2�
+(43a)
+R4b = ⟨F1⟩2
+�
+⟨F1⟩ − 1
+2
+�
+(1 − ⟨F1⟩) .
+(43b)
+A possible reason may be that, if one considers the four-tangle as an entanglement measure, although it is not a
+measure of genuine multipartite entanglement, the first set of state has zero four-tangle, whereas for the second one
+it is non-zero. In Figure 4 (left panel) we report the reduction factors in Equation (43), while in the right panel we
+report the average fidelity of Equation (42).
+IV.
+DISCUSSION
+We have shown that the QST of an entangled n ≥ 2 quantum state across parallel, independent U(1)-symmetric
+quantum channels, as, e.g., embodied by an XXZ spin- 1
+2 Hamiltonian, leads to a lower average fidelity than that of
+the QST of a product state at fixed one-qubit QST average fidelity, or, equivalently, at fixed transition amplitude.
+For the case of n = 2, we have expressed the average fidelity reduction in terms of the squared concurrence times a
+reduction factor. Similarly, for n = 3, we obtained that the presence of entanglement, both bipartite and multipartite,
+has a detrimental effect on the average fidelity. In particular, we obtained that the reduction factor has a greater
+weight in the presence of genuine three-partite entanglement, i.e., three-tangle and GME concurrence, than in the
+presence of two-qubit squared concurrence for specific canonical classes of the three-qubit pure state. Finally, we have
+considered specific cases of 4-qubit entangled states, which, again, result in an average fidelity reduction due to the
+presence of entanglement in the initial state.
+Our work clearly shows that for entanglement distribution in a routing configuration, where parties are sent over
+independent quantum channels, the single-qubit average fidelity is not a reliable figure of merit. This calls for more
+investigations into the properties of quantum channels able to faithfully distribute multipartite entangled states.
+ACKNOWLEDGEMENTS
+TJGA acknowledges funding through the IPAS+ (Internationalisation Partnership Awards Scheme +) QUEST
+project by the MCST (The Malta Council for Science & Technology). MC acknowledges funding from the Tertiary
+Education Scholarships Scheme and from the Project QVAQT financed by the Malta Council for Science & Technology,
+for and on behalf of the Foundation for Science and Technology, through the FUSION: R&I Research Excellence
+Programme REP-2022-003.
+K ˙Z acknowledges support by Narodowe Centrum Nauki under the Quantera project
+
+10
+number 2021/03/Y/ST2/00193 and by the Foundation for Polish Science under the Team-Net project POIR.04.04.00-
+00-17C1/18-00. SL acknowledges support by MUR under PRIN Project No. 2017 SRN-BRK QUSHIP
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+

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@@ -0,0 +1,3093 @@
+arXiv:2301.01716v1  [math.OC]  4 Jan 2023
+First-order penalty methods for bilevel optimization
+Zhaosong Lu ∗
+Sanyou Mei ∗
+January 4, 2023
+Abstract
+In this paper we study a class of unconstrained and constrained bilevel optimization problems in
+which the lower-level part is a convex optimization problem, while the upper-level part is possibly
+a nonconvex optimization problem. In particular, we propose penalty methods for solving them,
+whose subproblems turn out to be a structured minimax problem and are suitably solved by a first-
+order method developed in this paper. Under some suitable assumptions, an operation complexity
+of O(ε−4 log ε−1) and O(ε−7 log ε−1), measured by their fundamental operations, is established for
+the proposed penalty methods for finding an ε-KKT solution of the unconstrained and constrained
+bilevel optimization problems, respectively.
+To the best of our knowledge, the methodology and
+results in this paper are new.
+Keywords: bilevel optimization, minimax optimization, penalty methods, first-order methods, opera-
+tion complexity
+Mathematics Subject Classification: 90C26, 90C30, 90C47, 90C99, 65K05
+1
+Introduction
+Bilevel optimization is a two-level hierarchical optimization in which partial or full decision variables in
+the upper level are also involved in the lower level. Generically, it can be written in the following form:
+min
+x,y
+f(x, y)
+s.t.
+g(x, y) ≤ 0,
+y ∈ Argmin
+z
+{ ˜f(x, z)|˜g(x, z) ≤ 0}.1
+(1)
+Bilevel optimization has found a variety of important applications, including adversarial training [36,
+37, 46], continual learning [32], hyperparameter tuning [3, 17], image reconstruction [9], meta-learning
+[4, 23, 42], neural architecture search [15, 30], reinforcement learning [20, 27], and Stackelberg games [48].
+More applications about it can be found in [2, 8, 10, 11, 12, 44] and the references therein. Theoretical
+properties including optimality conditions of (1) have been extensively studied in the literature (e.g., see
+[12, 13, 34, 47, 50]).
+Numerous methods have been developed for solving some special cases of (1). For example, constraint-
+based methods [19, 43], deterministic gradient-based methods [16, 17, 21, 35, 41, 42], and stochastic
+gradient-based methods [6, 18, 20, 24, 26] were proposed for solving (1) with g ≡ 0, ˜g ≡ 0, f, ˜f being
+smooth, and ˜f being strongly convex with respect to y. Besides, when all the functions involved in
+(1) are smooth and ˜f, ˜g are convex with respect to y, gradient-type methods were proposed by solving
+the mathematical program with equilibrium constraints (MPEC) resulting from replacing the lower-level
+optimization problem of (1) by its first-order optimality conditions (e.g., see [1, 33, 40]).
+Recently,
+difference-of-convex (DC) algorithms were developed in [51] for solving (1) with g ≡ 0, f being a DC
+function, and ˜f, ˜g being convex functions. In addition, a double penalty method [22] was proposed for
+(1), which solves a sequence of bilevel optimization problems of the form
+min
+x,y
+f(x, y) + ρkΨ(x, y)
+s.t.
+y ∈ Argmin
+z
+˜f(x, z) + ρk ˜Ψ(x, z),
+(2)
+∗Department of Industrial and Systems Engineering, University of Minnesota, USA (email:
+zhaosong@umn.edu,
+mei00035@umn.edu). This work was partially supported by NSF Award IIS-2211491.
+1For ease of reading, throughout this paper the tilde symbol is particularly used for the functions related to the lower-level
+optimization problem. Besides, “Argmin” denotes the set of optimal solutions of the associated problem.
+1
+
+where {ρk} is a sequence of penalty parameters, and Ψ and ˜Ψ are a penalty function associated with the
+sets {(x, y)|g(x, y) ≤ 0} and {(x, z)|˜g(x, z) ≤ 0}, respectively. Though problem (2) appears to be simpler
+than (1), there is no method available for finding an approximate solution of (2) in general. Conse-
+quently, the double penalty method [22] is typically not implementable. More discussion on algorithmic
+development for bilevel optimization can be found in [2, 8, 12, 31, 45, 47]) and the references therein.
+It has long been known that the notorious challenge of bilevel optimization (1) mainly comes from the
+lower level part, which requires that the variable y be a solution of another optimization problem. Due
+to this, for the sake of simplicity, we only consider a subclass of bilevel optimization with the constraint
+g(x, y) ≤ 0 being excluded, namely,
+min
+x,y
+f(x, y)
+s.t.
+y ∈ Argmin
+z
+{ ˜f(x, z)|˜g(x, z) ≤ 0}.
+(3)
+Nevertheless, the results in this paper can be possibly extended to problem (1).
+The main goal of this paper is to develop a first-order penalty method for solving problem (3). Our
+key observations toward this development are: (i) problem (3) can be approximately solved as a penalty
+problem (see (49)); (ii) such a penalty problem is equivalent to a structured minimax problem (see
+(50)), which can be suitably solved by a first-order method proposed in Section 2. As a result, these
+observations lead to development of a novel first-order penalty method for solving (3) (see Sections 3
+and 4), which enjoys the following appealing features.
+• It uses only the first-order information of the problem. Specifically, its fundamental operations
+consist only of evaluations of the gradient of ˜g and the smooth component of f and ˜f and also
+the proximal operator of the nonsmooth component of f and ˜f. Thus, it is suitable for solving
+large-scale problems (see Sections 3 and 4).
+• It has theoretical guarantees on operation complexity, which is measured by the aforementioned
+fundamental operations, for finding an ε-KKT solution of (3).
+In particular, when ˜g ≡ 0, it
+enjoys an operation complexity of O(ε−4 log ε−1). Otherwise, it enjoys an operation complexity of
+O(ε−7 log ε−1) (see Theorems 4 and 6).
+• It is applicable to a broader class of problems than existing methods.
+For example, it can be
+applied to (3) with f, ˜f being nonsmooth and ˜f, ˜g being nonconvex with respect to x, which is
+however not suitable for existing methods.
+To the best of our knowledge, the methodology and results in this paper are new.
+The rest of this paper is organized as follows. In Subsection 1.1 we introduce some notation and
+terminology. In Section 2 we propose a first-order method for solving a nonconvex-concave minimax
+problem and study its complexity.
+In Sections 3 and 4, we propose first-order penalty methods for
+unconstrained and constrained bilevel optimization and study their complexity, respectively. In Section
+5 we present the proofs of the main results. Finally, we make some concluding remarks in Section 6.
+1.1
+Notation and terminology
+The following notation will be used throughout this paper.
+Let Rn denote the Euclidean space of
+dimension n and Rn
++ denote the nonnegative orthant in Rn. The standard inner product and Euclidean
+norm are denoted by ⟨·, ·⟩ and ∥ · ∥, respectively. For any v ∈ Rn, let v+ denote the nonnegative part of
+v, that is, (v+)i = max{vi, 0} for all i. For any two vectors u and v, (u; v) denotes the vector resulting
+from stacking v under u. Given a point x and a closed set S in Rn, let dist(x, S) = minx′∈S ∥x′ − x∥ and
+IS denote the indicator function associated with S.
+A function or mapping φ is said to be Lφ-Lipschitz continuous on a set S if ∥φ(x)−φ(x′)∥ ≤ Lφ∥x−x′∥
+for all x, x′ ∈ S. In addition, it is said to be L∇φ-smooth on S if ∥∇φ(x) − ∇φ(x′)∥ ≤ L∇φ∥x − x′∥ for
+all x, x′ ∈ S. For a closed convex function p : Rn → R ∪ {∞},2 the proximal operator associated with p
+is denoted by proxp, that is,
+proxp(x) = arg min
+x′∈Rn
+�1
+2∥x′ − x∥2 + p(x′)
+�
+∀x ∈ Rn.
+(4)
+2For convenience, ∞ stands for +∞.
+2
+
+Given that evaluation of proxγp(x) is often as cheap as proxp(x), we count the evaluation of proxγp(x)
+as one evaluation of proximal operator of p for any γ > 0 and x ∈ Rn.
+For a lower semicontinuous function φ : Rn → R∪{∞}, its domain is the set dom φ := {x|φ(x) < ∞}.
+The upper subderivative of φ at x ∈ dom φ in a direction d ∈ Rn is defined by
+φ′(x; d) = lim sup
+x′ φ
+→x, t↓0
+inf
+d′→d
+φ(x′ + td′) − φ(x′)
+t
+,
+where t ↓ 0 means both t > 0 and t → 0, and x′
+φ→ x means both x′ → x and φ(x′) → φ(x). The
+subdifferential of φ at x ∈ dom φ is the set
+∂φ(x) = {s ∈ Rn��sT d ≤ φ′(x; d) ∀d ∈ Rn}.
+We use ∂xiφ(x) to denote the subdifferential with respect to xi. In addition, for an upper semicontinuous
+function φ, its subdifferential is defined as ∂φ = −∂(−φ). If φ is locally Lipschitz continuous, the above
+definition of subdifferential coincides with the Clarke subdifferential. Besides, if φ is convex, it coincides
+with the ordinary subdifferential for convex functions. Also, if φ is continuously differentiable at x , we
+simply have ∂φ(x) = {∇φ(x)}, where ∇φ(x) is the gradient of φ at x. In addition, it is not hard to
+verify that ∂(φ1 + φ2)(x) = ∇φ1(x) + ∂φ2(x) if φ1 is continuously differentiable at x and φ2 is lower or
+upper semicontinuous at x. See [7, 49] for more details.
+Finally, we introduce two types of approximate solutions for a general minimax problem
+Ψ∗ = min
+x max
+y
+Ψ(x, y),
+(5)
+where Ψ(·, y) : Rn → R ∪ {∞} is a lower semicontinuous function, Ψ(x, ·) : Rm → R ∪ {−∞} is an upper
+semicontinuous function, and Ψ∗ is finite.
+Definition 1. A point (xǫ, yǫ) is called an ǫ-optimal solution of the minimax problem (5) if
+max
+y
+Ψ(xǫ, y) − Ψ(xǫ, yǫ) ≤ ǫ,
+Ψ(xǫ, yǫ) − Ψ∗ ≤ ǫ.
+Definition 2. A point (x, y) is called a stationary point of the minimax problem (5) if
+0 ∈ ∂xΨ(x, y),
+0 ∈ ∂yΨ(x, y).
+In addition, for any ǫ > 0, a point (xǫ, yǫ) is called an ǫ-stationary point of the minimax problem (5) if
+dist (0, ∂xΨ(xǫ, yǫ)) ≤ ǫ,
+dist (0, ∂yΨ(xǫ, yǫ)) ≤ ǫ.
+2
+A first-order method for nonconvex-concave minimax prob-
+lem
+In this section, we propose a first-order method for finding an approximate stationary point of a
+nonconvex-concave minimax problem, which will be used as a subproblem solver for the penalty methods
+proposed in Sections 3 and 4. In particular, we consider the minimax problem
+H∗ = min
+x max
+y
+{H(x, y) := h(x, y) + p(x) − q(y)} .
+(6)
+Assume that problem (6) has at least one optimal solution. In addition, h, p and q satisfy the following
+assumptions.
+Assumption 1.
+(i) p : Rn → R ∪ {∞} and q : Rm → R ∪ {∞} are proper convex functions and
+continuous on their domain, and moreover, their domain is compact.
+(ii) The proximal operator associated with p and q can be exactly evaluated.
+(iii) h is L∇h-smooth on dom p × dom q, and moreover, h(x, ·) is concave for any x ∈ dom p.
+3
+
+Recently, an accelerated inexact proximal point smoothing (AIPP-S) scheme was proposed in [28]
+for finding an approximate stationary point of a class of minimax composite nonconvex optimization
+problems, which includes (6) as a special case. When applied to (6), AIPP-S requires the availability of
+the oracle including exact evaluation of ∇xh(x, y) and
+arg min
+x
+�
+p(x) + 1
+2λ∥x − x′∥2
+�
+,
+arg max
+y
+�
+h(x′, y) − q(y) − 1
+2λ∥y − y′∥2
+�
+(7)
+for any λ > 0, x′ ∈ Rn and y′ ∈ Rm. Note that h is typically sophisticated and the exact solution of the
+second problem in (7) usually cannot be found. As a result, AIPP-S is generally not implementable for
+(6), though an oracle complexity of O(ǫ−5/2) was established in [28] for it to find an ǫ-stationary point
+of (6).
+In what follows, we first propose a modified optimal first-order method for solving a strongly-convex-
+strongly-concave minimax problem in Subsection 2.1. Using this method as a subproblem solver for an
+inexact proximal point scheme, we then propose a first-order method for (6) in Subsection 2.2, which
+enjoys an operation complexity of O(ǫ−5/2 log ǫ−1), measured by the amount of evaluations of ∇h and
+proximal operator of p and q, for finding an ǫ-stationary point of (6).
+2.1
+A modified optimal first-order method for strongly-convex-strongly-concave
+minimax problem
+In this subsection, we consider the strongly-convex-strongly-concave minimax problem
+¯H∗ = min
+x max
+y
+� ¯H(x, y) := ¯h(x, y) + p(x) − q(y)
+�
+,
+(8)
+where p and q satisfy Assumption 1 and ¯h satisfies the following assumption.
+Assumption 2. ¯h(x, y) is σx-strongly-convex-σy-strongly-concave and L∇¯h-smooth on dom p × dom q
+for some σx, σy > 0.
+The goal of this subsection is to propose a modified optimal first-order method for finding an approx-
+imate stationary point of problem (8) and study its complexity. Before proceeding, we introduce some
+more notation below, most of which is adopted from [29].
+Let (x∗, y∗) denote the optimal solution of (8), z∗ = −σxx∗, and
+Dp = max{∥u − v∥
+��u, v ∈ dom p},
+Dq = max{∥u − v∥
+��u, v ∈ dom q},
+(9)
+¯Hlow = min
+� ¯H(x, y)|
+�
+x, y) ∈ dom p × dom q},
+(10)
+ˆh(x, y) = ¯h(x, y) − σx∥x∥2/2 + σy∥y∥2/2,
+(11)
+G(z, y) = sup
+x {⟨x, z⟩ − p(x) − ˆh(x, y) + q(y)},
+(12)
+P(z, y) = σ−1
+x ∥z∥2/2 + σy∥y∥2/2 + G(z, y),
+(13)
+ϑk = η−1
+z ∥zk − z∗∥2 + η−1
+y ∥yk − y∗∥2 + 2¯α−1(P(zk
+f, yk
+f) − P(z∗, y∗)),
+(14)
+ak
+x(x, y) = ∇xˆh(x, y) + σx(x − σ−1
+x zk
+g)/2,
+ak
+y(x, y) = −∇yˆh(x, y) + σyy + σx(y − yk
+g)/8,
+where ¯α = min
+�
+1,
+�
+8σy/σx
+�
+, ηz = σx/2, ηy = min {1/(2σy), 4/(¯ασx)}, and yk, yk
+f, yk
+g, zk, zk
+f and zk
+g
+are generated at iteration k of Algorithm 1 below. By Assumptions 1 and 2, one can observe that Dp,
+Dq and ¯Hlow are finite.
+We are now ready to present a modified optimal first-order method for solving (8) in Algorithm 1. It is
+a slight modification of the novel optimal first-order method [29, Algorithm 4] by incorporating a forward-
+backward splitting scheme and also a verifiable termination criterion (see steps 23-25 in Algorithm 1) in
+order to find a τ-stationary point of (8) (see Definition 2) for any prescribed tolerance τ > 0.
+4
+
+Algorithm 1 A modified optimal first-order method for (8)
+Input: τ > 0, ¯z0 = z0
+f ∈ −σxdom p,3 ¯y0 = y0
+f ∈ dom q, (z0, y0) = (¯z0, ¯y0), ¯α = min
+�
+1,
+�
+8σy/σx
+�
+,
+ηz = σx/2, ηy = min {1/(2σy), 4/(¯ασx)}, βt = 2/(t + 3), ζ =
+�
+2
+√
+5(1 + 8L∇¯h/σx)
+�−1, γx = γy =
+8σ−1
+x , and ˆζ = min{σx, σy}/L2
+∇¯h.
+1: for k = 0, 1, 2, . . . do
+2:
+(zk
+g , yk
+g) = ¯α(zk, yk) + (1 − ¯α)(zk
+f, yk
+f).
+3:
+(xk,−1, yk,−1) = (−σ−1
+x zk
+g, yk
+g).
+4:
+xk,0 = proxζγxp(xk,−1 − ζγxak
+x(xk,−1, yk,−1)).
+5:
+yk,0 = proxζγyq(yk,−1 − ζγyak
+y(xk,−1, yk,−1)).
+6:
+bk,0
+x
+=
+1
+ζγx (xk,−1 − ζγxak
+x(xk,−1, yk,−1) − xk,0).
+7:
+bk,0
+y
+=
+1
+ζγy (yk,−1 − ζγyak
+y(xk,−1, yk,−1) − yk,0).
+8:
+t = 0.
+9:
+while
+γx∥ak
+x(xk,t, yk,t) + bk,t
+x ∥2 + γy∥ak
+y(xk,t, yk,t) + bk,t
+y ∥2 > γ−1
+x ∥xk,t − xk,−1∥2 + γ−1
+y ∥yk,t − yk,−1∥2
+do
+10:
+xk,t+1/2 = xk,t + βt(xk,0 − xk,t) − ζγx(ak
+x(xk,t, yk,t) + bk,t
+x ).
+11:
+yk,t+1/2 = yk,t + βt(yk,0 − yk,t) − ζγy(ak
+y(xk,t, yk,t) + bk,t
+y ).
+12:
+xk,t+1 = proxζγxp(xk,t + βt(xk,0 − xk,t) − ζγxak
+x(xk,t+1/2, yk,t+1/2)).
+13:
+yk,t+1 = proxζγyq(yk,t + βt(yk,0 − yk,t) − ζγyak
+y(xk,t+1/2, yk,t+1/2)).
+14:
+bk,t+1
+x
+=
+1
+ζγx (xk,t + βt(xk,0 − xk,t) − ζγxak
+x(xk,t+1/2, yk,t+1/2) − xk,t+1).
+15:
+bk,t+1
+y
+=
+1
+ζγy (yk,t + βt(yk,0 − yk,t) − ζγyak
+y(xk,t+1/2, yk,t+1/2) − yk,t+1).
+16:
+t ← t + 1.
+17:
+end while
+18:
+(xk+1
+f
+, yk+1
+f
+) = (xk,t, yk,t).
+19:
+(zk+1
+f
+, wk+1
+f
+) = (∇xˆh(xk+1
+f
+, yk+1
+f
+) + bk,t
+x , −∇yˆh(xk+1
+f
+, yk+1
+f
+) + bk,t
+y ).
+20:
+zk+1 = zk + ηzσ−1
+x (zk+1
+f
+− zk) − ηz(xk+1
+f
++ σ−1
+x zk+1
+f
+).
+21:
+yk+1 = yk + ηyσy(yk+1
+f
+− yk) − ηy(wk+1
+f
++ σyyk+1
+f
+).
+22:
+xk+1 = −σ−1
+x zk+1.
+23:
+ˆxk+1 = proxˆζp(xk+1 − ˆζ∇x¯h(xk+1, yk+1)).
+24:
+ˆyk+1 = proxˆζq(yk+1 + ˆζ∇y¯h(xk+1, yk+1)).
+25:
+Terminate the algorithm and output (ˆxk+1, ˆyk+1) if
+∥ˆζ−1(xk+1 − ˆxk+1, ˆyk+1 − yk+1) − (∇¯h(xk+1, yk+1) − ∇¯h(ˆxk+1, ˆyk+1))∥ ≤ τ.
+(15)
+26: end for
+The following theorem presents iteration and operation complexity of Algorithm 1 for finding a τ-
+stationary point of problem (8), whose proof is deferred to Subsection 5.1.
+Theorem 1 (Complexity of Algorithm 1). Suppose that Assumptions 1 and 2 hold. Let ¯H∗, Dp,
+Dq, ¯Hlow, and ϑ0 be defined in (8), (9), (10) and (14), σx, σy and L∇¯h be given in Assumption 2, ¯α,
+ηy, ηz, τ, ˆζ be given in Algorithm 1, and
+¯δ = (2 + ¯α−1)σxD2
+p + max{2σy, ¯ασx/4}D2
+q,
+(16)
+¯K =
+�
+max
+� 2
+¯α, ¯ασx
+4σy
+�
+log 4 max{ηzσ−2
+x , ηy}ϑ0
+(ˆζ−1 + L∇¯h)−2τ 2
+�
++
+,
+(17)
+¯N =
+�
+max
+�
+2,
+� σx
+2σy
+�
+log 4 max {1/(2σx), min {1/(2σy), 4/(¯ασx)}}
+�¯δ + 2¯α−1 � ¯H∗ − ¯Hlow
+��
+(L2
+∇¯h/ min{σx, σy} + L∇¯h)−2τ 2
+�
++
+×
+��
+96
+√
+2
+�
+1 + 8L∇¯hσ−1
+x
+��
++ 2
+�
+.
+(18)
+Then Algorithm 1 outputs a τ-stationary point of (8) in at most ¯K iterations.
+Moreover, the total
+3For convenience, −σxdom p stands for the set {−σxu|u ∈ dom p}.
+5
+
+number of evaluations of ∇¯h and proximal operator of p and q performed in Algorithm 1 is no more than
+¯N, respectively.
+Remark 1. It can be observed from Theorem 1 that Algorithm 1 enjoys an operation complexity of
+O(log τ−1), measured by the amount of evaluations of ∇¯h and proximal operator of p and q, for finding
+a τ-stationary point of the strongly-convex-strongly-concave minimax problem (8).
+2.2
+A first-order method for problem (6)
+In this subsection, we propose a first-order method for finding an ǫ-stationary point of problem (6) (see
+Definition 2) for any prescribed tolerance ǫ > 0. In particular, we first add a perturbation to the max
+part of (6) for obtaining an approximation of (6), which is given as follows:
+min
+x max
+y
+�
+h(x, y) + p(x) − q(y) −
+ǫ
+4Dq
+∥y − ˆy0∥2
+�
+(19)
+for some ˆy0 ∈ dom q. We then apply an inexact proximal point method [25] to (19), which consists of
+approximately solving a sequence of subproblems
+min
+x max
+y
+{Hk(x, y) := hk(x, y) + p(x) − q(y)} ,
+(20)
+where
+hk(x, y) = h(x, y) − ǫ∥y − ˆy0∥2/(4Dq) + L∇h∥x − xk∥2.
+(21)
+By Assumption 1, one can observe that (i) hk is L∇h-strongly convex in x and ǫ/(2Dq)-strongly concave
+in y on dom p × dom q; (ii) hk is (3L∇h + ǫ/(2Dq))-smooth on dom p × dom q. Consequently, problem
+(20) is a special case of (8) and we can apply Algorithm 1 to solve (20). The resulting first-order method
+for (6) is presented in Algorithm 2.
+Algorithm 2 A first-order method for problem (6)
+Input: ǫ > 0, ǫ0 ∈ (0, ǫ/2], (ˆx0, ˆy0) ∈ dom p × dom q, (x0, y0) = (ˆx0, ˆy0), and ǫk = ǫ0/(k + 1).
+1: for k = 0, 1, 2, . . . do
+2:
+Call Algorithm 1 with ¯h ← hk, τ ← ǫk, σx ← L∇h, σy ← ǫ/(2Dq), L∇¯h ← 3L∇h + ǫ/(2Dq),
+¯z0 = z0
+f ← −σxxk, ¯y0 = y0
+f ← yk, and denote its output by (xk+1, yk+1), where hk is given in (21).
+3:
+Terminate the algorithm and output (xǫ, yǫ) = (xk+1, yk+1) if
+∥xk+1 − xk∥ ≤ ǫ/(4L∇h).
+(22)
+4: end for
+Remark 2. It can be observed from step 2 of Algorithm 2 that (xk+1, yk+1) results from applying Algo-
+rithm 1 to the subproblem (20). As will be shown in Lemma 2, (xk+1, yk+1) is an ǫk-stationary point of
+(20).
+We next study complexity of Algorithm 2 for finding an ǫ-stationary point of problem (6). Before
+proceeding, we define
+Hlow := min {H(x, y)|(x, y) ∈ dom p × dom q} .
+(23)
+By Assumption 1, one can observe that Hlow is finite.
+The following theorem presents iteration and operation complexity of Algorithm 2 for finding an
+ǫ-stationary point of problem (6), whose proof is deferred to Subsection 5.2.
+Theorem 2 (Complexity of Algorithm 2). Suppose that Assumption 1 holds. Let H∗, H Dp, Dq,
+and Hlow be defined in (6), (9) and (23), L∇h be given in Assumption 1, ǫ, ǫ0 and ˆx0 be given in
+6
+
+Algorithm 2, and
+α = min
+�
+1,
+�
+4ǫ/(DqL∇h)
+�
+,
+(24)
+δ = (2 + α−1)L∇hD2
+p + max {ǫ/Dq, αL∇h/4} D2
+q,
+(25)
+K =
+�
+16(max
+y
+H(ˆx0, y) − H∗ + ǫDq/4)L∇hǫ−2 + 32ǫ2
+0(1 + 4D2
+qL2
+∇hǫ−2)ǫ−2 − 1
+�
++
+,
+(26)
+N =
+��
+96
+√
+2
+�
+1 + (24L∇h + 4ǫ/Dq) L−1
+∇h
+��
++ 2
+�
+max
+�
+2,
+�
+DqL∇hǫ−1
+�
+×
+�
+(K + 1)
+�
+log
+4 max
+�
+1
+2L∇h , min
+�
+Dq
+ǫ ,
+4
+αL∇h
+�� �
+δ + 2α−1(H∗ − Hlow + ǫDq/4 + L∇hD2
+p)
+�
+[(3L∇h + ǫ/(2Dq))2/ min{L∇h, ǫ/(2Dq)} + 3L∇h + ǫ/(2Dq)]−2 ǫ2
+0
+�
++
++ K + 1 + 2K log(K + 1)
+�
+.
+(27)
+Then Algorithm 2 terminates and outputs an ǫ-stationary point (xǫ, yǫ) of (6) in at most K + 1 outer
+iterations that satisfies
+max
+y
+H(xǫ, y) ≤ max
+y
+H(ˆx0, y) + ǫDq/4 + 2ǫ2
+0
+�
+L−1
+∇h + 4D2
+qL∇hǫ−2�
+.
+(28)
+Moreover, the total number of evaluations of ∇h and proximal operator of p and q performed in Algo-
+rithm 2 is no more than N, respectively.
+Remark 3. Since ǫ0 ∈ (0, ǫ/2], one can observe from Theorem 2 that α = O(ǫ1/2), δ = O(ǫ−1/2),
+K = O(ǫ−2), and N = O(ǫ−5/2 log(ǫ−1
+0 ǫ−1). Consequently, Algorithm 2 enjoys an operation complexity
+of O(ǫ−5/2 log(ǫ−1
+0 ǫ−1)), measured by the amount of evaluations of ∇h and proximal operator of p and
+q, for finding an ǫ-stationary point of the nonconvex-concave minimax problem (6).
+3
+Unconstrained bilevel optimization
+In this section, we consider an unconstrained bilevel optimization problem4
+f ∗ = min
+f(x, y)
+s.t.
+y ∈ Argmin
+z
+˜f(x, z).
+(29)
+Assume that problem (29) has at least one optimal solution. In addition, f and ˜f satisfy the following
+assumptions.
+Assumption 3.
+(i) f(x, y) = f1(x, y)+f2(x) and ˜f(x, y) = ˜f1(x, y)+ ˜f2(y) are continuous on X ×Y,
+where f2 : Rn → R ∪ {∞} and ˜f2 : Rm → R ∪ {∞} are proper closed convex functions, ˜f1(x, ·) is
+convex for any given x ∈ X, and f1, ˜f1 are respectively L∇f1- and L∇ ˜f1-smooth on X × Y with
+X := dom f2 and Y := dom ˜f2.
+(ii) The proximal operator associated with f2 and ˜f2 can be exactly evaluated.
+(iii) The sets X and Y (namely, dom f2 and dom ˜f2) are compact.
+For notational convenience, we define
+Dx := max{∥u − v∥
+��u, v ∈ X},
+Dy := max{∥u − v∥
+��u, v ∈ Y},
+(30)
+˜fhi := max{ ˜f(x, y)|(x, y) ∈ X × Y},
+˜flow := min{ ˜f(x, y)|(x, y) ∈ X × Y},
+(31)
+flow := min{f(x, y)|(x, y) ∈ X × Y}.
+(32)
+4For convenience, problem (29) is referred to as an unconstrained bilevel optimization problem since its lower level part
+does not have an explicit constraint. Strictly speaking, it can be a constrained bilevel optimization problem. For example,
+when part of f and/or ˜f is the indicator function of a closed convex set, (29) is essentially a constrained bilevel optimization
+problem.
+7
+
+By Assumption 3, one can observe that Dx, Dy, ˜fhi, ˜flow and flow are finite.
+The goal of this subsection is to propose penalty methods for solving problem for solving (29). To
+this end, we observe that problem (29) can be viewed as
+min
+x,y {f(x, y)| ˜f(x, y) ≤ min
+z
+˜f(x, z)}.
+(33)
+Notice that ˜f(x, y) − minz ˜f(x, z) ≥ 0 for all x, y. Consequently, a natural penalty problem associated
+with (33) is
+min
+x,y f(x, y) + ρ( ˜f(x, y) − min
+z
+˜f(x, z)),
+(34)
+where ρ > 0 is a penalty parameter. We further observe that (34) is equivalent to the minimax problem
+min
+x,y max
+z
+Pρ(x, y, z),
+where
+Pρ(x, y, z) := f(x, y) + ρ( ˜f(x, y) − ˜f(x, z)).
+(35)
+In view of Assumption 3(i), Pρ can be rewritten as
+Pρ(x, y, z) =
+�
+f1(x, y) + ρ ˜f1(x, y) − ρ ˜f1(x, z)
+�
++
+�
+f2(x) + ρ ˜f2(y) − ρ ˜f2(z)
+�
+.
+(36)
+By this and Assumption 3, one can observe that Pρ enjoys the following nice properties.
+• Pρ is the sum of smooth function f1(x, y)+ ρ ˜f1(x, y)− ρ ˜f1(x, z) with Lipschitz continuous gradient
+and possibly nonsmooth function f2(x)+ρ ˜f2(y)−ρ ˜f2(z) with exactly computable proximal operator.
+• Pρ is nonconvex in (x, y) but concave in z.
+Thanks to the nice structure of Pρ, an approximate stationary point of the minimax problem (35) can
+be found by Algorithm 2 proposed in Subsection 2.2.
+Based on the above observations, we are now ready to propose penalty methods for the unconstrained
+bilevel optimization problem (29) by solving either a sequence of or a single minimax problem in the
+form of (35). In particular, we first propose an ideal penalty method for (29) by solving a sequence of
+minimax problems (see Algorithm 3). Then we propose a practical penalty method for (29) by finding
+an approximate stationary point of a single minimax problem (see Algorithm 4).
+Algorithm 3 An ideal penalty method for problem (29)
+Input: positive sequences {ρk} and {ǫk} with limk→∞(ρk, ǫk) = (∞, 0).
+1: for k = 0, 1, 2, . . . do
+2:
+Find an ǫk-optimal solution (xk, yk, zk) of problem (35) with ρ = ρk.
+3: end for
+The following theorem states a convergence result of Algorithm 3, whose proof is deferred to Section
+5.3.
+Theorem 3 (Convergence of Algorithm 3). Suppose that Assumption 3 holds and that {(xk, yk, zk)}
+is generated by Algorithm 3. Then any accumulation point of {(xk, yk)} is an optimal solution of problem
+(29).
+Notice that (35) is a nonconvex-concave minimax problem. It is typically hard to find an ǫ-optimal
+solution of (35) for an arbitrary ǫ > 0. Consequently, Algorithm 3 is not implementable in general. We
+next propose a practical penalty method for problem (29) by finding an approximate stationary point of
+a single minimax problem (35) with a suitable choice of ρ.
+Algorithm 4 A practical penalty method for problem (29)
+Input: ε ∈ (0, 1/4], ρ = ε−1, (x0, y0) ∈ X × Y with ˜f(x0, y0) ≤ miny ˜f(x0, y) + ε.
+1: Call Algorithm 2 with ǫ ← ε, ǫ0 ← ε3/2, ˆx0 ← (x0, y0), ˆy0 ← y0, and L∇h ← L∇f1 + 2ε−1L∇ ˜
+f1 to
+find an ǫ-stationary point (xǫ, yǫ, zǫ) of problem (35) with ρ = ε−1.
+2: Output: (xǫ, yǫ).
+8
+
+Remark 4. (i) The initial point (x0, y0) of Algorithm 4 can be found by an additional procedure. Indeed,
+one can first choose any x0 ∈ X and then apply accelerated proximal gradient method [38] to the problem
+miny ˜f(x0, y) for finding y0 ∈ Y such that ˜f(x0, y0) ≤ miny ˜f(x0, y) + ε; (ii) As seen from Theorem 2,
+an ǫ-stationary point of (35) can be successfully found in step 1 of Algorithm 4 by applying Algorithm 2
+to (35); (iii) For the sake of simplicity, a single subproblem of the form (35) with static penalty and
+tolerance parameters is solved in Algorithm 4. Nevertheless, Algorithm 4 can be modified into a perhaps
+practically more efficient algorithm by solving a sequence of subproblems of the form (35) with dynamic
+penalty and tolerance parameters instead.
+In order to characterize the approximate solution found by Algorithm 4, we next introduce a termi-
+nology called an ε-KKT solution of problem (29).
+Recall that problem (29) can be viewed as problem (33). In the spirit of classical constrained opti-
+mization, one would naturally be interested in a KKT solution (x, y) of (33) or equivalently (29), namely,
+(x, y) satisfies ˜f(x, y) ≤ minz ˜f(x, z) and moreover (x, y) is a stationary point of the problem
+min
+x′,y′ f(x′, y′) + ρ
+� ˜f(x′, y′) − min
+z′
+˜f(x′, z′)
+�
+(37)
+for some ρ ≥ 0. Yet, due to the sophisticated problem structure, characterizing a stationary point of (37)
+is generally difficult. On another hand, notice that problem (37) is equivalent to the minimax problem
+min
+x′,y′ max
+z′
+f(x′, y′) + ρ( ˜f(x′, y′) − ˜f(x′, z′)),
+whose stationary point (x, y, z) according to Definition 2 satisfies
+0 ∈ ∂f(x, y) + ρ∂ ˜f(x, y) − (ρ∇x ˜f(x, z); 0),
+0 ∈ ρ∂z ˜f(x, z).
+(38)
+Based on this observation, we are instead interested in a (weak) KKT solution of problem (29) and its
+inexact counterpart that are defined below.
+Definition 3. The pair (x, y) is said to be a KKT solution of problem (29) if there exists (z, ρ) ∈ Rm×R+
+such that (38) and ˜f(x, y) ≤ minz′ ˜f(x, z′) hold. In addition, for any ε > 0, (x, y) is said to be an ε-KKT
+solution of problem (29) if there exists (z, ρ) ∈ Rm × R+ such that
+dist
+�
+0, ∂f(x, y) + ρ∂ ˜f(x, y) − (ρ∇x ˜f(x, z); 0)
+�
+≤ ε,
+dist
+�
+0, ρ∂z ˜f(x, z)
+�
+≤ ε,
+˜f(x, y) − min
+z′
+˜f(x, z′) ≤ ε.
+We are now ready to present a theorem regarding operation complexity of Algorithm 4, measured by
+the amount of evaluations of ∇f1, ∇ ˜f1 and proximal operator of f2 and ˜f2, for finding an O(ε)-KKT
+solution of (29), whose proof is deferred to Subsection 5.3.
+Theorem 4 (Complexity of Algorithm 4). Suppose that Assumption 3 holds. Let f ∗, f, ˜f, Dx, Dy,
+˜fhi, ˜flow and flow be defined in (29), (30), (31) and (32), L∇f1 and L∇ ˜
+f1 be given in Assumption 3, ε,
+ρ, x0, y0 and zǫ be given in Algorithm 4, and
+�L = L∇f1 + 2ε−1L∇ ˜
+f1, ˆα = min
+�
+1,
+�
+4ε/(Dy�L)
+�
+,
+(39)
+ˆδ = (2 + ˆα−1)(D2
+x + D2
+y)�L + max
+�
+ε/Dy, ˆα�L/4
+�
+D2
+y,
+�C =
+4 max
+�
+1
+2�L, min
+�
+Dy
+ε ,
+4
+ˆα�L
+�� �
+ˆδ + 2ˆα−1(f ∗ − flow + ε−1( ˜fhi − ˜flow) + εDy/4 + �L(D2
+x + D2
+y))
+�
+�
+(3�L + ε/(2Dy))2/ min{�L, ε/(2Dy)} + 3�L + ε/(2Dy)
+�−2
+ε3
+,
+�K =
+�
+16(1 + f(x0, y0) − flow + εDy/4)�Lε−2 + 32(1 + 4D2
+y�L2ε−2)ε − 1
+�
++ ,
+�
+N =
+��
+96
+√
+2(1 + (24�L + 4ε/Dy)�L−1)
+�
++ 2
+�
+max
+�
+2,
+�
+Dy�Lε−1
+�
+× (( �
+K + 1)(log �C)+ + �K + 1 + 2 �K log( �K + 1)).
+9
+
+Then Algorithm 4 outputs an approximate solution (xǫ, yǫ) of (29) satisfying
+dist
+�
+0, ∂f(xǫ, yǫ) + ρ∂ ˜f(xǫ, yǫ) − (ρ∇x ˜f(xǫ, zǫ); 0)
+�
+≤ ε,
+dist
+�
+0, ρ∂ ˜f(xǫ, zǫ)
+�
+≤ ε,
+(40)
+˜f(xǫ, yǫ) ≤ min
+z
+˜f(xǫ, z) + ε
+�
+1 + f(x0, y0) − flow + 2ε3(�L−1 + 4D2
+y�Lε−2) + Dyε/4
+�
+,
+(41)
+after at most �
+N evaluations of ∇f1, ∇ ˜f1 and proximal operator of f2 and ˜f2, respectively.
+Remark 5. One can observe from Theorem 4 that �L = O(ε−1), ˆα = O(ε), ˆδ = O(ε−2), �C = O(ε−11),
+�K = O(ε−3), and �
+N = O(ε−4 log ε−1). Consequently, Algorithm 4 enjoys an operation complexity of
+O(ε−4 log ε−1), measured by the amount of evaluations of ∇f1, ∇ ˜f1 and proximal operator of f2 and ˜f2,
+for finding an O(ε)-KKT solution (xǫ, yǫ) of (29) satisfying
+dist
+�
+0, ∂f(xǫ, yǫ) + ρ∂ ˜f(xǫ, yǫ) − (ρ∇x ˜f(xǫ, zǫ); 0)
+�
+≤ ε,
+dist
+�
+0, ρ∂ ˜f(xǫ, zǫ)
+�
+≤ ε,
+˜f(xǫ, yǫ) − min
+z
+˜f(xǫ, z) = O(ε),
+where zǫ is given in Algorithm 4 and ρ = ε−1.
+4
+Constrained bilevel optimization
+In this section, we consider a constrained bilevel optimization problem5
+f ∗ = min
+f(x, y)
+s.t.
+y ∈ Argmin
+z
+{ ˜f(x, z)|˜g(x, z) ≤ 0},
+(42)
+where f and ˜f satisfy Assumption 3. Recall from Assumption 3 that X = dom f2 and Y = dom ˜f2. We
+now make some additional assumptions for problem (42).
+Assumption 4.
+(i) f and ˜f are Lf- and L ˜
+f-Lipschitz continuous on X × Y, respectively.
+(ii) ˜g : Rn × Rm → Rl is L∇˜g-smooth and L˜g-Lipschitz continuous on X × Y.
+(iii) ˜gi(x, ·) is convex and there exists ˆzx ∈ Y for each x ∈ X such that ˜gi(x, ˆzx) < 0 for all i = 1, 2, . . ., l
+and G := min{−˜gi(x, ˆzx)|x ∈ X, i = 1, . . . , l} > 0.6
+For notational convenience, we define
+˜f ∗(x) := min
+z { ˜f(x, z)|˜g(x, z) ≤ 0},
+(43)
+˜f ∗
+hi := sup{ ˜f ∗(x)|x ∈ X},
+(44)
+˜ghi := max{∥˜g(x, y)∥
+��(x, y) ∈ X × Y},
+(45)
+It then follows from Assumption 4(ii) that
+∥∇˜g(x, y)∥ ≤ L˜g
+∀(x, y) ∈ X × Y.
+(46)
+In addition, by Assumptions 3 and 4 and the compactness of X and Y, one can observe that ˜ghi and G
+are finite. Besides, as will be shown in Lemma 6(ii), ˜f ∗
+hi is finite.
+The goal of this subsection is to propose penalty methods for solving problem (42). To this end, let
+us introduce a penalty function for the lower level optimization problem y ∈ Argmin
+z
+{ ˜f(x, z)|˜g(x, z) ≤ 0}
+of (42), which is given by
+�Pµ(x, z) = ˜f(x, z) + µ ∥[˜g(x, z)]+∥2
+(47)
+5For convenience, problem (42) is referred to as a constrained bilevel optimization problem since its lower level part has
+at least one explicit constraint.
+6The latter part of this assumption can be weakened to the one that the pointwise Slater’s condition holds for the lower
+level part of (42), that is, there exists ˆzx ∈ Y such that ˜g(x, ˆzx) < 0 for each x ∈ X. Indeed, if G > 0, Assumption 4(iii)
+clearly holds. Otherwise, one can solve the perturbed counterpart of (42) with ˜g(x, z) being replaced by ˜g(x, z) − ǫ for
+some suitable ǫ > 0 instead, which satisfies Assumption 4(iii).
+10
+
+for a penalty parameter µ > 0. Observe that problem (42) can be approximately solved as the uncon-
+strained bilevel optimization problem
+f ∗
+µ = min
+x,y
+�
+f(x, y)|y ∈ Argmin
+z
+�Pµ(x, z)
+�
+.
+(48)
+Further, by the study in Section 3, problem (48) can be approximately solved as the penalty problem
+min
+x,y f(x, y) + ρ
+�
+�Pµ(x, y) − min
+z
+�Pµ(x, z)
+�
+(49)
+for some suitable ρ > 0. One can also observe that problem (49) is equivalent to the minimax problem
+min
+x,y max
+z
+Pρ,µ(x, y, z),
+where
+Pρ,µ(x, y, z) := f(x, y) + ρ( �Pµ(x, y) − �Pµ(x, z)).
+(50)
+In view of (47), (50) and Assumption 3(i), Pρ,µ can be rewritten as
+Pρ,µ(x, y, z) =
+�
+f1(x, y) + ρ ˜f1(x, y) + ρµ ∥[˜g(x, y)]+∥2 − ρ ˜f1(x, z) − ρµ ∥[˜g(x, z)]+∥2 �
++
+�
+f2(x) + ρ ˜f2(y) − ρ ˜f2(z)
+�
+.
+(51)
+By this and Assumptions 3 and 4, one can observe that Pρ,µ enjoys the following nice properties.
+• Pρ,µ is the sum of smooth function f1(x, y)+ρ ˜f1(x, y)+ρµ ∥[˜g(x, y)]+∥2−ρ ˜f1(x, z)−ρµ ∥[˜g(x, z)]+∥2
+with Lipschitz continuous gradient and possibly nonsmooth function f2(x) + ρ ˜f2(y) − ρ ˜f2(z) with
+exactly computable proximal operator;
+• Pρ,µ is nonconvex in (x, y) but concave in z.
+Due to the nice structure of Pρ,µ, an approximate stationary point of the minimax problem (50) can be
+found by Algorithm 2 proposed in Subsection 2.2.
+Based on the above observations, we are now ready to propose penalty methods for the constrained
+bilevel optimization problem (42) by solving a sequence of or a single minimax problem of the form (50).
+In particular, we first propose an ideal penalty method for (42) by solving a sequence of minimax problems
+(see Algorithm 5). Then we propose a practical penalty method for (42) by finding an approximate
+stationary point of a single minimax problem (see Algorithm 6).
+Algorithm 5 An ideal penalty method for problem (42)
+Input: positive sequences {ρk}, {µk} and {ǫk} with limk→∞(ρk, µk, ǫk) = (∞, ∞, 0).
+1: for k = 0, 1, 2, . . . do
+2:
+Find an ǫk-optimal solution (xk, yk, zk) of problem (50) with (ρ, µ) = (ρk, µk).
+3: end for
+To study convergence of Algorithm 5, we make the following error bound assumption on the solution
+set of the lower level optimization problem of (42). This type of error bounds has been considered in the
+context of set-value mappings in the literature (e.g., see [14]).
+Assumption 5. There exists a non-decreasing function ω : R+ → R+ with limθ↓0 ω(θ) = 0 and ¯θ > 0
+such that dist(z, Sθ(x)) ≤ ω(θ) for all x ∈ X, z ∈ S0(x) and θ ∈ [0, ¯θ], where
+Sθ(x) := Argmin
+z
+{ ˜f(x, z) : ∥[˜g(x, z)]+∥ ≤ θ}.
+We are now ready to state a convergence result of Algorithm 5, whose proof is deferred to Section
+5.4.
+Theorem 5 (Convergence of Algorithm 5). Suppose that Assumptions 3-5 hold and that {(xk, yk, zk)}
+is generated by Algorithm 5. Then any accumulation point of {(xk, yk)} is an optimal solution of problem
+(42).
+Notice that (50) is a nonconvex-concave minimax problem. It is generally hard to find an ǫ-optimal
+solution of (50) for an arbitrary ǫ > 0. As a result, Algorithm 5 is generally not implementable. We next
+propose a practical penalty method for problem (42) by finding an approximate stationary point of (50)
+with a suitable choice of ρ and µ.
+11
+
+Algorithm 6 A practical penalty method for problem (42)
+Input: ε ∈ (0, 1/4], ρ = ε−1, µ = ε−2, (x0, y0) ∈ X × Y with �Pµ(x0, y0) ≤ miny �Pµ(x0, y) + ε.
+1: Call Algorithm 2 with ǫ ← ε, ǫ0 ← ε5/2, ˆx0 ← (x0, y0), ˆy0 ← y0, and L∇h ← L∇f1 + 2ρL∇ ˜f1 +
+4ρµ(˜ghiL∇˜g +L2
+˜g) to find an ǫ-stationary point (xǫ, yǫ, zǫ) of problem (50) with ρ = ε−1 and µ = ε−2.
+2: Output: (xǫ, yǫ).
+Remark 6. (i) The initial point (x0, y0) of Algorithm 6 can be found by the similar procedure as described
+in Remark 4 with ˜f being replaced by �Pµ; (ii) As seen from Theorem 2, an ǫ-stationary point of (50)
+can be successfully found in step 1 of Algorithm 6 by applying Algorithm 2 to (50); (iii) For the sake of
+simplicity, a single subproblem of the form (50) with static penalty and tolerance parameters is solved in
+Algorithm 6. Nevertheless, Algorithm 6 can be modified into a perhaps practically more efficient algorithm
+by solving a sequence of subproblems of the form (50) with dynamic penalty and tolerance parameters
+instead.
+In order to characterize the approximate solution found by Algorithm 6, we next introduce a termi-
+nology called an ε-KKT solution of problem (42).
+By the definition of ˜f ∗ in (43), problem (42) can be viewed as
+min
+x,y {f(x, y)| ˜f(x, y) ≤ ˜f ∗(x), ˜g(x, y) ≤ 0}.
+(52)
+Its associated Lagrangian function is given by
+L(x, y, ρ, λ) = f(x, y) + ρ( ˜f(x, y) − ˜f ∗(x)) + ⟨λ, ˜g(x, y)⟩.
+(53)
+In the spirit of classical constrained optimization, one would naturally be interested in a KKT solution
+(x, y) of (52) or equivalently (42), namely, (x, y) satisfies
+˜f(x, y) ≤ ˜f ∗(x),
+˜g(x, y) ≤ 0,
+ρ( ˜f(x, y) − ˜f ∗(x)) = 0,
+⟨λ, ˜g(x, y)⟩ = 0,
+(54)
+and moreover (x, y) is a stationary point of the problem
+min
+x′,y′ L(x′, y′, ρ, λ)
+(55)
+for some ρ ≥ 0 and λ ∈ Rl
++. Yet, due to the sophisticated problem structure, characterizing a stationary
+point of (55) is generally difficult. On another hand, notice from Lemma 6 and (53) that problem (55)
+is equivalent to the minimax problem
+min
+x′,y′,˜λ′ max
+z′
+�
+f(x′, y′) + ρ
+� ˜f(x′, y′) − ˜f(x′, z′) − ⟨˜λ′, ˜g(x′, z′)⟩
+�
++ ⟨λ, ˜g(x′, y′)⟩ + IRl
++(˜λ′)
+�
+,
+whose stationary point (x, y, ˜λ, z) according to Definition 2 satisfies
+0 ∈ ∂f(x, y) + ρ∂ ˜f(x, y) − ρ(∇x ˜f(x, z) + ∇x˜g(x, z)˜λ; 0) + ∇˜g(x, y)λ,
+(56)
+0 ∈ ρ(∂z ˜f(x, z) + ∇z˜g(x, z)˜λ),
+(57)
+˜λ ∈ Rl
++,
+˜g(x, z) ≤ 0,
+⟨˜λ, ˜g(x, z)⟩ = 0.
+(58)
+Based on this observation and also the fact that (54) is equivalent to
+˜f(x, y) = ˜f ∗(x),
+˜g(x, y) ≤ 0,
+⟨λ, ˜g(x, y)⟩ = 0,
+(59)
+we are instead interested in a (weak) KKT solution of problem (42) and its inexact counterpart that are
+defined below.
+Definition 4. The pair (x, y) is said to be a KKT solution of problem (42) if there exists (z, ρ, λ, ˜λ) ∈
+Rm × R+ × Rl
++ × Rl
++ such that (56)-(59) hold. In addition, for any ε > 0, (x, y) is said to be an ε-KKT
+solution of problem (42) if there exists (z, ρ, λ, ˜λ) ∈ Rm × R+ × Rl
++ × Rl
++ such that
+dist
+�
+0, ∂f(x, y) + ρ∂ ˜f(x, y) − ρ(∇x ˜f(x, z) + ∇x˜g(x, z)˜λ; 0) + ∇˜g(x, y)λ
+�
+≤ ε,
+dist
+�
+0, ρ(∂z ˜f(x, z) + ∇z˜g(x, z)˜λ)
+�
+≤ ε,
+∥[˜g(x, z)]+∥ ≤ ε,
+|⟨˜λ, ˜g(x, z)⟩| ≤ ε,
+| ˜f(x, y) − ˜f ∗(x)| ≤ ε,
+∥[˜g(x, y)]+∥ ≤ ε,
+|⟨λ, ˜g(x, y)⟩| ≤ ε,
+where ˜f ∗ is defined in (43).
+12
+
+We are now ready to present an operation complexity of Algorithm 6, measured by the amount of
+evaluations of ∇f1, ∇ ˜f1, ∇˜g and proximal operator of f2 and ˜f2, for finding an O(ε)-KKT solution of
+(42), whose proof is deferred to Subsection 5.4.
+Theorem 6 (Complexity of Algorithm 6). Suppose that Assumptions 3 and 4 hold. Let f ∗, f, ˜f,
+˜g, Dx, Dy, ˜fhi, ˜flow, flow, ˜f ∗, ˜f ∗
+hi, and ˜ghi be defined in (29), (30), (31), (32), (43), (44) and (45),
+L∇f1, L∇ ˜
+f1, L ˜
+f, L∇˜g, L˜g and G be given in Assumptions 3 and 4, ε, ρ, µ, x0, y0 and zǫ be given in
+Algorithm 6, and
+˜λ = 2ε−1[˜g(xǫ, zǫ)]+,
+ˆλ = 2ε−3[˜g(xǫ, yǫ)]+,
+(60)
+�L = L∇f1 + 2ε−1L∇ ˜
+f1 + 4ε−3(˜ghiL∇˜g + L2
+˜g),
+(61)
+˜α = min
+�
+1,
+�
+4ε/(Dy�L)
+�
+, ˜δ = (2 + ˜α−1)(D2
+x + D2
+y)�L + max
+�
+ε/Dy, ˜α�L/4
+�
+D2
+y,
+�C =
+4 max{1/(2�L), min{Dyε−1, 4/(˜α�L)}}
+[(3�L + ε/(2Dy))2/ min{�L, ε/(2Dy)} + 3�L + ε/(2Dy)]−2ε5
+×
+�
+˜δ + 2˜α−1[f ∗ − flow + 2ε−1( ˜fhi − ˜flow) + ε−3˜g2
+hi + εDy/4 + �L(D2
+x + D2
+y)]
+�
+,
+�K =
+�
+32(1 + f(x0, y0) − flow + εDy/4)�Lε−2 + 32ε3 �
+1 + 4D2
+y�L2ε−2�
+− 1
+�
++ ,
+�
+N =
+��
+96
+√
+2
+�
+1 + (24�L + 4ε/Dy)�L−1��
++ 2
+�
+max
+�
+2,
+�
+Dy�Lε−1
+�
+× [( �K + 1)(log �C)+ + �K + 1 + 2 �K log( �K + 1)].
+Then Algorithm 6 outputs an approximate solution (xǫ, yǫ) of (42) satisfying
+dist
+�
+∂f(xǫ, yǫ) + ρ∂ ˜f(xǫ, yǫ) − ρ(∇x ˜f(xǫ, zǫ) + ∇x˜g(xǫ, zǫ)˜λ; 0) + ∇˜g(xǫ, yǫ)ˆλ
+�
+≤ ε,
+(62)
+dist
+�
+0, ρ(∂z ˜f(xǫ, zǫ) + ∇z˜g(xǫ, zǫ)˜λ)
+�
+≤ ε,
+(63)
+∥[˜g(xǫ, zǫ)]+∥ ≤ ε2G−1Dy(ε2 + L ˜
+f)/2,
+(64)
+|⟨˜λ, ˜g(xǫ, zǫ)⟩| ≤ ε2G−2D2
+y(ρ−1ǫ + L ˜
+f)2/2,
+(65)
+| ˜f(xǫ, yǫ) − ˜f ∗(xǫ)| ≤ max
+�
+ε
+�
+1 + f(x0, y0) − flow + 2ε5(�L−1 + 4D2
+y�Lε−2) + Dyε/4
+�
+,
+ε2G−2D2
+yL ˜
+f(ε2 + εLf + L ˜
+f)/2
+�
+,
+(66)
+∥[˜g(xǫ, yǫ)]+∥ ≤ ε2G−1Dy(ε2 + εLf + L ˜
+f)/2,
+(67)
+|⟨ˆλ, ˜g(xǫ, yǫ)⟩| ≤ εG−2D2
+y(ε2 + εLf + L ˜
+f)2/2,
+(68)
+after at most �
+N evaluations of ∇f1, ∇ ˜f1, ∇˜g and proximal operator of f2 and ˜f2, respectively.
+Remark 7. One can observe from Theorem 6 that �L = O(ε−3), ˜α = O(ε2), ˜δ = O(ε−5), �C = O(ε−23),
+�K = O(ε−5), and �
+N = O(ε−7 log ε−1). Consequently, Algorithm 6 enjoys an operation complexity of
+O(ε−7 log ε−1), measured by the amount of evaluations of ∇f1, ∇ ˜f1, ∇˜g and proximal operator of f2
+and ˜f2, for finding an O(ε)-KKT solution (xǫ, yǫ) of (42) satisfying
+dist
+�
+0, ∂f(xǫ, yǫ) + ρ∂ ˜f(xǫ, yǫ) − ρ(∇x ˜f(xǫ, zǫ) + ∇x˜g(xǫ, zǫ)˜λ; 0) + ∇˜g(xǫ, yǫ)ˆλ
+�
+≤ ε,
+dist
+�
+0, ρ(∂z ˜f(xǫ, zǫ) + ∇z˜g(xǫ, zǫ)˜λ)
+�
+≤ ε,
+∥[˜g(xǫ, zǫ)]+∥ = O(ε2),
+|⟨˜λ, ˜g(xǫ, zǫ)⟩| = O(ε2),
+| ˜f(xǫ, yǫ) − ˜f ∗(xǫ)| = O(ε),
+∥[˜g(xǫ, yǫ)]+∥ = O(ε2),
+|⟨ˆλ, ˜g(xǫ, yǫ)⟩| = O(ε),
+where ˜f ∗ is defined in (43), ˆλ, ˜λ ∈ Rl
++ are defined in (60), zǫ is given in Algorithm 6 and ρ = ε−1.
+5
+Proof of the main results
+In this section we provide a proof of our main results presented in Sections 2, 3 and 4, which are
+particularly Theorems 1-6.
+13
+
+5.1
+Proof of the main results in Subsection 2.1
+In this subsection we prove Theorem 1. Before proceeding, we establish a lemma below.
+Lemma 1. Suppose that Assumptions 1 and 2 hold. Let ¯H∗, ¯Hlow, ϑ0 and ¯δ be defined in (8), (10),
+(14) and (16), and ¯α be given in Algorithm 1. Then we have
+ϑ0 ≤ ¯δ + 2¯α−1 � ¯H∗ − ¯Hlow
+�
+.
+(69)
+Proof. By (8), (10), (11) and (12), one has
+G(¯z0, ¯y0)
+(12)
+=
+sup
+x
+�
+⟨x, ¯z0⟩ − p(x) − ˆh(x, ¯y0) + q(¯y0)
+�
+(11)
+=
+max
+x∈dom p
+�
+⟨x, ¯z0⟩ − p(x) − ¯h(x, ¯y0) + σx
+2 ∥x∥2 − σy
+2 ∥¯y0∥2 + q(¯y0)
+�
+(8)(10)
+≤
+max
+x∈dom p
+�
+⟨x, ¯z0⟩ + σx
+2 ∥x∥2�
+− σy
+2 ∥¯y0∥2 − ¯Hlow
+=
+max
+x∈dom p
+σx
+2 ∥x + σ−1
+x ¯z0∥2 − σ−1
+x
+2 ∥¯z0∥2 − σy
+2 ∥¯y0∥2 − ¯Hlow
+≤ σxD2
+p
+2
+− σ−1
+x
+2 ∥¯z0∥2 − σy
+2 ∥¯y0∥2 − ¯Hlow,
+(70)
+where the last inequality follows from (9) and the fact that z0 ∈ −σxdom p.
+Recall that (x∗, y∗) is the optimal solution of (8) and z∗ = −σxx∗. It follows from (8), (11) and (12)
+that
+G(z∗, y∗)
+(12)
+=
+sup
+x
+�
+⟨x, z∗⟩ − p(x) − ˆh(x, y∗) + q(y∗)
+�
+≥ ⟨x∗, z∗⟩ − p(x∗) − ˆh(x∗, y∗) + q(y∗)
+(11)
+= ⟨x∗, z∗⟩ + σx
+2 ∥x∗∥2 − σy
+2 ∥y∗∥2 − p(x∗) − ¯h(x∗, y∗) + q(y∗)
+=
+− σ−1
+x
+2 ∥z∗∥2 − σy
+2 ∥y∗∥2 − ¯H∗,
+where the last equality follows from (8), the definition of (x∗, y∗), and z∗ = −σxx∗. This together with
+(13) and (70) implies that
+P(¯z0, ¯y0) − P(z∗, y∗) = σ−1
+x
+2 ∥¯z0∥2 + σy
+2 ∥¯y0∥2 + G(¯z0, ¯y0) − σ−1
+x
+2 ∥z∗∥2 − σy
+2 ∥y∗∥2 − G(z∗, y∗)
+≤ σxD2
+p/2 − ¯Hlow + ¯H∗.
+Notice from Algorithm 1 that z0 = z0
+f = ¯z0 ∈ −σxdom p and y0 = y0
+f = ¯y0 ∈ dom q.
+By these,
+z∗ = −σxx∗, (9), (14), and the above inequality, one has
+ϑ0
+(14)
+= η−1
+z ∥¯z0 − z∗∥2 + η−1
+y ∥¯y0 − y∗∥2 + 2¯α−1(P(¯z0, ¯y0) − P(z∗, y∗))
+≤ η−1
+z σ2
+xD2
+p + η−1
+y D2
+q + 2¯α−1 �
+σxD2
+p/2 − ¯Hlow + ¯H∗�
+= η−1
+z σ2
+xD2
+p + ¯α−1σxD2
+p + η−1
+y D2
+q + 2¯α−1 � ¯H∗ − ¯Hlow
+�
+.
+Hence, the conclusion follows from this, (16), ηz = σx/2 and ηy = min {1/(2σy), 4/(¯ασx)}.
+We are now ready to prove Theorem 1.
+Proof of Theorem 1. Suppose for contradiction that Algorithm 1 runs for more than ¯K outer itera-
+tions, where ¯K is given in (17). By this and Algorithm 1, one can assert that (15) does not hold for
+k = ¯K − 1. On the other hand, by (17) and [29, Theorem 3], one has
+∥(x
+¯
+K, y
+¯
+K) − (x∗, y∗)∥ ≤ (ˆζ−1 + L∇¯h)−1τ/2,
+(71)
+where (x∗, y∗) is the optimal solution of problem (8) and ˆζ is an input of Algorithm 1. Notice from
+Algorithm 1 that (ˆx ¯
+K, ˆy ¯
+K) results from the forward-backward splitting (FBS) step applied to the strongly
+monotone inclusion problem 0 ∈ (∇x¯h(x, y), −∇y¯h(x, y)) + (∂p(x), ∂q(y)) at the point (x ¯
+K, y ¯
+K). It then
+14
+
+follows from this, ˆζ = min{σx, σy}/L2
+∇¯h (see Algorithm 1), and the contraction property of FBS [5,
+Corollary 2.5] that ∥(ˆx ¯
+K, ˆy ¯
+K) − (x∗, y∗)∥ ≤ ∥(x ¯
+K, y ¯
+K) − (x∗, y∗)∥. Using this and (71), we have
+∥ˆζ−1(x
+¯
+K − ˆx
+¯
+K, ˆy
+¯
+K − y
+¯
+K) − (∇¯h(x
+¯
+K, y
+¯
+K) − ∇¯h(ˆx
+¯
+K, ˆy
+¯
+K))∥
+≤ ˆζ−1∥(x
+¯
+K, y
+¯
+K) − (ˆx
+¯
+K, ˆy
+¯
+K)∥ + ∥∇¯h(x
+¯
+K, y
+¯
+K) − ∇¯h(ˆx
+¯
+K, ˆy
+¯
+K)∥
+≤ (ˆζ−1 + L∇¯h)∥(x
+¯
+K, y
+¯
+K) − (ˆx
+¯
+K, ˆy
+¯
+K)∥
+≤ (ˆζ−1 + L∇¯h)(∥(x
+¯
+K, y
+¯
+K) − (x∗, y∗)∥ + ∥(ˆx
+¯
+K, ˆy
+¯
+K) − (x∗, y∗)∥)
+≤ 2(ˆζ−1 + L∇¯h)∥(x
+¯
+K, y
+¯
+K) − (x∗, y∗)∥
+(71)
+≤ τ,
+where the second inequality uses the fact that ¯h is L∇¯h-smooth on dom p × dom q. It follows that (15)
+holds for k = ¯K − 1, which contradicts the above assertion. Hence, Algorithm 1 must terminate in at
+most ¯K outer iterations.
+We next show that the output of Algorithm 1 is a τ-stationary point of (8). To this end, suppose
+that Algorithm 1 terminates at some iteration k at which (15) is satisfied. Then by (4) and the definition
+of ˆxk+1 and ˆyk+1 (see steps 23 and 24 of Algorithm 1), one has
+0 ∈ ˆζ∂p(ˆxk+1) + ˆxk+1 − xk+1 + ˆζ∇x¯h(xk+1, yk+1),
+0 ∈ ˆζ∂q(ˆyk+1) + ˆyk+1 − yk+1 − ˆζ∇y¯h(xk+1, yk+1),
+which yield
+ˆζ−1(xk+1 − ˆxk+1) − ∇x¯h(xk+1, yk+1) ∈ ∂p(ˆxk+1), ˆζ−1(yk+1 − ˆyk+1) + ∇y¯h(xk+1, yk+1) ∈ ∂q(ˆyk+1).
+These together with the definition of ¯H in (8) imply that
+∇x¯h(ˆxk+1, ˆyk+1) + ˆζ−1(xk+1 − ˆxk+1) − ∇x¯h(xk+1, yk+1) ∈ ∂x ¯H(ˆxk+1, ˆyk+1),
+∇y¯h(ˆxk+1, ˆyk+1) − ˆζ−1(yk+1 − ˆyk+1) − ∇y¯h(xk+1, yk+1) ∈ ∂y ¯H(ˆxk+1, ˆyk+1).
+Using these and (15), we obtain
+dist(0, ∂x ¯H(ˆxk+1, ˆyk+1))2 + dist(0, ∂y ¯H(ˆxk+1, ˆyk+1))2
+≤ ∥ˆζ−1(xk+1 − ˆxk+1) + ∇x¯h(ˆxk+1, ˆyk+1) − ∇x¯h(xk+1, yk+1)∥2
++ ∥ˆζ−1(ˆyk+1 − yk+1) + ∇y¯h(ˆxk+1, ˆyk+1) − ∇y¯h(xk+1, yk+1)∥2
+= ∥ˆζ−1(xk+1 − ˆxk+1, ˆyk+1 − yk+1) − (∇¯h(xk+1, yk+1) − ∇¯h(ˆxk+1, ˆyk+1))∥2 (15)
+≤ τ2,
+which implies that dist(0, ∂x ¯H(ˆxk+1, ˆyk+1)) ≤ τ and dist(0, ∂y ¯H(ˆxk+1, ˆyk+1)) ≤ τ. It then follows from
+these and Definition 2 that the output (ˆxk+1, ˆyk+1) of Algorithm 1 is a τ-stationary point of (8).
+Finally, we show that the total number of evaluations of ∇¯h and proximal operator of p and q
+performed in Algorithm 1 is no more than ¯N, respectively.
+Indeed, notice from Algorithm 1 that
+¯α = min
+�
+1,
+�
+8σy/σx
+�
+, which implies that 2/¯α = max{2,
+�
+σx/(2σy)} and ¯α ≤
+�
+8σy/σx. By these,
+one has
+max
+� 2
+¯α, ¯ασx
+4σy
+�
+≤ max
+�
+2,
+� σx
+2σy
+,
+�
+8σy
+σx
+σx
+4σy
+�
+= max
+�
+2,
+� σx
+2σy
+�
+.
+(72)
+In addition, by [29, Lemma 4], the number of inner iterations performed in each outer iteration of
+Algorithm 1 is at most
+T =
+�
+48
+√
+2
+�
+1 + 8L∇¯hσ−1
+x
+��
+− 1.
+Then one can observe that the number of evaluations of ∇¯h and proximal operator of p and q performed
+15
+
+in Algorithm 1 is at most
+(2T + 3) ¯K ≤
+��
+96
+√
+2
+�
+1 + 8L∇¯hσ−1
+x
+��
++ 2
+� �
+max
+� 2
+¯α, ¯ασx
+4σy
+�
+log 4 max{ηzσ−2
+x , ηy}ϑ0
+(ˆζ−1 + L∇¯h)−2τ 2
+�
++
+(72)
+≤
+��
+96
+√
+2
+�
+1 + 8L∇¯hσ−1
+x
+��
++ 2
+� �
+max
+�
+2,
+� σx
+2σy
+�
+log 4 max{ηzσ−2
+x , ηy}ϑ0
+(ˆζ−1 + L∇¯h)−2τ 2
+�
++
+≤
+��
+96
+√
+2
+�
+1 + 8L∇¯hσ−1
+x
+��
++ 2
+�
+×
+�
+max
+�
+2,
+� σx
+2σy
+�
+log 4 max{1/(2σx), min {1/(2σy), 4/(¯ασx)}} ϑ0
+(L2
+∇¯h/ min{σx, σy} + L∇¯h)−2τ 2
+�
++
+(69)(18)
+≤
+¯N,
+where the second last inequality follows from the definition of ηy, ηz and ˆζ in Algorithm 1. Hence, the
+conclusion holds as desired.
+5.2
+Proof of the main results in Subsection 2.2
+In this subsection we prove Theorem 2.
+Before proceeding, let {(xk, yk)}k∈K denote all the iterates
+generated by Algorithm 2, where K is a subset of consecutive nonnegative integers starting from 0. Also,
+we define K − 1 = {k − 1 : k ∈ K}. We first establish two lemmas and then use them to prove Theorem
+2 subsequently.
+Lemma 2. Suppose that Assumption 1 holds. Let {(xk, yk)}k∈K be generated by Algorithm 2, H∗, Dp,
+Dq, Hlow, α, δ be defined in (6), (9), (23), (24) and (25), L∇h be given in Assumption 1, ǫ, ǫk be given
+in Algorithm 2, and
+Nk =
+��
+96
+√
+2
+�
+1 + (24L∇h + 4ǫ/Dq) L−1
+∇h
+��
++ 2
+�
+×
+�
+max
+�
+2,
+�
+DqL∇h
+ǫ
+�
+× log
+4 max
+�
+1
+2L∇h , min
+�
+Dq
+ǫ ,
+4
+αL∇h
+�� �
+δ + 2α−1(H∗ − Hlow + ǫDq/4 + L∇hD2
+p)
+�
+[(3L∇h + ǫ/(2Dq))2/ min{L∇h, ǫ/(2Dq)} + 3L∇h + ǫ/(2Dq)]−2 ǫ2
+k
+�
++
+.
+(73)
+Then for all 0 ≤ k ∈ K−1, (xk+1, yk+1) is an ǫk-stationary point of (20). Moreover, the total number of
+evaluations of ∇h and proximal operator of p and q performed at iteration k of Algorithm 2 for generating
+(xk+1, yk+1) is no more than Nk, respectively.
+Proof. Let (x∗, y∗) be an optimal solution of (6). Recall that H, Hk and hk are given in (6), (20) and
+(21), respectively. Then we have
+Hk,∗ := min
+x max
+y
+Hk(x, y) = min
+x max
+y
+�
+H(x, y) −
+ǫ
+4Dq
+∥y − ˆy0∥2 + L∇h∥x − xk∥2
+�
+≤ max
+y {H(x∗, y) + L∇h∥x∗ − xk∥2}
+(6)(9)
+≤
+H∗ + L∇hD2
+p.
+(74)
+Moreover, by (9) and (23), one has
+Hk,low :=
+min
+(x,y)∈dom p×dom q Hk(x, y) =
+min
+(x,y)∈dom p×dom q
+�
+H(x, y) −
+ǫ
+4Dq
+∥y − ˆy0∥2 + L∇h∥x − xk∥2
+�
+(23)
+≥ Hlow − max
+y∈dom q
+ǫ
+4Dq
+∥y − ˆy0∥2 (9)
+≥ Hlow − ǫDq/4.
+(75)
+In addition, by Assumption 1 and the definition of hk in (21), it is not hard to verify that hk(x, y) is
+L∇h-strongly-convex in x, ǫ/(2Dq)-strongly-concave in y, and (3L∇h + ǫ/(2Dq))-smooth on its domain.
+Also, recall from Remark 2 that (xk+1, yk+1) results from applying Algorithm 1 to problem (20). The
+conclusion of this lemma then follows by using (74) and (75) and applying Theorem 1 to (20) with τ = ǫk,
+σx = L∇h, σy = ǫ/(2Dq), L∇¯h = 3L∇h + ǫ/(2Dq), ¯α = α, ¯δ = δ, ¯Hlow = Hk,low, and ¯H∗ = Hk,∗.
+16
+
+Lemma 3. Suppose that Assumption 1 holds. Let {xk}k∈K be generated by Algorithm 2, H, H∗ and
+Dq be defined in (6) and (9), L∇h be given in Assumption 1, and ǫ, ǫ0 and ˆx0 be given in Algorithm 2.
+Then for all 0 ≤ K ∈ K − 1, we have
+min
+0≤k≤K ∥xk+1 − xk∥ ≤ maxy H(ˆx0, y) − H∗ + ǫDq/4
+L∇h(K + 1)
++ 2ǫ2
+0(1 + 4D2
+qL2
+∇hǫ−2)
+L2
+∇h(K + 1)
+,
+(76)
+max
+y
+H(xK+1, y) ≤ max
+y
+H(ˆx0, y) + ǫDq/4 + 2ǫ2
+0
+�
+L−1
+∇h + 4D2
+qL∇hǫ−2�
+.
+(77)
+Proof. For convenience of the proof, let
+H∗
+ǫ (x) = max
+y
+�
+H(x, y) − ǫ∥y − ˆy0∥2/(4Dq)
+�
+,
+(78)
+H∗
+k(x) = max
+y
+Hk(x, y),
+yk+1
+∗
+= arg max
+y
+Hk(xk+1, y).
+(79)
+One can observe from these, (20) and (21) that
+H∗
+k(x) = H∗
+ǫ (x) + L∇h∥x − xk∥2.
+(80)
+By this and Assumption 1, one can also see that H∗
+k is L∇h-strongly convex on dom p. In addition,
+recall from Lemma 2 that (xk+1, yk+1) is an ǫk-stationary point of problem (20) for all 0 ≤ k ∈ K − 1.
+It then follows from Definition 2 that there exist some u ∈ ∂xHk(xk+1, yk+1) and v ∈ ∂yHk(xk+1, yk+1)
+with ∥u∥ ≤ ǫk and ∥v∥ ≤ ǫk.
+Also, by (79), one has 0 ∈ ∂yHk(xk+1, yk+1
+∗
+), which together with
+v ∈ ∂yHk(xk+1, yk+1) and ǫ/(2Dq)-strong concavity of Hk(xk+1, ·), implies that ⟨−v, yk+1 − yk+1
+∗
+⟩ ≥
+ǫ∥yk+1 − yk+1
+∗
+∥2/(2Dq). This and ∥v∥ ≤ ǫk yield
+∥yk+1 − yk+1
+∗
+∥ ≤ 2ǫkDq/ǫ.
+(81)
+In addition, by u ∈ ∂xHk(xk+1, yk+1), (20) and (21), one has
+u ∈ ∇xh(xk+1, yk+1) + ∂p(xk+1) + 2L∇h(xk+1 − xk).
+(82)
+Also, observe from (20), (21) and (79) that
+∂H∗
+k(xk+1) = ∇xh(xk+1, yk+1
+∗
+) + ∂p(xk+1) + 2L∇h(xk+1 − xk),
+which together with (82) yields
+u + ∇xh(xk+1, yk+1
+∗
+) − ∇xh(xk+1, yk+1) ∈ ∂H∗
+k(xk+1).
+By this and L∇h-strong convexity of H∗
+k, one has
+H∗
+k(xk) ≥ H∗
+k(xk+1) + ⟨u + ∇xh(xk+1, yk+1
+∗
+) − ∇xh(xk+1, yk+1), xk − xk+1⟩ + L∇h∥xk − xk+1∥2/2. (83)
+Using this, (80), (81), (83), ∥u∥ ≤ ǫk, and the Lipschitz continuity of ∇h, we obtain
+H∗
+ǫ (xk) − H∗
+ǫ (xk+1)
+(80)
+= H∗
+k(xk) − H∗
+k(xk+1) + L∇h∥xk − xk+1∥2
+(83)
+≥ ⟨u + ∇xh(xk+1, yk+1
+∗
+) − ∇xh(xk+1, yk+1), xk − xk+1⟩ + 3L∇h∥xk − xk+1∥2/2
+≥
+�
+− ∥u + ∇xh(xk+1, yk+1
+∗
+) − ∇xh(xk+1, yk+1)∥∥xk − xk+1∥ + L∇h∥xk − xk+1∥2/2
+�
++ L∇h∥xk − xk+1∥2
+≥ −(2L∇h)−1∥u + ∇xh(xk+1, yk+1
+∗
+) − ∇xh(xk+1, yk+1)∥2 + L∇h∥xk − xk+1∥2
+≥ −L−1
+∇h∥u∥2 − L−1
+∇h∥∇xh(xk+1, yk+1
+∗
+) − ∇xh(xk+1, yk+1)∥2 + L∇h∥xk − xk+1∥2
+≥ −L−1
+∇hǫ2
+k − L∇h∥yk+1 − yk+1
+∗
+∥2 + L∇h∥xk − xk+1∥2
+(81)
+≥ −(L−1
+∇h + 4D2
+qL∇hǫ−2)ǫ2
+k + L∇h∥xk − xk+1∥2,
+where the second and fourth inequalities follow from Cauchy-Schwartz inequality, and the third inequal-
+ity is due to Young’s inequality, and the fifth inequality follows from L∇h-Lipschitz continuity of ∇h.
+Summing up the above inequality for k = 0, 1, . . ., K yields
+L∇h
+K
+�
+k=0
+∥xk − xk+1∥2 ≤ H∗
+ǫ (x0) − H∗
+ǫ (xK+1) + (L−1
+∇h + 4D2
+qL∇hǫ−2)
+K
+�
+k=0
+ǫ2
+k.
+(84)
+17
+
+In addition, it follows from (6), (9) and (78) that
+H∗
+ǫ (xK+1) = max
+y
+�
+H(xK+1, y) − ǫ∥y − ˆy0∥2/(4Dq)
+�
+≥ min
+x max
+y
+H(x, y) − ǫDq/4 = H∗ − ǫDq/4,
+H∗
+ǫ (x0) = max
+y
+�
+H(x0, y) − ǫ∥y − ˆy0∥2/(4Dq)
+�
+≤ max
+y
+H(x0, y).
+(85)
+These together with (84) yield
+L∇h(K + 1) min
+0≤k≤K ∥xk+1 − xk∥2 ≤ L∇h
+K
+�
+k=0
+∥xk − xk+1∥2
+≤ max
+y
+H(x0, y) − H∗ + ǫDq/4 + (L−1
+∇h + 4D2
+qL∇hǫ−2)
+K
+�
+k=0
+ǫ2
+k,
+which together with x0 = ˆx0, ǫk = ǫ0(k + 1)−1 and �K
+k=0(k + 1)−2 < 2 implies that (76) holds.
+Finally, we show that (77) holds. Indeed, it follows from (9), (78), (84), (85), ǫk = ǫ0(k + 1)−1, and
+�K
+k=0(k + 1)−2 < 2 that
+max
+y
+H(xK+1, y)
+(9)
+≤
+max
+y
+�
+H(xK+1, y) − ǫ∥y − ˆy0∥2/(4Dq)
+�
++ ǫDq/4
+(78)
+= H∗
+ǫ (xK+1) + ǫDq/4
+(84)
+≤ H∗
+ǫ (x0) + ǫDq/4 + (L−1
+∇h + 4D2
+qL∇hǫ−2)
+K
+�
+k=0
+ǫ2
+k
+(85)
+≤
+max
+y
+H(x0, y) + ǫDq/4 + 2ǫ2
+0(L−1
+∇h + 4D2
+qL∇hǫ−2).
+It then follows from this and x0 = ˆx0 that (77) holds.
+We are now ready to prove Theorem 2.
+Proof of Theorem 2. Suppose for contradiction that Algorithm 2 runs for more than K + 1 outer
+iterations, where K is given in (26). By this and Algorithm 2, one can then assert that (22) does not
+hold for all 0 ≤ k ≤ K. On the other hand, by (26) and (76), one has
+min
+0≤k≤K ∥xk+1 − xk∥2
+(76)
+≤
+maxy H(ˆx0, y) − H∗ + ǫDq/4
+L∇h(K + 1)
++ 2ǫ2
+0(1 + 4D2
+qL2
+∇hǫ−2)
+L2
+∇h(K + 1)
+(26)
+≤
+ǫ2
+16L2
+∇h
+,
+which implies that there exists some 0 ≤ k ≤ K such that ∥xk+1 − xk∥ ≤ ǫ/(4L∇h), and thus (22) holds
+for such k, which contradicts the above assertion. Hence, Algorithm 2 must terminate in at most K + 1
+outer iterations.
+Suppose that Algorithm 2 terminates at some iteration 0 ≤ k ≤ K, namely, (22) holds for such k. We
+next show that its output (xǫ, yǫ) = (xk+1, yk+1) is an ǫ-stationary point of (6) and moreover it satisfies
+(28). Indeed, recall from Lemma 2 that (xk+1, yk+1) is an ǫk-stationary point of (20), namely, it satisfies
+dist(0, ∂xHk(xk+1, yk+1)) ≤ ǫk and dist(0, ∂yHk(xk+1, yk+1)) ≤ ǫk. By these, (6), (20) and (21), there
+exists (u, v) such that
+u ∈ ∂xH(xk+1, yk+1) + 2L∇h(xk+1 − xk),
+∥u∥ ≤ ǫk,
+v ∈ ∂yH(xk+1, yk+1) − ǫ(yk+1 − ˆy0)/(2Dq),
+∥v∥ ≤ ǫk.
+It then follows that u−2L∇h(xk+1−xk) ∈ ∂xH(xk+1, yk+1) and v+ǫ(yk+1−ˆy0)/(2Dq) ∈ ∂yH(xk+1, yk+1).
+These together with (9), (22), and ǫk ≤ ǫ0 ≤ ǫ/2 (see Algorithm 2) imply that
+dist
+�
+0, ∂xH(xk+1, yk+1)
+�
+≤ ∥u − 2L∇h(xk+1 − xk)∥ ≤ ∥u∥ + 2L∇h∥xk+1 − xk∥
+(22)
+≤ ǫk + ǫ/2 ≤ ǫ,
+dist
+�
+0, ∂yH(xk+1, yk+1)
+�
+≤ ∥v + ǫ(yk+1 − ˆy0)/(2Dq)∥ ≤ ∥v∥ + ǫ∥yk+1 − ˆy0∥/(2Dq)
+(9)
+≤ ǫk + ǫ/2 ≤ ǫ.
+Hence, the output (xk+1, yk+1) of Algorithm 2 is an ǫ-stationary point of (6). In addition, (28) holds
+due to Lemma 3.
+18
+
+Recall from Lemma 2 that the number of evaluations of ∇h and proximal operator of p and q
+performed at iteration k of Algorithm 2 is at most Nk, respectively, where Nk is defined in (73). Also,
+one can observe from the above proof and the definition of K that |K| ≤ K + 2. It then follows that the
+total number of evaluations of ∇h and proximal operator of p and q in Algorithm 2 is respectively no
+more than �|K|−2
+k=0
+Nk. Consequently, to complete the rest of the proof of Theorem 2, it suffices to show
+that �|K|−2
+k=0
+Nk ≤ N, where N is given in (27). Indeed, by (27), (73) and |K| ≤ K + 2, one has
+|K|−2
+�
+k=0
+Nk
+(73)
+≤
+K
+�
+k=0
+��
+96
+√
+2
+�
+1 + (24L∇h + 4ǫ/Dq) L−1
+∇h
+��
++ 2
+�
+×
+�
+max
+�
+2,
+�
+DqL∇h
+ǫ
+�
+× log
+4 max
+�
+1
+2L∇h , min
+�
+Dq
+ǫ ,
+4
+αL∇h
+�� �
+δ + 2α−1(H∗ − Hlow + ǫDq/4 + L∇hD2
+p)
+�
+[(3L∇h + ǫ/(2Dq))2/ min{L∇h, ǫ/(2Dq)} + 3L∇h + ǫ/(2Dq)]−2 ǫ2
+k
+�
++
+≤
+��
+96
+√
+2
+�
+1 + (24L∇h + 4ǫ/Dq) L−1
+∇h
+��
++ 2
+�
+max
+�
+2,
+�
+DqL∇h
+ǫ
+�
+×
+K
+�
+k=0
+
+
+
+log
+4 max
+�
+1
+2L∇h , min
+�
+Dq
+ǫ ,
+4
+αL∇h
+�� �
+δ + 2α−1(H∗ − hlow + ǫDq/4 + L∇hD2
+p)
+�
+[(3L∇h + ǫ/(2Dq))2/ min{L∇h, ǫ/(2Dq)} + 3L∇h + ǫ/(2Dq)]−2 ǫ2
+k
+
+
++
++ 1
+
+
+≤
+��
+96
+√
+2
+�
+1 + (24L∇h + 4ǫ/Dq) L−1
+∇h
+��
++ 2
+�
+max
+�
+2,
+�
+DqL∇h
+ǫ
+�
+×
+�
+(K + 1)
+�
+log
+4 max
+�
+1
+2L∇h , min
+�
+Dq
+ǫ ,
+4
+αL∇h
+�� �
+δ + 2α−1(H∗ − Hlow + ǫDq/4 + L∇hD2
+p)
+�
+[(3L∇h + ǫ/(2Dq))2/ min{L∇h, ǫ/(2Dq)} + 3L∇h + ǫ/(2Dq)]−2 ǫ2
+0
+�
++
++ K + 1 + 2
+K
+�
+k=0
+log(k + 1)
+�
+(27)
+≤ N,
+where the last inequality is due to (27) and �K
+k=0 log(k + 1) ≤ K log(K + 1). This completes the proof
+of Theorem 2.
+5.3
+Proof of the main results in Section 3
+In this subsection we prove Theorems 3 and 4. We first establish a lemma below, which will be used to
+prove Theorem 3 subsequently.
+Lemma 4. Suppose that Assumption 3 holds and (xǫ, yǫ, zǫ) is an ǫ-optimal solution of problem (35) for
+some ǫ > 0. Let f, ˜f, f ∗, flow and ρ be given in (29), (32) and (35), respectively. Then we have
+˜f(xǫ, yǫ) ≤ min
+z
+˜f(xǫ, z) + ρ−1(f ∗ − flow + 2ǫ),
+f(xǫ, yǫ) ≤ f ∗ + 2ǫ.
+Proof. Since (xǫ, yǫ, zǫ) is an ǫ-optimal solution of (35), it follows from Definition 1 that
+max
+z
+Pρ(xǫ, yǫ, z) ≤ Pρ(xǫ, yǫ, zǫ) + ǫ,
+Pρ(xǫ, yǫ, zǫ) ≤ min
+x,y max
+z
+Pρ(x, y, z) + ǫ.
+Summing up these inequalities yields
+max
+z
+Pρ(xǫ, yǫ, z) ≤ min
+x,y max
+z
+Pρ(x, y, z) + 2ǫ.
+(86)
+Let (x∗, y∗) be an optimal solution of (29).
+It then follows that f(x∗, y∗) = f ∗ and ˜f(x∗, y∗) =
+minz ˜f(x∗, z). By these and the definition of Pρ in (35), one has
+max
+z
+Pρ(x∗, y∗, z) = f(x∗, y∗) + ρ( ˜f(x∗, y∗) − min
+z
+˜f(x∗, z)) = f(x∗, y∗) = f ∗,
+which implies that
+min
+x,y max
+z
+Pρ(x, y, z) ≤ max
+z
+Pρ(x∗, y∗, z) = f ∗.
+(87)
+19
+
+It then follows from (35), (86) and (87) that
+f(xǫ, yǫ) + ρ( ˜f(xǫ, yǫ) − min
+z
+˜f(xǫ, z))
+(35)
+= max
+z
+Pρ(xǫ, yǫ, z)
+(86)(87)
+≤
+f ∗ + 2ǫ,
+which together with ˜f(xǫ, yǫ) − minz ˜f(xǫ, z) ≥ 0 implies that
+f(xǫ, yǫ) ≤ f ∗ + 2ǫ,
+˜f(xǫ, yǫ) ≤ min
+z
+˜f(xǫ, z) + ρ−1 (f ∗ − f(xǫ, yǫ) + 2ǫ) .
+The conclusion of this lemma directly follows from these and (32).
+We are now ready to prove Theorem 3.
+Proof of Theorem 3. Let {(xk, yk, zk)} be generated by Algorithm 3 with limk→∞(ρk, ǫk) = (∞, 0).
+By considering a convergent subsequence if necessary, we assume without loss of generality that limk→∞(xk, yk) =
+(x∗, y∗). we now show that (x∗, y∗) is an optimal solution of problem (29). Indeed, since (xk, yk, zk)
+is an ǫk-optimal solution of (35) with ρ = ρk, it follows from Lemma 4 with (ρ, ǫ) = (ρk, ǫk) and
+(xǫ, yǫ) = (xk, yk) that
+˜f(xk, yk) ≤ min
+z
+˜f(xk, z) + ρ−1
+k (f ∗ − flow + 2ǫk),
+f(xk, yk) ≤ f ∗ + 2ǫk.
+By the continuity of f and ˜f, limk→∞(xk, yk) = (x∗, y∗), limk→∞(ρk, ǫk) = (∞, 0), and taking limits as
+k → ∞ on both sides of the above relations, we obtain that ˜f(x∗, y∗) ≤ minz ˜f(x∗, z) and f(x∗, y∗) ≤ f ∗,
+which clearly imply that y∗ ∈ Argminz ˜f(x∗, z) and f(x∗, y∗) = f ∗. Hence, (x∗, y∗) is an optimal solution
+of (29) as desired.
+We next prove Theorem 4. Before proceeding, we establish a lemma below, which will be used to
+prove Theorem 4 subsequently.
+Lemma 5. Suppose that Assumption 3 holds and (xǫ, yǫ, zǫ) is an ǫ-stationary point of (35). Let Dy,
+flow, ˜f, ρ, and Pρ be given in (30), (32) and (35), respectively. Then we have
+dist
+�
+0, ∂f(xǫ, yǫ) + ρ∂ ˜f(xǫ, yǫ) − (ρ∇x ˜f(xǫ, zǫ); 0)
+�
+≤ ǫ,
+dist
+�
+0, ρ∂ ˜f(xǫ, zǫ)
+�
+≤ ǫ,
+˜f(xǫ, yǫ) ≤ min
+z
+˜f(xǫ, z) + ρ−1(max
+z
+Pρ(xǫ, yǫ, z) − flow).
+Proof. Since (xǫ, yǫ, zǫ) is an ǫ-stationary point of (35), it follows from Definition 2 that
+dist
+�
+0, ∂x,yPρ(xǫ, yǫ, zǫ)
+�
+≤ ǫ,
+dist
+�
+0, ∂zPρ(xǫ, yǫ, zǫ)
+�
+≤ ǫ.
+Using these and the definition of Pρ in (35), we have
+dist
+�
+0, ∂f(xǫ, yǫ) + ρ∂ ˜f(xǫ, yǫ) − (ρ∇x ˜f(xǫ, zǫ); 0)
+�
+≤ ǫ,
+dist
+�
+0, ρ∂ ˜f(xǫ, zǫ)
+�
+≤ ε.
+In addition, by (35), we have
+f(xǫ, yǫ) + ρ( ˜f(xǫ, yǫ) − min
+z
+˜f(xǫ, z)) = max
+z
+Pρ(xǫ, yǫ, z),
+which along with (32) implies that
+˜f(xǫ, yǫ) − min
+z
+˜f(xǫ, z) ≤ ρ−1(max
+z
+Pρ(xǫ, yǫ, z) − flow).
+This completes the proof of this lemma.
+We are now ready to prove Theorem 4.
+Proof of Theorem 4. Observe from (36) that problem (35) can be viewed as
+min
+x,y max
+z
+{Pρ(x, y, z) = h(x, y, z) + p(x, y) − q(z)} ,
+where h(x, y, z) = f1(x, y) + ρ ˜f1(x, y) − ρ ˜f1(x, z), p(x, y) = f2(x) + ρ ˜f2(y), and q(z) = ρ ˜f2(z). Hence,
+problem (35) is in the form of (6) with H = Pρ. By Assumption 3 and ρ = ε−1, one can see that h is
+20
+
+�L-smooth on its domain, where �L is given in (39). Also, notice from Algorithm 4 that ǫ0 = ε3/2 ≤ ε/2
+due to ε ∈ (0, 1/4]. Consequently, Algorithm 2 can be suitably applied to problem (35) with ρ = ε−1 for
+finding an ǫ-stationary point (xǫ, yǫ, zǫ) of it.
+In addition, notice from Algorithm 4 that ˜f(x0, y0) ≤ miny ˜f(x0, y)+ε. Using this, (35) and ρ = ε−1,
+we obtain
+max
+z
+Pρ(x0, y0, z) = f(x0, y0) + ρ( ˜f(x0, y0) − min
+z
+˜f(x0, z)) ≤ f(x0, y0) + ρε = f(x0, y0) + 1.
+(88)
+By this and (28) with H = Pρ, ǫ = ε, ǫ0 = ε3/2, ˆx0 = (x0, y0), Dq = Dy, and L∇h = �L, one has
+Pρ(xǫ, yǫ, zǫ) ≤ max
+z
+Pρ(x0, y0, z) + εDy/4 + 2ε3(�L−1 + 4D2
+y�Lε−2)
+(88)
+≤ 1 + f(x0, y0) + εDy/4 + 2ε3(�L−1 + 4D2
+y�Lε−2).
+It then follows from this and Lemma 5 with ǫ = ε and ρ = ε−1 that (xǫ, yǫ, zǫ) satisfies (40) and (41).
+We next show that at most �
+N evaluations of ∇f1, ∇ ˜f1, and proximal operator of f2 and ˜f2 are
+respectively performed in Algorithm 4. Indeed, by (31), (32) and (35), one has
+min
+x,y max
+z
+Pρ(x, y, z)
+(35)
+= min
+x,y {f(x, y) + ρ( ˜f(x, y) − min
+z
+˜f(x, z))} ≥
+min
+(x,y)∈X ×Y f(x, y)
+(32)
+= flow,
+(89)
+min
+(x,y,z)∈X ×Y×Y Pρ(x, y, z)
+(35)
+=
+min
+(x,y,z)∈X ×Y×Y{f(x, y) + ρ( ˜f(x, y) − ˜f(x, z))}
+(31)(32)
+≥
+flow + ρ( ˜flow − ˜fhi).
+(90)
+For convenience of the rest proof, let
+H = Pρ,
+H∗ = min
+x,y max
+z
+Pρ(x, y, z),
+Hlow = min{Pρ(x, y, z)|(x, y, z) ∈ X × Y × Y}.
+(91)
+In view of these, (87), (88), (89), (90), and ρ = ε−1, we obtain that
+max
+z
+H(x0, y0, z)
+(88)
+≤ f(x0, y0) + 1,
+flow
+(89)
+≤ H∗ (87)
+≤ f ∗,
+Hlow
+(90)
+≥ flow + ρ( ˜flow − ˜fhi) = flow + ε−1( ˜flow − ˜fhi).
+Using these and Theorem 2 with ǫ = ε, ˆx0 = (x0, y0), Dp =
+�
+D2x + D2y, Dq = Dy, ǫ0 = ε3/2, L∇h = �L,
+α = ˆα, δ = ˆδ, and H, H∗, Hlow given in (91), we can conclude that Algorithm 4 performs at most
+�
+N evaluations of ∇f1, ∇ ˜f1 and proximal operator of f2 and ˜f2 respectively for finding an approximate
+solution (xǫ, yǫ) of problem (29) satisfying (40) and (41).
+5.4
+Proof of the main results in Section 4
+In this subsection we prove Theorems 5 and 6. Before proceeding, we define
+r = G−1Dy(ρ−1ǫ + L ˜
+f),
+B+
+r = {λ ∈ Rl
++ : ∥λ∥ ≤ r},
+(92)
+where Dy is defined in (30), G is given in Assumption 4(iii), and ǫ and ρ are given in Algorithm 6. In
+addition, one can observe from (43) and (47) that
+min
+z
+�Pµ(x, z) ≤ ˜f ∗(x)
+∀x ∈ X,
+(93)
+which will be frequently used later.
+We next establish several technical lemmas that will be used to prove Theorem 5 subsequently.
+Lemma 6. Suppose that Assumptions 3 and 4 hold. Let Dy, L ˜
+f, G, ˜f ∗, ˜f ∗
+hi and B+
+r be given in (30),
+(43), (44), (92) and Assumption 4, respectively. Then the following statements hold.
+(i) ∥λ∗∥ ≤ G−1L ˜
+fDy and λ∗ ∈ B+
+r for all λ∗ ∈ Λ∗(x) and x ∈ X, where Λ∗(x) denotes the set of
+optimal Lagrangian multipliers of problem (43) for any x ∈ X.
+21
+
+(ii) The function ˜f ∗ is Lipschitz continuous on X and ˜f ∗
+hi is finite.
+(iii) It holds that
+˜f ∗(x) = max
+λ
+min
+z
+˜f(x, z) + ⟨λ, ˜g(x, z)⟩ − IRl
++(λ)
+∀x ∈ X,
+(94)
+where IRl
++(·) is the indicator function associated with Rl
++.
+Proof. (i) Let x ∈ X and λ∗ ∈ Λ∗(x) be arbitrarily chosen, and let z∗ ∈ Y be such that (z∗, λ∗) is a pair
+of primal-dual optimal solutions of (43). It then follows that
+z∗ ∈ Argmin
+z
+˜f(x, z) + ⟨λ∗, ˜g(x, z)⟩,
+⟨λ∗, ˜g(x, z∗)⟩ = 0,
+˜g(x, z∗) ≤ 0,
+λ∗ ≥ 0.
+The first relation above yields
+˜f(x, z∗) + ⟨λ∗, ˜g(x, z∗)⟩ ≤ ˜f(x, ˆzx) + ⟨λ∗, ˜g(x, ˆzx)⟩,
+where ˆzx is given in Assumption 4(iii). By this and ⟨λ∗, ˜g(x, z∗)⟩ = 0, one has
+⟨λ∗, −˜g(x, ˆzx)⟩ ≤ ˜f(x, ˆzx) − ˜f(x, z∗),
+which together with λ∗ ≥ 0, (30) and Assumption 4 implies that
+G
+l
+�
+i=1
+λ∗
+i ≤ ⟨λ∗, −˜g(x, ˆzx)⟩ ≤ ˜f(x, ˆzx) − ˜f(x, z∗) ≤ L ˜
+f∥ˆzx − z∗∥ ≤ L ˜
+fDy,
+(95)
+where the first inequality is due to Assumption 4(iii), and the third inequality follows from (30) and L ˜
+f-
+Lipschitz continuity of ˜f (see Assumption 4(i)). By (92), (95) and λ∗ ≥ 0, we have ∥λ∗∥ ≤ �l
+i=1 λ∗
+i ≤
+G−1L ˜
+fDy and λ∗ ∈ B+
+r .
+(ii) Recall from Assumptions 3(i) and 4(iii) that ˜f(x, ·) and ˜gi(x, ·), i = 1, . . . , l, are convex for any
+given x ∈ X. Using this, (43) and the first statement of this lemma, we observe that
+˜f ∗(x) = min
+z
+max
+λ∈B+
+r
+˜f(x, z) + ⟨λ, ˜g(x, z)⟩
+∀x ∈ X.
+(96)
+Notice from Assumption 4 that ˜f and ˜g are Lipschitz continuous on their domain. Then it is not hard to
+observe that max{ ˜f(x, z)+⟨λ, ˜g(x, z)⟩|λ ∈ B+
+r } is a Lipschitz continuous function of (x, z) on its domain.
+By this and (96), one can easily verify that ˜f ∗ is Lipschitz continuous on X. In addition, the finiteness
+of ˜f ∗
+hi follows from (44), the continuity of ˜f ∗, and the compactness of X.
+(iii) One can observe from (43) that for all x ∈ X,
+˜f ∗(x) = min
+z
+max
+λ
+˜f(x, z) + ⟨λ, ˜g(x, z)⟩ − IRl
++(λ) ≥ max
+λ
+min
+z
+˜f(x, z) + ⟨λ, ˜g(x, z)⟩ − IRl
++(λ)
+where the inequality follows from the weak duality. In addition, it follows from Assumption 3 that the
+domain of ˜f(x, ·) is compact for all x ∈ X. By this, (96) and the strong duality, one has
+˜f ∗(x) = max
+λ∈B+
+r
+min
+z
+˜f(x, z) + ⟨λ, ˜g(x, z)⟩ − IRl
++(λ)
+∀x ∈ X,
+which together with the above inequality implies that (94) holds.
+Lemma 7. Suppose that Assumptions 3 and 4 hold and that (xǫ, yǫ, zǫ) is an ǫ-optimal solution of
+problem (50) for some ǫ > 0. Let flow, f, �Pµ, f ∗
+µ, ρ and µ be given in (32), (42), (47), (48) and (50),
+respectively. Then we have
+�Pµ(xǫ, yǫ) ≤ min
+z
+�Pµ(xǫ, z) + ρ−1(f ∗
+µ − flow + 2ǫ),
+f(xǫ, yǫ) ≤ f ∗
+µ + 2ǫ.
+(97)
+Proof. The proof follows from the same argument as the one for Lemma 4 with f ∗ and ˜f being replaced
+by f ∗
+µ and �Pµ, respectively.
+Lemma 8. Suppose that Assumptions 3-5 hold. Let ˜flow, f ∗, ˜f ∗
+hi, f ∗
+µ be defined in (31), (42), (44) and
+(48), and Lf, ω and ¯θ be given in Assumptions 4 and 5. Suppose that µ ≥ ( ˜f ∗
+hi − ˜flow)/¯θ2. Then we have
+f ∗
+µ ≤ f ∗ + Lfω
+��
+µ−1( ˜f ∗
+hi − ˜flow)
+�
+.
+(98)
+22
+
+Proof. Let x ∈ X, y ∈ Argminz{ ˜f(x, z)|˜g(x, z) ≤ 0} and z∗ ∈ Argminz �Pµ(x, z) be arbitrarily chosen.
+One can easily see from (47) and (93) that ˜f(x, z∗) + µ ∥[˜g(x, z∗)]+∥2 ≤ ˜f ∗(x), which together with (31)
+and (44) implies that
+∥[˜g(x, z∗)]+∥2 ≤ µ−1( ˜f ∗
+hi − ˜flow).
+(99)
+Since µ ≥ ( ˜f ∗
+hi− ˜flow)/¯θ2, it follows from (99) that ∥[˜g(x, z∗)]+∥ ≤ ¯θ. By this relation, y ∈ Argmin
+z
+{ ˜f(x, z)|˜g(x, z) ≤
+0} and Assumption 5, there exists some ˆz∗ such that
+∥y − ˆz∗∥ ≤ ω(∥[˜g(x, z∗)]+∥),
+ˆz∗ ∈ Argmin
+z
+�
+˜f(x, z)
+�� ∥[˜g(x, z)]+∥ ≤ ∥[˜g(x, z∗)]+∥
+�
+.
+(100)
+In view of (47), z∗ ∈ Argminz �Pµ(x, z) and the second relation in (100), one can observe that ˆz∗ ∈
+Argminz �Pµ(x, z), which along with (48) yields f(x, ˆz∗) ≥ f ∗
+µ. Also, using (100) and Lf-Lipschitz conti-
+nuity of f (see Assumption 4), we have
+f(x, y) − f(x, ˆz∗) ≥ −Lf∥y − ˆz∗∥
+(100)
+≥
+−Lfω(∥[˜g(x, z∗)]+∥).
+Taking minimum over x ∈ X and y ∈ Argminz{ ˜f(x, z)|˜g(x, z) ≤ 0} on both sides of this relation, and
+using (42), (99), f(x, ˆz∗) ≥ f ∗
+µ and the monotonicity of ω, we can conclude that (98) holds.
+Lemma 9. Suppose that Assumptions 3-5 hold. Let ˜flow, flow, f, ˜f, f ∗, ˜f ∗, ˜f ∗
+hi, ρ and µ be given in
+(31), (32), (42), (43), (44) and (50), and Lf, ω and ¯θ be given in Assumptions 4 and 5, respectively.
+Suppose that µ ≥ ( ˜f ∗
+hi − ˜flow)/¯θ2 and (xǫ, yǫ, zǫ) is an ǫ-optimal solution of problem (50) for some ǫ > 0.
+Then we have
+f(xǫ, yǫ) ≤ f ∗ + Lfω
+��
+µ−1( ˜f ∗
+hi − ˜flow)
+�
++ 2ǫ,
+˜f(xǫ, yǫ) ≤ ˜f ∗(xǫ) + ρ−1�
+f ∗ − flow + Lfω
+��
+µ−1( ˜f ∗
+hi − ˜flow)
+�
++ 2ǫ
+�
+,
+∥[˜g(xǫ, yǫ)]+∥2 ≤ µ−1�
+˜f ∗(xǫ) − ˜flow + ρ−1�
+f ∗ − flow + Lfω
+��
+µ−1( ˜f ∗
+hi − ˜flow)
+�
++ 2ǫ
+��
+.
+Proof. By (47), (93), and the first relation in (97), one has
+˜f(xǫ, yǫ) + µ ∥[˜g(xǫ, yǫ)]+∥2 (47)
+=
+�Pµ(xǫ, yǫ)
+(93)(97)
+≤
+˜f ∗(xǫ) + ρ−1(f ∗
+µ − flow + 2ǫ).
+It then follows from this and (31) that
+˜f(xǫ, yǫ) ≤ ˜f ∗(xǫ) + ρ−1(f ∗
+µ − flow + 2ǫ),
+∥[˜g(xǫ, yǫ)]+∥2 ≤ µ−1( ˜f ∗(xǫ) − ˜flow + ρ−1(f ∗
+µ − flow + 2ǫ)).
+In addition, recall from (97) that f(xǫ, yǫ) ≤ f ∗
+µ + 2ǫ. The conclusion of this lemma then follows from
+these three relations and (98).
+We are now ready to prove Theorem 5.
+Proof of Theorem 5. Let {(xk, yk, zk)} be generated by Algorithm 5 with limk→∞(ρk, µk, ǫk) = (∞, ∞, 0).
+By considering a convergent subsequence if necessary, we assume without loss of generality that limk→∞(xk, yk) =
+(x∗, y∗). We now show that (x∗, y∗) is an optimal solution of problem (42). Indeed, since (xk, yk, zk) is
+an ǫk-optimal solution of (50) with (ρ, µ) = (ρk, µk) and limk→∞ µk = ∞, it follows from Lemma 9 with
+(ρ, µ, ǫ) = (ρk, µk, ǫk) and (xǫ, yǫ) = (xk, yk) that for all sufficiently large k, one has
+f(xk, yk) ≤ f ∗ + Lfω
+��
+µ−1
+k ( ˜f ∗
+hi − ˜flow)
+�
++ 2ǫk,
+˜f(xk, yk) ≤ ˜f ∗(xk) + ρ−1
+k
+�
+f ∗ − flow + Lfω
+��
+µ−1
+k ( ˜f ∗
+hi − ˜flow)
+�
++ 2ǫk
+�
+,
+��[˜g(xk, yk)]+
+��2 ≤ µ−1
+k
+�
+˜f ∗(xk) − ˜flow + ρ−1
+k
+�
+f ∗ − flow + Lfω
+��
+µ−1
+k ( ˜f ∗
+hi − ˜flow)
+�
++ 2ǫk
+��
+.
+By the continuity of f, ˜f and ˜f ∗ (see Assumption 3(i) and Lemma 6(ii)), limk→∞(xk, yk) = (x∗, y∗),
+limk→∞(ρk, µk, ǫk) = (∞, ∞, 0), limθ↓0 ω(θ) = 0, and taking limits as k → ∞ on both sides of the above
+relations, we obtain that f(x∗, y∗) ≤ f ∗, ˜f(x∗, y∗) ≤ ˜f ∗(x∗) and [˜g(x∗, y∗)]+ = 0, which along with
+(42) and (43) imply that f(x∗, y∗) = f ∗ and y∗ ∈ Argminz{ ˜f(x∗, z)|˜g(x∗, z) ≤ 0}. Hence, (x∗, y∗) is an
+optimal solution of (42) as desired.
+23
+
+We next prove Theorem 6. Before proceeding, we establish several technical lemmas below, which
+will be used to prove Theorem 6 subsequently.
+Lemma 10. Suppose that Assumptions 3 and 4 hold and that (xǫ, yǫ, zǫ) is an ǫ-stationary point of
+problem (50) for some ǫ > 0. Let Dy, ˜g, ρ, µ, Lf, L ˜
+f and G be given in (30), (42), (50) and Assumption
+4, respectively. Then we have
+∥[˜g(xǫ, zǫ)]+∥ ≤ (2µG)−1Dy(ρ−1ǫ + L ˜
+f),
+(101)
+∥[˜g(xǫ, yǫ)]+∥ ≤ (2µG)−1Dy(ρ−1ǫ + ρ−1Lf + L ˜
+f).
+(102)
+Proof. We first prove (101). Since (xǫ, yǫ, zǫ) is an ǫ-stationary point of (50), it follows from Definition
+2 that dist(0, ∂zPρ,µ(xǫ, yǫ, zǫ)) ≤ ǫ. Also, by (47) and (50), one has
+Pρ,µ(x, y, z) = f(x, y) + ρ( ˜f(x, y) + µ ∥[˜g(x, y)]+∥2) − ρ( ˜f(x, z) + µ ∥[˜g(x, z)]+∥2).
+(103)
+Using these relations, we have
+dist
+�
+0, ∂z ˜f(xǫ, zǫ) + 2µ
+l
+�
+i=1
+[˜gi(xǫ, zǫ)]+∇z˜gi(xǫ, zǫ)
+�
+≤ ρ−1ǫ.
+Hence, there exists s ∈ ∂z ˜f(xǫ, zǫ) such that
+���s + 2µ
+l
+�
+i=1
+[˜gi(xǫ, zǫ)]+∇z˜gi(xǫ, zǫ)
+��� ≤ ρ−1ǫ.
+(104)
+Let ˆzxǫ and G be given in Assumption 4(iii). It then follows that ˆzxǫ ∈ Y and −˜gi(xǫ, ˆzxǫ) ≥ G > 0 for
+all i. Notice that [˜gi(xǫ, zǫ)]+˜gi(xǫ, zǫ) ≥ 0 for all i and ∥zǫ − ˆzxǫ∥ ≤ Dy due to (30). Using these, (104),
+and the convexity of ˜f(xǫ, ·) and ˜gi(xǫ, ·) for all i, we have
+˜f(xǫ, zǫ) − ˜f(xǫ, ˆzxǫ) + 2µG
+l
+�
+i=1
+[˜gi(xǫ, zǫ)]+ ≤ ˜f(xǫ, zǫ) − ˜f(xǫ, ˆzxǫ) − 2µ
+l
+�
+i=1
+[˜gi(xǫ, zǫ)]+˜gi(xǫ, ˆzxǫ)
+≤ ˜f(xǫ, zǫ) − ˜f(xǫ, ˆzxǫ) + 2µ
+l
+�
+i=1
+[˜gi(xǫ, zǫ)]+(˜gi(xǫ, zǫ) − ˜gi(xǫ, ˆzxǫ))
+≤ ⟨s, zǫ − ˆzxǫ⟩ + 2µ
+l
+�
+i=1
+[˜gi(xǫ, zǫ)]+⟨∇z˜gi(xǫ, zǫ), zǫ − ˆzxǫ⟩
+= ⟨s + 2µ
+l
+�
+i=1
+[˜g(xǫ, zǫ)]+∇z˜gi(xǫ, zǫ), zǫ − ˆzxǫ⟩ ≤ ρ−1Dyǫ,
+(105)
+where the first inequality is due to −˜gi(xǫ, ˆzxǫ) ≥ G for all i, the second inequality follows from
+[˜gi(xǫ, zǫ)]+˜gi(xǫ, zǫ) ≥ 0 for all i, the third inequality is due to s ∈ ∂z ˜f(xǫ, zǫ) and the convexity of
+˜f(xǫ, ·) and ˜gi(xǫ, ·) for all i, and the last inequality follows from (30) and (104). In view of (30), (105),
+and L ˜
+f-Lipschitz continuity of ˜f(x, y) (see Assumption 4), one has
+∥[˜g(xǫ, zǫ)]+∥ ≤
+l
+�
+i=1
+[˜gi(xǫ, zǫ)]+
+(105)
+≤
+(2µG)−1(ρ−1Dyǫ + ˜f(xǫ, ˆzxǫ) − ˜f(xǫ, zǫ))
+≤ (2µG)−1(ρ−1Dyǫ + L ˜
+f∥ˆzxǫ − zǫ∥)
+(30)
+≤ (2µG)−1Dy(ρ−1ǫ + L ˜
+f).
+Hence, (101) holds.
+We next prove (102). Since (xǫ, yǫ, zǫ) is an ǫ-stationary point of (50), it follows from Definition 2
+that dist(0, ∂yPρ,µ(xǫ, yǫ, zǫ)) ≤ ǫ. This together with (103) implies that
+dist
+�
+0, ∂yf(xǫ, yǫ) + ρ∂y ˜f(xǫ, yǫ) + 2ρµ∇y˜g(xǫ, yǫ)[˜g(xǫ, yǫ)]+
+�
+≤ ǫ.
+Hence, there exists s ∈ ∂yf(xǫ, yǫ) and ˜s ∈ ∂y ˜f(xǫ, yǫ) such that
+∥s + ρ˜s + 2ρµ∇y˜g(xǫ, yǫ)[˜g(xǫ, yǫ)]+∥ ≤ ǫ.
+(106)
+24
+
+Let ¯
+A(xǫ, yǫ) = {i|˜gi(xǫ, yǫ) > 0, 1 ≤ i ≤ l}, ˆzxǫ and G be given in Assumption 4(iii). It then follows
+that ˆzxǫ ∈ Y and −˜gi(xǫ, ˆzxǫ) ≥ G > 0 for all i. Using these and the convexity of ˜gi(xǫ, ·) for all i, we
+have
+⟨∇y˜g(xǫ, yǫ)[˜g(xǫ, yǫ)]+, yǫ − ˆzxǫ⟩ =
+�
+i∈ ¯
+A(xǫ,yǫ)
+⟨∇y˜gi(xǫ, yǫ), yǫ − ˆzxǫ⟩[gi(xǫ, yǫ)]+
+≥
+�
+i∈ ¯
+A(xǫ,yǫ)
+(˜gi(xǫ, yǫ) − ˜gi(xǫ, ˆzxǫ))[˜gi(xǫ, yǫ)]+
+≥
+�
+i∈ ¯
+A(xǫ,yǫ)
+G[˜gi(xǫ, yǫ)]+ = G
+l
+�
+i=1
+[˜gi(xǫ, yǫ)]+ ≥ G ∥[˜g(xǫ, yǫ)]+∥ ,
+(107)
+where the first inequality follows from the convexity of ˜g(xǫ, ·) and the second inequality is due to
+−˜gi(xǫ, ˆzxǫ) ≥ G. It then follows from this, (106) and (107) that
+Dyǫ ≥ ∥s + ρ˜s + 2ρµ∇y˜g(xǫ, yǫ)[˜g(xǫ, yǫ)]+∥ · ∥yǫ − ˆzxǫ∥
+≥ ⟨s + ρ˜s + 2ρµ∇y˜g(xǫ, yǫ)[˜g(xǫ, yǫ)]+, yǫ − ˆzxǫ⟩
+= ��s + ρ˜s, yǫ − ˆzxǫ⟩ + 2ρµ⟨∇y˜g(xǫ, yǫ)[˜g(xǫ, yǫ)]+, yǫ − ˆzxǫ⟩
+(107)
+≥ − (∥s∥ + ρ∥˜s∥) ∥yǫ − ˆzxǫ∥ + 2ρµG ∥[˜g(xǫ, yǫ)]+∥
+≥ −(Lf + ρL ˜
+f)Dy + 2ρµG ∥[˜g(xǫ, yǫ)]+∥ ,
+(108)
+where the last inequality follows from ∥yǫ − ˆzxǫ∥ ≤ Dy and the fact that ∥s∥ ≤ Lf and ∥˜s∥ ≤ L ˜
+f, which
+are due to (30), s ∈ ∂yf(xǫ, yǫ), ˜s ∈ ∂y ˜f(xǫ, yǫ) and Assumption 4(i). By (108), one can immediately see
+that (102) holds.
+Lemma 11. Suppose that Assumptions 3 and 4 hold. Let f, ˜f, ˜g, Dy, flow, ˜f ∗ and Pρ,µ be given in (29),
+(30), (32), (43) and (50), Lf, L ˜
+f and G be given in Assumptions 3 and 4, (xǫ, yǫ, zǫ) be an ǫ-stationary
+point of (50) for some ǫ > 0, and
+˜λ = 2µ[˜g(xǫ, zǫ)]+,
+ˆλ = 2ρµ[˜g(xǫ, yǫ)]+.
+(109)
+Then we have
+dist
+�
+∂f(xǫ, yǫ) + ρ∂ ˜f(xǫ, yǫ) − ρ(∇x ˜f(xǫ, zǫ) + ∇x˜g(xǫ, zǫ)˜λ; 0) + ∇˜g(xǫ, yǫ)ˆλ
+�
+≤ ǫ,
+(110)
+dist
+�
+0, ρ(∂z ˜f(xǫ, zǫ) + ∇z˜g(xǫ, zǫ)˜λ)
+�
+≤ ǫ,
+(111)
+∥[˜g(xǫ, zǫ)]+∥ ≤ (2µG)−1Dy(ρ−1ǫ + L ˜
+f),
+(112)
+|⟨˜λ, ˜g(xǫ, zǫ)⟩| ≤ (2µ)−1G−2D2
+y(ρ−1ǫ + L ˜
+f)2,
+(113)
+| ˜f(xǫ, yǫ) − ˜f ∗(xǫ)| ≤ max
+�
+ρ−1(max
+z
+Pρ,µ(xǫ, yǫ, z) − flow), (2µ)−1G−2D2
+yL ˜
+f(ρ−1ǫ + ρ−1Lf + L ˜
+f)
+�
+,
+(114)
+∥[˜g(xǫ, yǫ)]+∥ ≤ (2µG)−1Dy(ρ−1ǫ + ρ−1Lf + L ˜
+f),
+(115)
+|⟨ˆλ, ˜g(xǫ, yǫ)⟩| ≤ (2µ)−1ρG−2D2
+y(ρ−1ǫ + ρ−1Lf + L ˜
+f)2.
+(116)
+Proof. Since (xǫ, yǫ, zǫ) is an ǫ-stationary point of (50), it easily follows from (103), (109) and Definition
+2 that (110) and (111) hold. Also, it follows from (101) and (102) that (112) and (115) hold. In addition,
+in view of (109), (112) and (115), one has
+|⟨˜λ, ˜g(xǫ, zǫ)⟩|
+(109)
+=
+2µ ∥[˜g(xǫ, zǫ)]+∥2 (112)
+≤
+(2µ)−1G−2D2
+y(ρ−1ǫ + L ˜
+f)2,
+|⟨ˆλ, ˜g(xǫ, yǫ)⟩|
+(109)
+=
+2ρµ ∥[˜g(xǫ, yǫ)]∥+∥2 (115)
+≤
+(2µ)−1ρG−2D2
+y(ρ−1ǫ + ρ−1Lf + L ˜
+f)2,
+and hence (113) and (116) hold. Also, observe from the definition of Pρ,µ in (50) that
+�Pµ(xǫ, yǫ) − min
+z
+�Pµ(xǫ, z) = ρ−1(max
+z
+Pρ,µ(xǫ, yǫ, z) − f(xǫ, yǫ)).
+25
+
+Using this, (32), (47) and (93), we obtain that
+˜f(xǫ, yǫ) + µ ∥[˜g(xǫ, yǫ)]+∥2 (47)
+=
+�Pµ(xǫ, yǫ) =
+min
+z
+�Pµ(xǫ, z) + ρ−1(max
+z
+Pρ,µ(xǫ, yǫ, z) − f(xǫ, yǫ))
+(32)(93)
+≤
+˜f ∗(xǫ) + ρ−1(max
+z
+Pρ,µ(xǫ, yǫ, z) − flow).
+(117)
+On the other hand, let λ∗ ∈ Rl
++ be an optimal Lagrangian multiplier of problem (43) with x = xǫ. It
+then follows from Lemma 6(i) that ∥λ∗∥ ≤ G−1L ˜
+fDy. Using these and (115), we have
+˜f ∗(xǫ) = min
+y
+�
+˜f(xǫ, y) + ⟨λ∗, ˜g(xǫ, y)⟩
+�
+≤ ˜f(xǫ, yǫ) + ⟨λ∗, ˜g(xǫ, yǫ)⟩
+≤ ˜f(xǫ, yǫ) + ∥λ∗∥∥[˜g(xǫ, yǫ)]+∥ ≤ ˜f(xǫ, yǫ) + (2µ)−1G−2D2
+yL ˜
+f(ρ−1ǫ + ρ−1Lf + L ˜
+f).
+By this and (117), one can see that (114) holds.
+We are now ready to prove Theorem 6.
+Proof of Theorem 6. Observe from (51) that problem (50) can be viewed as
+min
+x,y max
+z
+{Pρ,µ(x, y, z) = h(x, y, z) + p(x, y) − q(z)} ,
+where h(x, y, z) = f1(x, y) + ρ ˜f1(x, y) + ρµ ∥[˜g(x, y)]+∥2 − ρ ˜f1(x, z) − ρµ ∥[˜g(x, z)]+∥2, p(x, y) = f2(x) +
+ρ ˜f2(y) and q(z) = ρ ˜f2(z). Hence, problem (50) is in the form of (6) with H = Pρ,µ. By Assumption 3,
+(45), (46), ρ = ε−1 and µ = ε−2, one can see that h is �L-smooth on its domain, where �L is given in (61).
+Also, notice from Algorithm 6 that ǫ0 = ε5/2 ≤ ε/2 = ǫ/2 due to ε ∈ (0, 1/4]. Consequently, Algorithm 2
+can be suitably applied to problem (50) with ρ = ε−1 and µ = ε−2 for finding an ǫ-stationary point
+(xǫ, yǫ, zǫ) of it.
+In addition, notice from Algorithm 6 that �Pµ(x0, y0) ≤ miny �Pµ(x0, y) + ε. Using this, (50) and
+ρ = ε−1, we obtain
+max
+z
+Pρ,µ(x0, y0, z)
+(50)
+= f(x0, y0) + ρ( �Pµ(x0, y0) − min
+z
+�Pµ(x0, z)) ≤ f(x0, y0) + ρε = f(x0, y0) + 1. (118)
+By this and (28) with H = Pρ,µ, ǫ = ε, ǫ0 = ε5/2, ˆx0 = (x0, y0), Dq = Dy and L∇h = �L, one has
+Pρ,µ(xǫ, yǫ, zǫ) ≤
+max
+z
+Pρ,µ(x0, y0, z) + εDy/4 + 2ε5(�L−1 + 4D2
+y�Lε−2)
+(118)
+≤
+1 + f(x0, y0) + εDy/4 + 2ε5(�L−1 + 4D2
+y�Lε−2).
+It then follows from this and Lemma 11 with ǫ = ε, ρ = ε−1 and µ = ε−2 that (xǫ, yǫ, zǫ) satisfies the
+relations (62)-(68).
+We next show that at most �
+N evaluations of ∇f1, ∇ ˜f1, ∇˜g and proximal operator of f2 and ˜f2 are
+respectively performed in Algorithm 6. Indeed, by (31), (32), (45), (47) and (50), one has
+min
+x,y max
+z
+Pρ,µ(x, y, z)
+(50)
+= min
+x,y {f(x, y) + ρ( �Pµ(x, y) − min
+z
+�Pµ(x, z))} ≥
+min
+(x,y)∈X ×Y f(x, y)
+(32)
+= flow, (119)
+min{Pρ,µ(x, y, z)|(x, y, z) ∈ X × Y × Y}
+(50)
+= min{f(x, y) + ρ( �Pµ(x, y) − �Pµ(x, z))|(x, y, z) ∈ X × Y × Y}
+(47)
+= min{f(x, y) + ρ( ˜f(x, y) + µ∥[˜g(x, y)]+∥2 − ˜f(x, z) − µ∥[˜g(x, z)]+∥2)|(x, y, z) ∈ X × Y × Y}
+≥ flow + ρ( ˜flow − ˜fhi) − ρµ˜g2
+hi,
+(120)
+where the last inequality follows from (31), (32) and (45). In addition, let (x∗, y∗) be an optimal solution
+of (42). It then follows that f(x∗, y∗) = f ∗ and [˜g(x∗, y∗)]+ = 0. By these, (31), (47) and (50), one has
+min
+x,y max
+z
+Pρ,µ(x, y, z) ≤ max
+z
+Pρ,µ(x∗, y∗, z)
+(50)
+= f(x∗, y∗) + ρ
+�
+�Pµ(x∗, y∗) − min
+z
+�Pµ(x∗, z)
+�
+(47)
+= f(x∗, y∗) + ρ( ˜f(x∗, y∗) + µ∥[˜g(x∗, y∗)]+∥2 − min
+z { ˜f(x∗, z) + µ∥[˜g(x∗, z)]+∥2})
+(31)
+≤ f ∗ + ρ( ˜fhi − ˜flow).
+(121)
+26
+
+For convenience of the rest proof, let
+H = Pρ,µ,
+H∗ = min
+x,y max
+z
+Pρ,µ(x, y, z),
+Hlow = min{Pρ,µ(x, y, z)|(x, y, z) ∈ X × Y × Y}.
+(122)
+In view of these, (118), (119), (120), (121), ρ = ε−1 and µ = ε−2, we obtain that
+max
+z
+H(x0, y0, z)
+(118)
+≤ f(x0, y0) + 1,
+flow
+(119)
+≤
+H∗ (121)
+≤
+f ∗ + ρ( ˜fhi − ˜flow) = f ∗ + ε−1( ˜fhi − ˜flow),
+Hlow
+(120)
+≥
+flow + ρ( ˜flow − ˜fhi) − ρµ˜g2
+hi = flow + ε−1( ˜flow − ˜fhi) − ε−3˜g2
+hi.
+Using these and Theorem 2 with ǫ = ε, ˆx0 = (x0, y0), Dp =
+�
+D2x + D2y, Dq = Dy, ǫ0 = ε5/2, L∇h = �L,
+α = ˜α, δ = ˜δ, and H, H∗, Hlow given in (122), we can conclude that Algorithm 6 performs at most �
+N
+evaluations of ∇f1, ∇ ˜f1, ∇˜g and proximal operator of f2 and ˜f2 for finding an approximate solution
+(xǫ, yǫ) of problem (42) satisfying (62)-(68).
+6
+Concluding remarks
+For the sake of simplicity, first-order penalty methods are proposed only for problem (3) in this paper.
+It would be interesting to extend them to problem (1) by using a standard technique (e.g., see [39]) for
+handling the constraint g(x, y) ≤ 0. In addition, a single subproblem with static penalty and tolerance
+parameters is solved in our methods (Algorithms 4 and 6), which may be conservative in practice. To
+make the methods possibly practically more efficient, it would be natural to modify them by solving
+a sequence of subproblems with dynamic penalty and tolerance parameters instead. These along with
+numerical experiments will be left for the future research.
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+

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+	
+Machine learning prediction of the MJO extends beyond one month 
+Tamaki Suematsu1*, Kengo Nakai2*, Tsuyoshi Yoneda3, Daisuke Takasuka4,5, Takuya Jinno6,3, 
+Yoshitaka Saiki7, Hiroaki Miura6 
+ 
+1RIKEN Center for Computational Science; Kobe, Hyogo, 650-0047, Japan. 
+*Tamaki Suematsu. E-mail: tamaki.suematsu@riken.jp 
+2Faculty of Marine Technology, Tokyo University of Marine Science and Technology; Tokyo 135-8533, 
+Japan. 
+*Kengo Nakai. E-mail: knakai0@kaiyodai.ac.jp 
+3Graduate School of Economics, Hitotsubashi University; Kunitachi, Tokyo, 186-8601; Japan. 
+4Atmosphere and Ocean Research Institute, The University of Tokyo; Kashiwa, Chiba, 277-0882, Japan. 
+5Japan Agency for Marine-Earth Science and Technology; Yokohama, Kanagawa, 236-0001, Japan. 
+6Graduate School of Science, The University of Tokyo; Bunkyo-ku, Tokyo, 113-0033, Japan. 
+7Graduate School of Business Administration, Hitotsubashi University; Kunitachi, Tokyo, 186-8601, 
+Japan. 
+*These authors contributed equally to this work 
+ Abstract 
+The prediction of the Madden-Julian Oscillation (MJO), a massive tropical weather 
+event with vast global socio-economic impacts1,2, has been infamously difficult with 
+physics-based weather prediction models3–5. Here we construct a machine learning 
+model using reservoir computing technique that forecasts the real-time multivariate 
+MJO index (RMM)6, a macroscopic variable that represents the state of the MJO. 
+The training data was refined by developing a novel filter that extracts the 
+recurrency of MJO signals from the raw atmospheric data and selecting a suitable 
+
+2	
+time-delay coordinate of the RMM. The model demonstrated the skill to forecast 
+the state of MJO events for a month from the pre-developmental stages. Best-
+performing cases predicted the RMM sequence over two months, which exceeds the 
+expected inherent predictability limit of the MJO.   
+ 
+Main text 
+The Madden–Julian Oscillation (MJO) 7 is a massive cluster of convective activities in the tropics that 
+spans thousands of kilometers traveling slowly eastward from the Indian Ocean to the central Pacific in 
+approximately 20 to 60 days. It has far reaching influence on the global weather 1,8 and is recognized to 
+be one of the most important sources of predictability in extended-range weather forecast longer than 
+weeks 9,10. However, simulation of the MJO by physics-based dynamical numerical models (hereafter 
+dynamical models) has been shown to be notoriously difficult 3,11,12. It has only been since the mid 2000s 
+that the predictability of the MJO by dynamical models 13,14 exceeded that of empirical statistical models, 
+such as atmospheric-only linear inverse models, at two to three weeks 15. The current forecast skills of 
+dynamical models for MJO prediction lie between two to five weeks 16, which falls short of the expected 
+inherent predictability of the MJO estimated from multi-model ensemble studies at six to seven weeks 17. 
+Weather forecasts by dynamical models require physical parameterizations that incorporate the mean 
+effects of the sub-grid scale processes on the evolution of the grid-scale flows. However, parameterizations 
+are prone to empirical tuning and are inevitable sources of uncertainty of the dynamical models 18–20 
+because we have yet to determine the correct theoretical formulations and parameters of the statistical mean 
+states of microscopic processes. The difficulties of reducing the ambiguities in parameterizations continue 
+to be a major limiting factor for improving dynamical models and for identifying processes essential for 
+successful weather predictions. In contrast, machine learning models with relatively small number of neural 
+networks trained only by the time series of macroscopic variables has the potential to implicitly incorporate 
+
+3	
+the influence of microscopic variables on the macroscopic variables and to eliminate the parameterizations 
+that unsatisfactorily replicate the multiscale interactions between the unresolved and the resolved processes.  
+The effectiveness of the machine learning methods has been demonstrated in the fields of atmospheric 
+and climate science. Considerable progress has been achieved in areas for forecasting phenomena with 
+large socio-economic impacts such as the El Niño Southern Oscillation 21,22, Asian summer monsoons 23, 
+and hurricanes 24 at incomparably small computational costs compared to the dynamical models of the 
+atmosphere and the ocean. However, phenomena at the intraseasonal time scale have been difficult to 
+predict with the use of machine learning methods because complex interactions between processes with 
+various spatio-temporal scales that range from the convective to seasonal scales play important role in 
+determining its time evolution. Particularly with regards to the MJO forecasts, machine learning models 
+have been outperformed by dynamical models 25,26. 
+Here, we employ the reservoir computing method to advance the machine learning prediction of the MJO. 
+The reservoir computing method is a brain-inspired machine-learning technique that constructs a data-
+driven dynamical model (hereafter reservoir computing models) 27–31. By training on a time series of 
+macroscopic variables of high-dimensional dynamics, the method can efficiently predict time series and 
+frequency spectra of its chaotic behaviors 32,33. For example, it is useful for predicting the statistical 
+quantities of a chaotic fluid flow, which cannot be calculated directly from a numerical simulation of the 
+Navier–Stokes equation due to its high computational cost 34. In this study, we construct a reservoir 
+computing MJO prediction model, trained only by the time series of a macroscopic variable, with a 
+performance competitive with the state-of-the-art physics based dynamical models. Our results 
+demonstrate that the inherent predictability of some MJO cases is longer than have been expected from 
+studies by dynamical models 17.   
+
+4	
+ 
+Fig. 1. Schematic picture of reservoir computing. (A) In the training phase, the input data 𝒖(𝑡)	for time 
+𝑡 is fed to the reservoir state vector 𝒓(𝑡) through input weight matrix 𝑾!" and the output weight matrix 
+𝑾#$%	 is determined by reservoir computing. (B) In the prediction phase, the time evolution of 𝒖(𝑡) for 
+time 𝑡 + Δ	𝑡 is predicted as 𝒖*(𝑡 + Δ	𝑡) from the 𝑾∗
+#$% determined in the training phase.  
+A reservoir is a recurrent neural network whose internal parameters are adjusted to fit the data in the 
+training process 27,28. It is trained by feeding an input time series and fitting a linear function of the high 
+dimensional reservoir state vector to the desired output time series (Fig. 1). The construction of a reservoir 
+computing model simply assumes recurrent and deterministic property of the input time series and does 
+not involve any physical knowledge of the input data. The reservoir computing model of this study is 
+described by: 
+
+ATraining phase
+Reservoir
+statevector
+r(t)
+Training
+Input data
+Outputdata
+()n
+(+)n
+Win
+r(t+△t)
+W
+out
+B Prediction phase
+Reservoir
+statevector
+r(t
+Predicted
+Input data
+outputdata
+u(t)
+(+)
+Determined
+Win
+r(t+△t)
+W
+Nout5	
++𝒓(𝑡 + Δ	𝑡) = (1 − 𝛼)	𝒓(𝑡) + 𝛼 tanh4𝑨	𝒓	(𝑡) + 𝑾!"	𝒖(𝑡)6	
+𝒖*(𝑡 + Δ	𝑡) = 𝑾∗
+#$%	𝒓(𝑡 + Δ	𝑡)
+				
+(1) 
+where 𝒖(𝑡) ∈	ℝ(	  is both the input variable vector, 𝒓(𝑡) ∈	ℝ)	(𝑁 ≫ 𝑀)	 is the reservoir state vector, 
+𝑨 ∈	ℝ)×) , 𝑾!"	 ∈	ℝ)×(  , and 𝑾∗
+#$% ∈	ℝ(×)   are reservoir, input, and output weight matrices, 
+respectively, 𝛼	(0 < 𝛼 ≤ 1)		is the coefficient that adjusts the nonlinearity of the dynamics of 𝒓, and Δ	𝑡 
+is the time step. We define tanh(𝐪) = (tanh(q+) , tanh(q,) , … tanh(q)))- , for a vector 𝐪 =
+(q+, q,, … q))-	, where 𝑇 represents the transpose of a vector. 𝑾∗
+#$% is determined to satisfy 𝒖(𝑡) ≈
+𝑾#$%		𝒓(𝑡) using the training data 𝒖(𝑡), where 𝑾#$%	is the output weight matrix in the training phase. 
+Further details on the construction of the reservoir computing model are provided in the supplementary 
+materials. In the prediction phase, the predicted variable 𝒖*(𝑡 + Δ	𝑡) is obtained from 𝒖(𝑡) and 𝒓	(𝑡), using 
+eqn. (1) with fixed 𝑨, 𝑾!"	, and 𝑾∗#$%. A successful training will give 𝒖*(𝑡 + Δ	𝑡) that approximates the 
+desired unmeasured quantity 𝒖(𝑡 + Δ	𝑡).  
+The objective of our reservoir computing model is to predict the sequence of the Realtime Multi-variate 
+MJO (RMM) index 6, which is widely accepted as the standard proxy for diagnosing the state of an MJO1. 
+It captures the signals of the MJO as an envelope of convective activities coupled to planetary-scale 
+circulation from the leading pair of principal components (RMM1, RMM2) of the equatorial outgoing 
+longwave radiation and zonal winds at 850 hPa and 200 hPa. The RMM calculated from data without 
+smoothing in time 6 has been applied to machine learning prediction of the MJO 25,26; however, their 
+machine learning predictions were susceptible to degradation ascribed to noises in unsmoothed data from 
+atmospheric variabilities outside of the MJO timescale. Moreover, signals at time scales longer than the 
+MJO needs to be removed from the training data for the machine learning to identify recurring patterns. 
+Thus, to refine the RMM time series to train our reservoir computing model for MJO prediction, signals 
+outside of the MJO frequency range were removed from the raw data by an application of a filter that 
+approximately retains signals only between 20 days and 120 days frequency range 35. 
+
+6	
+The Lanczos filter 36, which is conventionally used to filter MJO signals, cannot be employed as the filter 
+to generate the training data for the machine learning. This is because the Lanczos filter, whose weights 
+are symmetric in time, requires data from both the past and the future to calculate a filtered value at a 
+certain point in time (Fig. 2 A). To resolve this problem, we design a novel filter, applicable for real-time 
+use, that does not require data from the future. The filter Ψ.!,.",0 is defined as: 
+Ψ.!,.",0	(𝑡)	= F.!,."	(𝑡)	
+sin(𝑡
+𝑐 − 𝜋)	
+(𝑡
+𝑐 − 𝜋)
+,	 
+where 
+F.!,."	(𝑡)	=		K
+sin L 𝑡
+𝑟1N
+𝑡
+	−	
+sin L 𝑡
+𝑟2N
+𝑡
+						(𝑡 ≤ 0)
+																		0																								(𝑡 > 0)
+				,										 
+and 𝑐 is a parameter that adjusts the center of the weights. We set the parameters as (𝑟1, 𝑟2, 𝑐) = (
+,3
+4 ,
++,3
+4 , 14) to remove the signals at frequencies lower than 120 days and higher than 20 days. The shape of 
+the filter function in real-space and in Fourier space is compared against that of the Lanczos filter in Fig. 2 
+A, B. In contrast to the Lanczos filter, the weights of the new filter vanish at 𝑡 = 0 and require only the 
+data from the past. The asymmetric weights of the new filter make it suitable for its application to real-
+time use such as filtering the input variable data for machine learning predictions. Due to the asymmetry, 
+the center of the weight of the new filter shifts backward only by approximately eight days. Hereafter, this 
+filter will be referred to as the real-time band-pass filter (RB filter). The RMM time series is calculated 
+from data filtered by the RB filter in this study (see methods for details). 
+
+7	
+ 
+Fig. 2. Comparison of Real-time Band-pass filter and Lanczos filter. The shape of the Lanczos (red) 
+and RB (blue) filters are shown in (A) real space and in (B) Fourier space. (C) Sample trajectories, from 
+1st December 2018 (indicated by circles) to 9th January 2019, for the original Wheeler and Hendon 2004 
+RMM index (WH04, grey), and RMM index filtered by Lanczos (red) and RB (blue) filters and RMM2 
+replaced by 12-day time-delay coordinate of RMM1. The axis for both RMM2 and 12-day time-delay 
+coordinate of RMM1 is labeled as RMM2 for brevity.  
+Furthermore, the MJO prediction is refined by employing the RMM phase space spanned by RMM1 and 
+its delay coordinate to diagnose the state of the MJO. That is, we replace RMM2 with the delay coordinate 
+of RMM1 and eliminate the model prediction of RMM2. This enhances the recurrency of the input data 
+and contributes to the robustness of the trained model. The modification is founded on the expectation that 
+RMM2 can be reconstructed from the delay coordinate of RMM1, since RMM1 and RMM2 are orthogonal 
+by definition and the trajectories of the projections of MJO events on the RMM phase space are near 
+circular. The delay time of the delay coordinate is chosen at 12 days, when the auto-correlation of RMM1 
+crosses zero for the first time. The correlation coefficient of RMM2 and 12-day delay coordinate variable 
+of RMM1 is 0.75. The trajectories of the RB-filtered and Lanczos filtered RMM sequences with RMM2 
+replaced by the 12-day delay coordinate of RMM1 is compared with the original Wheeler and Hendon 
+RMM (WH04) 6 in Fig. 2C. We confirm that the RB filter removes signals from slow variabilities and 
+noises as effectively as the Lanczos filter and that the trajectory of the RMM sequence on the phase space 
+with RMM2 replaced by the 12-day delay coordinate of RMM1 is similar to that of the WH04 RMM on 
+phase space spanned by RMM1 and RMM2. Thus, we focus on the RMM phase space spanned by RMM1 
+
+8	
+and its 12-day delay coordinate, which we will refer to as the machine learning RMM (ML-RMM) phase 
+space. The relevance of ML-RMM phase space to the conventional one spanned by RMM1 and RMM2 is 
+further discussed in the supplementary materials (Fig. S1). We will denote RMM1 and its time-delay 
+coordinate at time 𝑡 as 𝑎(𝑡) ≔ RMM1(𝑡) and 𝑏(𝑡): = 𝑎(𝑡 − 12).  
+It is known that a chaotic attractor can be reconstructed by some observable variables and its delay 
+coordinates 37,38. For the construction of a reservoir computing model, it is efficient to take the delay 
+coordinate variable with an appropriate delay time as the input when the number of observable variable is 
+smaller than the effective dimension of the attractor 33. A suitable delay time and dimension of the delay 
+coordinate of RMM1 is inferred by computing its auto-correlation function. Thus, an M-dimensional 
+delay coordinate vector of RMM1 is introduced as the input and output variable vector 𝒖 in Eq. (1): 
+𝒖(𝑡) = 4RMM1(𝑡), RMM1(𝑡 − 1Δ	𝜏), … , RMM1(𝑡 − (𝑀 − 1)Δ	𝜏)6,	 
+where ∆τ is the delay time, and (Δ	𝜏, 𝑀) = (6, 7). The pair of parameters are chosen so that the behavior 
+of 𝒖(𝑡) is deterministic and has recurrency, which are essential properties for successful modelling. The 
+reservoir model (Eq. (1)) of the RMM1 time sequence is constructed by determining 𝑾∗
+#$%. The time series 
+of the RMM1 data from 30th December 1986 to 29th December 2011 was used as the training data. The 
+optimal reservoir computing model was selected from evaluation of test cases of RMM1 forecasts 
+initialized from every day between 8th April 2014 and 6th July 2014. The selected model is used throughout 
+this study for all predictions. 
+The predicted variables at time t initialized from time p are denoted as 𝑎\(𝑡, 𝑝) and 𝑏^(𝑡, 𝑝). We note that 
+𝑏^(𝑡, 𝑝) is predicted by the reservoir computing model simultaneously with 𝑎\(𝑡, 𝑝). The relationship 
+𝑏^(𝑡, 𝑝) =	𝑎\(𝑡 − 12, 𝑝) would hold only in an ideal case in which the model learns the delay coordinate of 
+the RMM1 perfectly. The reference time series in this case are 𝑎(𝑡) and 𝑏(𝑡). We compare the time series 
+of predicted variables against the reference time series using the bivariate correlation coefficient (COR) 
+16,39, defined by the equation:  
+
+9	
+COR(𝑞) ≔	
+∑
+L𝑎(𝑝 + 𝑞)𝑎\(𝑝 + 𝑞, 𝑝) + 	𝑏(𝑝 + 𝑞)𝑏^(𝑝 + 𝑞, 𝑝)N
+)
+56+
+c∑
+((𝑎(𝑝 + 𝑞))𝟐 + 4𝑏(𝑝 + 𝑞)6
+𝟐)
+)
+56+
++	c∑
+((𝑎\(𝑝 + 𝑞, 𝑝))𝟐 + (𝑏^(𝑝 + 𝑞, 𝑝))𝟐)
+)
+56+
+			,	 
+where 𝑞 is the forecast lead time. The COR corresponds to a covariance between the actual vector 
+(𝑎(𝑡), 𝑏(𝑡)) and the predicted vector  (𝑎\(𝑡, 𝑝), 𝑏^(𝑡, 𝑝)), and is conventionally used to evaluate the MJO 
+prediction skills of dynamical and statistical models 40,41. Here, the 𝑁 = 2010 is the number of 
+predictions initialized for all days between 28th July 2014 and 28th January 2020. In Fig. 3, we show the 
+time series of COR(𝑞) for all predictions and three cases, the details of which will be described next. The 
+COR(𝑞) stays above 0.5 for 28 days for all predictions. The threshold value 0.5 is customarily adopted 
+for MJO prediction skill score 16. This signifies that the expectancy of the skill score of the model is at 
+three weeks for all days, including periods devoid of MJO activity. The forecast skill was evaluated as 
+three weeks in consideration of the approximate 8-day shift by the RB filter as discussed above.  
+It is customary to evaluate the skill of MJO predictions from the forecasts of periods when MJO events 
+are identified 14. We reevaluate the forecast skill of the reservoir model following the custom. Here, the 
+MJO events were identified as continuous sequences from phase 2 to phase 7 on the RMM phase space 
+spanned by RMM1 and its delay coordinate of 12 days 8,35 (See methods for details). For the predictions 
+initialized on three, five, and seven days before the onsets of MJO events, the COR remains above 0.5 for 
+38 days for all three cases. Considering the 8-day shift by the RB filter, this signifies that the constructed 
+model has the potential to skillfully forecast the time evolutions of the MJO events for 30 days. 
+Counterintuitively, the COR decays below 0.5 faster for the forecasts initialized three days before the 
+MJO onsets than those for five and seven days before the MJO onsets. We note however, that this is 
+consistent with the fact that the predictions reach the terminations of MJO events faster for predictions 
+that are initiated closer to the onsets.  
+The performance of the MJO prediction on individual cases are examined to illuminate the similarity 
+between the predicted and the actual trajectories of the RMM. Figure 4 compares the actual and predicted 
+
+10	
+trajectories on the ML-RMM phase, prediction errors, and the phase difference for the 10th (A, B, C), 26th 
+(D, E, F), and 50th (G, H, I) best performing cases in terms of mean error over the first 60 days of the 
+prediction. The three samples are chosen so that there are no overlaps in the forecast lead times. The errors 
+are measured by the distance between the actual (𝑎(𝑡), 𝑏(𝑡)) and the predicted  (𝑎\(𝑡, 𝑝), 𝑏^(𝑡, 𝑝)) vectors. 
+The phase difference is evaluated from the cosine of the angle between (𝑎(𝑡), 𝑏(𝑡)) and (𝑎\(𝑡, 𝑝), 𝑏^(𝑡, 𝑝)) 
+(cos(𝜃5(𝑡))). In all three cases, the error remains below 1.4, the threshold of the root mean square errors 
+of the predicted RMM adopted to evaluate the skills of climate simulations 42, well beyond two months (>
+75 days). The prediction also stays in phase (cos(𝜃5(𝑡)) > 0.7) for nearly two months (58, 83, and 76 
+days for the 10th, 26th, and 50th best case, respectively). We note that the rapid increases in phase differences 
+occur when the amplitude of the RMM1 decreases. This is reasonable considering that the RMM phases 
+become physically meaningless with diminishment of its amplitude. These results indicate that our 
+reservoir computing model can predict the state of some MJO events well beyond the estimated inherent 
+predictability limit of 7 weeks from dynamical model studies 17. This inference is supported by cases of 
+RMM1 predictions that skillfully forecast RMM1 phases for longer than 120 days (see Fig. S2).  
+
+11	
+ 
+  
+Fig. 3. Bivariate correlation coefficient. The mean bivariate correlation coefficient (COR) as a function 
+of forecast lead time (days) for all 2010 predictions (red), and for predictions initialized on 3 (navy), 5 
+(blue), and 7 (light blue) days before onsets of MJO events. The dash-dot and the dotted lines indicate 28th 
+and the 38th day in the forecast lead time, respectively.   
+We constructed a computationally inexpensive machine learning model, using the reservoir computing 
+technique, that is capable of month-long forecasts of the state of the MJO. This prediction skill is superior 
+to that of preceding machine learning methods and is matched only by physics-based dynamical models 
+that inevitably demand the state-of-the-art supercomputers 13,14,40,43. It is remarkable that our model was 
+trained only by the macroscopic time series of the RMM1. This signifies that intricate information of the 
+atmospheric and oceanic states that influences the MJO 44,45 were implicitly incorporated into the reservoir 
+state variables of the neural network. The extended prediction skill of our reservoir model is attributed to 
+the refinement of the training data. The signals from slow variability and high frequency noise were filtered 
+out from the input data with the RB filter to restrict the degrees of freedom of the training. This was 
+
+12	
+essential because it was necessary for the model to efficiently learn from merely 26 years of RMM1 data 
+with a limited number (< 100) of MJO events. To further enhance the efficacy of the reservoir computing, 
+we introduced the delay coordinate variable of RMM1 to employ suitably correlated variables as our 
+training data 33. It is of interest how the extension of the training data with accumulation of observational 
+data in the future will enhance the performance of the reservoir model.  
+The best performing forecasts by our reservoir model predicted the RMM time series for more than two 
+months. These results indicate that some MJO events are inherently predictable beyond the potential 
+predictability estimates made from dynamical model studies at seven weeks 17. This implies a possibility 
+for significant improvements in dynamical models to extend their lead time in MJO prediction, which is 
+crucial for reliable global weather forecasts. However, observations suggest that global warming alters the 
+characteristics of the MJO 8, meaning that the applicability of machine learning models trained on historical 
+data for MJO predictions could be undermined by climate change in the future. Furthermore, the reservoir 
+model of this study can only forecast the RMM sequence and cannot directly assess the impact of the MJO 
+on the midlatitude weather. Thus, dynamical models are expected to continue to be an imperative tool for 
+predicting and understanding the behaviors of our atmosphere and it is important to make the efforts to 
+exploit machine learning weather predictions to advance the dynamical models. 
+ 
+
+13	
+ 
+Fig. 4. Samples of best performing cases of RMM1 predictions and their errors. The (A, B, C) 10th, 
+(D, E, F) 26th, and (G, H, I) 50th best performing cases of RMM1 predictions initialized from 19th June 
+2019, 14th April 2018, and 9th December 2015 (indicated by the red dots), respectively. (A, D, G) The 
+trajectories of the actual (red) and the predicted (blue) RMM1 (𝑎(𝑡)	and 𝑎\(𝑡, 𝑝)) and its time-delay 
+coordinate (𝑏(𝑡, 𝑝) and 𝑏^(𝑡, 𝑝)) are shown on the RMM phase space and (B, E, H) as a function of the 
+
+14	
+forecast lead time with the errors shown as the width of the gray shade. (C, F, I) The time series of 
+prediction errors measured by the cosines of the angles between  (𝑎(𝑡), 𝑏(𝑡)) and (𝑎\(𝑡, 𝑝), 𝑏^(𝑡, 𝑝)) 
+(cos(𝜃5(𝑡))	). The gray lines at ±0.7 in panels B, E, H indicate the threshold for the error = 1.4 and 
+cos(𝜃5(𝑡)) = 0.7 in panels C, F, I. 
+Acknowledgments 
+Funding: 	
+Japan Society for the Promotion of Science KAKENHI Grants, 21K13991 and 20H05730 (TS) 
+Japan Society for the Promotion of Science KAKENHI Grants, 22K17965 (KN) 
+Japan Science and Technology Agency PRESTO, 22724051(KN) 
+Japan Society for the Promotion of Science KAKENHI Grants, 20H01819 (TY) 
+Japan Society for the Promotion of Science KAKENHI Grants, 20H05728 and 22H01297 (DT) 
+MEXT Program for Promoting Researches on the Supercomputer Fugaku, hp210166 and hp220167 (DT) 
+Japan Society for the Promotion of Science KAKENHI Grants, 20J11246 (TJ)  
+Japan Society for the Promotion of Science KAKENHI Grants, 19KK0067 and 21K18584 (YS) 
+"Joint Usage/Research Center for Interdisciplinary Large-scale Information Infrastructures" in Japan, 
+jh210027and jh220007, (YS) 
+HPCI System Research Project, hp210072 (YS) 
+Japan Society for the Promotion of Science KAKENHI Grants, 16H04048 and 20H05729 (HM) 
+ 
+Author contributions: 
+Conceptualization: KN, TS, DT, TY, HM, YS 
+Methodology: KN, TY, TS, HM, YS 
+Investigation: KN, TS, DT, HM, YS 
+Visualization: TS, KN 
+
+15	
+Funding acquisition: TS, HM, KN, DT, TJ, TY, YS 
+Supervision: HM, YS 
+Writing – original draft: TS, KN, HM, YS  
+Writing – review and editing: TS, HM, KN, DT, TJ, TY, YS 
+ 
+Competing interests: Authors declare that they have no competing interests.  
+ 
+Data availability  
+NOAA-OLR data are available at https://www.psl.noaa.gov/data/gridded/data.olrcdr.interp.html . 
+NCEP-DOE reanalysis data for zonal wind data are available at 
+https://psl.noaa.gov/data/gridded/data.ncep.reanalysis2.html . 
+The original Wheeler and Hendon 2004 RMM time series are available at 
+http://www.bom.gov.au/climate/mjo/ .  
+ 
+Code availability 
+All source codes of our reservoir model, filter-function of the RB filter, input and output data of the 
+reservoir computing, and the list of MJO events will be provided via zenodo before publication of this 
+work. 
+ 
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+ 
+ 
+ 
+
+21	
+Methods  
+ 
+Reservoir computing technique  
+The method of determining the output weight matrix  𝑾∗
+#$%
+ of our reservoir computing machine learning 
+model is described. The time development of the reservoir state vector 𝒓(𝑙	Δ	𝑡) is determined by: 
+𝒓(𝑡 + Δ	𝑡) = (1 − 𝛼)	𝒓(𝑡) + 𝛼 tanh4𝑨	𝒓	(𝑡) + 𝑾!"	𝒖(𝑡)6	,
+(1 − M) 
+where {𝒖(𝑙	Δ	𝑡)}	(−𝐿3 ≤ 𝑙	 ≤ 𝐿) is the training time series data, 𝐿3 is the transient time, and 𝐿 is the time 
+length to determine 𝑾∗
+#$%
+ . For given random matrices  𝑨 and 𝑾!"	, we determine  𝑾#$%	 so that the 
+following quadratic form takes the minimum: 
+lm𝑾#$%		𝒓	(𝑙Δ𝑡) − 𝒖4(𝑙 + 1)Δ𝑡6m
+, + 𝛽[𝑇𝑟	(𝑾#$%	𝑾#$%	
+-
+)],
+8
+963
+	
+(2 − M) 
+where ‖𝒒‖, =	𝒒𝑻𝒒 for a vector 𝒒.  The minimizer is 
+𝑾∗
+#$% = 𝛿𝑼𝛿𝑹-(𝛿𝑹𝛿𝑹- + 𝛽	𝑰)2+		
+(3 − M) 
+where 𝑰 is the 𝑁 × 𝑁 identity matrix, 𝛿𝑹 and 𝛿𝑼 are the matrices whose 𝑙-th column is 𝒓	(𝑙Δ𝑡) and  
+𝒖	(𝑙Δ𝑡), respectively. (see Lukosevivcius and Jaeger (2009)46 P.140 and Tikhonov and Arsenin (1977)47 
+Chapter 1 for details).  
+Note that 𝑨 is chosen to have a maximum eigenvalue 𝜌	(|𝜌| < 1) in order for eqn. (2-M) to satisfy the 
+so called echo state property. It is known that addition of noises to the training time series data is potentially 
+useful in the construction of a data-driven model 22. Further details on the reservoir computing can be found 
+in preceding literatures 32–34,48. 
+The set of parameter values used to construct the reservoir computing model is shown in Table 1. We 
+determine 𝑾#$%	 using the training time series data 𝒖, which in this case is the RMM1 data from 30th 
+December 1986 (𝑡 = 0) to 29th December 2011 (𝑡 = 9131). The optimal reservoir computing model was 
+
+22	
+selected based on predictions of the RMM1 initialized every day between 8th April 2014 and 16th July 2014 
+by using 𝑾#$%	  for a given 𝑨  and 𝑾!"	 . We selected a model with the smallest prediction error 
+max
+;∈[+,>]|𝑢+(𝑡) − 𝑢\+(𝑡)| and max
+;∈[+,@3]|𝑢+(𝑡) − 𝑢\+(𝑡)|, where 𝑢+(𝑡) is the first component of 𝒖 and 𝑢\+(𝑡) is the 
+predicted variables of 𝑢+(𝑡). 
+ 
+parameter 
+value 
+𝑀 
+Dimension of input and output variables  
+7 
+𝑁 
+Dimension of reservoir state vector 
+1000 
+Δt 
+Time step of the model 
+1 (day) 
+ρ 
+Maximal eigenvalue of 𝑨  
+0.8 
+α 
+Nonlinearity degree in the model 
+0.7 
+β 
+Regularization parameter  
+0.01 
+Δτ 
+Delay-time for input and output variables  
+6 (day)  
+ 
+Table 1. The list of parameters and their values for the selected reservoir computing model 
+ 
+ 
+MJO detection method  
+ 
+The RMM is calculated from the combined empirical orthogonal functions of the outgoing 
+longwave radiation data from National Oceanic and Atmospheric Administration 49, and zonal wind data 
+from National Centers for Environmental Prediction-Department of Energy reanalysis 50.  With the 
+exception of replacing RMM2 with the 12-day time delay coordinate of RMM1, the orientation of RMM1 
+and definitions of the RMM phases follow the convention introduced by Wheeler and Hendon 6. The MJO 
+events were identified from time sequences that were projected on to the RMM index from phase 2 to phase 
+7, while satisfying the following four conditions employed by Suematsu and Miura (2018) 35: (1) Phases 
+do not skip forward nor recede backward by more than one phase. (2) The average amplitude is greater 
+than the critical value of 0.8. (3) Period of consecutive days with amplitude below 0.8 is less than 15. (4) 
+Transition from phase 2 to phase 7 takes 20 to 90 days.  
+ 
+
+23	
+Supplementary materials 
+ 
+Validity of the Machine Learning-RMM Phase Space  
+ 
+ 
+The relevance of employing the RMM phase space spanned by RMM1 and its delay coordinate, the 
+machine learning RMM (ML-RMM) phase space, to describe the MJO instead of that spanned by RMM1 
+and RMM2 is discussed. Conventionally, MJO events are identified using RMM phase space spanned by 
+the first two orthogonal functions, RMM1 and RMM2, of 20 - 120 day Lanczos bandpass filtered 36 
+outgoing longwave radiation and zonal winds at 850hPa and 200 hPa. Figure S1 compares the composites 
+of 1979 – 2020 December to February outgoing longwave radiation for each of the RMM phases spanned 
+by the conventional RMM1 and RMM2 with ML-RMM phase space.  
+The composites indicate that the definition of the ML-RMM phases (Fig. S1 B) can capture the 
+characteristic of the MJO convection to shift eastward from the Indian Ocean to the Western Pacific as 
+the conventional method (Fig. S1 A). We note however, that compared to the conventional method, the 
+convective signals over the Indian Ocean in the ML-RMM phase 2 is weaker. This may be a caveat to our 
+method that arises from replacing the RMM2 with the delay coordinate of RMM1, since the structure of 
+the eigenvector of RMM2 reflects the state of the atmosphere with deep convection over the Indian 
+Ocean (see Fig. 1 in 30). Despite the abovementioned concern, the method employed in this study is 
+capable of adequately tracking MJO events on the RMM phase space (Fig. 2C).   
+ 
+
+24	
+ 
+ 
+ 
+ 
+Fig. S1. Composites of 1979 – 2020 December to February outgoing longwave radiation for each of the 
+RMM phases on the (A) conventional RMM1 and RMM2 phase space calculated from 20-120 days 
+Lanczos bandpass filtered data and (B) on the ML-RMM calculated from the 20-120 days RB filtered 
+data. 
+ 
+ 
+
+25	
+Examples of best performing cases in terms of phase prediction 
+The best performing prediction cases in terms of ML-RMM phase predictions are examined. Figure S2 
+shows the best three cases evaluated by the first day the cosine of the phase difference between the actual 
+(𝑎(𝑡), 𝑏(𝑡)) and the predicted (𝑎\(𝑡, 𝑝), 𝑏^(𝑡, 𝑝))	 vector, cos(𝜃5(𝑡)), becomes less than 0.7. The first (Fig. 
+S2. A, D), second (Fig. S2. B, E) and third (Fig. S2.C, F) best performing cases are the predictions of ML-
+RMM initiated on 10th October 2017, 26th March 2019, and 18th April 2018, respectively. In all three 
+cases, the predictions stay in phase (cos(𝜃5(𝑡)) > 0.7) for longer than 120 days. However, there is a 
+tendency for the amplitudes to be underestimated in these cases, which leads to growth in error as measured 
+by the distance between (𝑎(𝑡), 𝑏(𝑡))  and (𝑎\(𝑡, 𝑝), 𝑏^(𝑡, 𝑝)) from early stages of the predictions. Thus, 
+while the long predictability of the RMM phases over 120 days suggest the possibility of predicting the 
+MJO over a season (three months), overcoming the difficulty of accurately predicting the RMM phase and 
+amplitude simultaneously remains a challenge.   
+ 
+Fig. S2. The three best performing prediction of ML-RMM time series in terms of phase 
+prediction. (A, B, C) Predictions of ML-RMM initialized from (A, D) 2nd October 2017, (B, E) 18th 
+March 2019, and  (C, F) 10th April 2018, which are the three best prediction cases measured by the 
+cosine of the phase difference between the actual (𝑎(𝑡), 𝑏(𝑡)) and the predicted vectors (cos(𝜃5(𝑡))).  
+
+26	
+(D, E, F) show the time evolution of the cos(𝜃5(𝑡)). The width of the grey shades in A, B, C indicates 
+the error measured by the distance between (𝑎(𝑡), 𝑏(𝑡)) and (𝑎\(𝑡, 𝑝), 𝑏^(𝑡, 𝑝)). 
+ 
+

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+STABILIZED WEIGHTED REDUCED ORDER METHODS FOR
+PARAMETRIZED ADVECTION-DOMINATED OPTIMAL CONTROL
+PROBLEMS GOVERNED BY PARTIAL DIFFERENTIAL EQUATIONS WITH
+RANDOM INPUTS
+FABIO ZOCCOLAN1, MARIA STRAZZULLO2, AND GIANLUIGI ROZZA3
+Abstract. In this work, we analyze Parametrized Advection-Dominated distributed Optimal
+Control Problems with random inputs in a Reduced Order Model (ROM) context. All the simula-
+tions are initially based on a finite element method (FEM) discretization; moreover, a space-time
+approach is considered when dealing with unsteady cases. To overcome numerical instabilities that
+can occur in the optimality system for high values of the P´eclet number, we consider a Streamline
+Upwind Petrov–Galerkin technique applied in an optimize-then-discretize approach.
+We com-
+bine this method with the ROM framework in order to consider two possibilities of stabilization:
+Offline-Only stabilization and Offline-Online stabilization. Moreover we consider random parame-
+ters and we use a weighted Proper Orthogonal Decomposition algorithm in a partitioned approach
+to deal with the issue of uncertainty quantification. Several quadrature techniques are used to
+derive weighted ROMs: tensor rules, isotropic sparse grids, Monte-Carlo and quasi Monte-Carlo
+methods. We compare all the approaches analyzing relative errors between the FEM and ROM
+solutions and the computational efficiency based on the speedup-index.
+1. Introduction
+Here we present a numerical study concerning stabilized Parametrized Advection-Dominated Op-
+timal Control Problems (OCP(µ)s) with random inputs in a Reduced Order Methods (ROMs)
+framework. As a matter of fact, engineering and scientific applications often need very fast evalu-
+ations of the numerical solutions for many parameters that characterize the problem, for instance
+in real-time scenarios. A solution to these many-query situations can be to exploit the parameter
+dependence of the OCP(µ)s using ROMs [6, 24, 41, 40, 39]. This process is divided in two stages.
+The former is the offline phase, when many numerical solutions for different values of parameters
+are collected considering a first discretization of the OCP(µ), such a finite element method (FEM)
+one, called the high-fidelity or truth approximation. Then all parameter-independent components
+are calculated and stored, and reduced spaces are built. The latter is the online phase, when all
+parameter-dependent parts and, then, the reduced solutions are computed. More precisely, to deal
+with the randomness which is hidden in the parameters, we consider a modification of the Proper
+Orthogonal Decomposition (POD) that takes into account the probability distribution of the random
+inputs: the weighted POD (wPOD) [61, 60]. We apply this procedure in a partitioned approach,
+following good results shown in literature [30, 34, 53, 63]. As this algorithm aims to minimize the
+expectation of the square error between the truth and the ROM solutions, we can identify different
+types of weighted ROMs (wROMs) [11, 15, 13, 16, 17, 18, 49, 59, 61, 60] based on the chosen quad-
+rature rules. In this work, we will exploit Monte-Carlo and Quasi Monte-Carlo procedures, tensor
+rules based on Gauss-Jacobi and Clenshaw-Curtis quadrature techniques, and Smolyak isotropic
+sparse grids.
+1 Section de Math´ematiques, ´Ecole Polytechnique F´ed´erale de Lausanne, 1015 Lausanne, Switzerland,
+email: fabio.zoccolan@epfl.ch
+2 DISMA, Politecnico di Torino, Corso Duca degli Abruzzi 24, Turin, Italy.
+email: maria.strazzullo@polito.it
+3 mathLab, Mathematics Area, SISSA, via Bonomea 265, I-34136 Trieste, Italy.
+email: gianluigi.rozza@sissa.it
+1
+arXiv:2301.01975v1  [math.NA]  5 Jan 2023
+
+2
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+The optimization problem will always concern a linear-quadratic cost functional. We use FEM
+as the truth approximation, both for steady and unsteady problems. At a first level, FEM approx-
+imations of stochastic steady OCP(µ)s have been already presented, for example, in [26] consid-
+ering stochastic PDEs. In the parabolic case, we discretize time-dependency via a space-time ap-
+proach [25, 50, 51]. Concerning stabilization, we considered the Streamline Upwind Petrov–Galerkin
+(SUPG) [9, 28, 38] suitably combined with the ROM setting in an optimize-then-discretize approach.
+We exploit two possibilities: when stabilization only occurs in the offline phase, Offline-Only stabi-
+lization or when SUPG is applied in both phases, Offline-Online stabilization.
+Stabilized Advection-Dominated problems in a ROM framework without control are studied,
+for instance, in [37, 59], both for steady and unsteady cases. Instead, in [11] wROMs for generic
+OCP(µ)s are applied to experiments concerning environmental sciences. Instead, SUPG Advection-
+Dominated distributed OCP(µ)s are analyzed in a deterministic context in [63], both for elliptic and
+parabolic experiments. To the best of our knowledge, this is the first time that stabilized Advection-
+Dominated OCP(µ)s with random inputs are analyzed in a ROM context, both for elliptic and
+parabolic problems.
+This work is arranged as follows. In Section 2, we introduce linear-quadratic optimal control
+theory for PDEs and its FEM discretization. Section 3 firstly concerns the basic theory of SUPG
+stabilization for Advection-Dominated PDEs in an optimize-then-discretize approach [19], then the
+space-time procedure that will be used is presented. wROMs features will be illustrated in Sec-
+tion 4. Section 5 will concern numerical simulations. Two Advection-Dominated problems under
+distributed control and random inputs will be analyzed: the Graetz-Poiseuille Problem under ge-
+ometrical parametrization and the Propagating Front in a Square Problem. We will compare the
+wPOD procedures through relative errors between the FEM and the ROM solutions and computa-
+tional time considering the speedup-index. Finally, conclusions follow in Section 6.
+2. Problem Formulation for Random Input Optimal Control Problems
+2.1. Mathematical Setting. Let Ω be an open and bounded regular domain in R2, where ΓN and
+ΓD will indicate the portions of the boundary ∂Ω where Neumann and Dirichlet boundary conditions
+are imposed, respectively. With the symbol Ωobs ⊆ Ω the observation domain will be indicated, i.e.
+the subset of the domain where we seek the state variable to be similar to a desired solution profile
+yd ∈ Y , with Y Hilbert space, in a sense that will be specified later. For time-dependent problems we
+will also take into account the time interval (0, T) ⊂ R+. Let us consider a compact set P ⊆ Rp, for
+natural number p. We will call P and as the parameter space and a p-vector µ ∈ P is the parameter
+of our Parametric OCP(µ)s. As the setting is completely general, for instance µ can characterize
+our yd or geometrical and physical properties of the problem. Furthermore, we denote with B(Q, R)
+the space of linear continuous operators between Banach spaces Q and R.
+The triplet (A, F, P) will denote a complete probability space, composed by A, which is the set
+of outcomes ω ∈ A, F, that is a σ-algebra of events, and P : F → [0, 1] with P(A) = 1, which is
+the chosen probability measure. As dealing with random input OCP(µ)s, the parameter µ will be
+a real-valued random vector. In detail, µ : (A, F) → (Rp, B) is a measurable function, where B is
+the Borel σ-algebra on Rp. The distribution function of µ : A → P ⊂ Rp, being P the image of µ,
+is defined as Pµ : P → [0, 1] such that
+(1)
+∀µ ∈ P,
+Pµ(µ) = P(ω ∈ A : µ(ω) ≤ µ).
+Let dPµ(µ) denote the distribution measure of µ, i.e.,
+(2)
+∀H ⊂ P,
+P(µ ∈ H) =
+�
+H
+dPµ(µ).
+We assume that µ admits a Lebesgue density, i.e. dPµ(µ) is absolutely continuous with respect
+to the Lebesgue measure dµ. This practically means that there exists a probability density func-
+tion ρµ : P → R+ such that ρµ(µ)dµ = dPµ(µ). It is worth to notice that the measure space
+(P, B(P), ρµ(µ)dµ) is isometric to (A, F, P) under the random vector µ.
+The aim of this work is to analyze random input OCP(µ)s from the numerical point of view.
+
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+3
+Problem 2.1.1 (Random Input Parametric Optimal Control Problem). Consider the state equation
+E : Y × U → Q, with Y, U, and Q real Banach spaces, satisfying a set of boundary and/or initial
+conditions, and a real functional J : Y ×U → R. Then for Pµ-a.e. find the pair
+�
+y(µ), u(µ)
+�
+∈ X :=
+Y × U that minimizes cost functional J (y(µ), u(µ); µ) under the constraint E(y(µ), u(µ); µ) = 0.
+Let Xad be the set of all couples (y, u) solutions of E: we will only consider the case of full
+admissibility, i.e. when Xad = Y × U. Problem 2.1.1 looks for minimizers among all state-control
+pairs such that:
+min
+(y(µ),u(µ))∈Y ×U J (y(µ), u(µ); µ) s.t. E(y(µ), u(µ); µ) = 0.
+This can be achieved through the research of the critical points of the Lagrangian operator
+L : Y × U × Q∗ → R defined as:
+(3)
+L(y(µ), u(µ), p(µ); µ) = J (y(µ), u(µ); µ) + ⟨p(µ), E(y(µ), u(µ); µ)⟩Q∗Q,
+where p(µ) is a Lagrange multiplier belonging to the adjoint space Q∗, the dual space of Q. For
+the sake of notation we write y := y(µ), u := u(µ) and p := p(µ). In case that Pµ is the uniform
+distribution with support in P, then Problem 2.1.1 is called to be deterministic problem. In this
+work linear-quadratic problems will be involved.
+Definition 2.1.2 (Linear-Quadratic OCP(µ). Let us consider the bilinear forms m : Z × Z → R
+and n : U × U → R, which are symmetric and continuous, where Z is a Banach space called the the
+observation space. Fix α > 0, a constant called the penalization parameter and consider a quadratic
+objective functional J of the form
+(4)
+J (y, u; µ) = 1
+2m (Oy(µ) − zd(µ), Oy(µ) − zd(µ)) + α
+2 n(u(µ), u(µ)),
+where O : Y → Z is a linear and bounded operator called the observation map and zd(µ) ∈ Z is
+the observed desired solution profile. Consider an affine map E defined as
+(5)
+E(y, u; µ) = A(µ)y + C(µ)u − f(µ),
+∀
+�
+y(µ), u(µ)
+�
+∈ Y × U,
+where A(µ) ∈ B(Y, Q), C(µ) ∈ B(U, Q) and f(µ) ∈ Q.
+Then an OCP(µ)s with the above properties is said a Linear-Quadratic Optimal Control Problem.
+For Linear-Quadratic OCP(µ)s well-posedness of Problem 2.1.2 yields [7, 8]. More precisely, the
+reader can refer to [10] to a comparison between the Lagrangian approach for the full-admissibility
+case and the adjoint one. Via the functional derivative of L, we obtain a optimality system to be
+solved to find the unique solution. In this case, this reads as finding (y, u, p) ∈ Y × U × Q∗ that
+satisfies [10],
+(6)
+�
+�
+�
+�
+�
+DyL(y, u, p; µ)(¯y) = 0 =⇒ m(Oy, O¯y; µ) + ⟨A∗(µ)p, ¯y⟩Y ∗Y = m (O¯y, zd; µ) ,
+∀¯y ∈ Y,
+DuL(y, u, p; µ)(¯u) = 0 =⇒ αn(u, ¯u; µ) + ⟨C∗(µ)p, ¯u⟩U ∗U = 0,
+∀¯u ∈ U,
+DpL(y, u, p; µ)(¯p) = 0 =⇒ ⟨¯p, A(µ)y + C(µ)u⟩Q∗Q = ⟨¯p, f(µ)⟩Q∗Q,
+∀¯p ∈ Q∗.
+In system (6), the first equation is called the adjoint equation, the second one is the gradient
+equation and the last one is state equation.
+Remark 2.1.3 (Notation). For the sake of notation, when Hilbert spaces will be taken into account,
+bilinear forms A, B and their adjoint counterparts will be indicate uniquely as:
+⟨A(µ)y, p⟩QQ∗ := a(y, p; µ)
+⟨C(µ)u, p⟩QQ∗ := c(u, p; µ).
+Remark 2.1.4 (Parabolic Problems). Concerning unsteady problems, one must add more hypotheses
+to the mathematical setting of Linear-Quadratic OCP(µ)ss to reach well-posedness. We will consider
+the following Hilbert spaces Y = L2(0, T; Y ), Y∗ = L2 (0, T; Y ∗), Z = L2 (0, T; Z), U = L2(0, T; U)
+with respective norms given by
+(7) ∥y∥2
+Y :=
+T
+�
+0
+∥y∥2
+Y dt,
+∥y∥2
+Y∗ :=
+T
+�
+0
+∥y∥2
+Y ∗dt,
+∥z∥2
+Z :=
+T
+�
+0
+∥z∥2
+Zdt,
+and
+∥u∥2
+U :=
+T
+�
+0
+∥u∥2
+Udt.
+
+4
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+Then we define the Hilbert space Yt := {y ∈ Y
+s.t.
+∂ty ∈ Y∗}. For parabolic problems we will
+also consider the case of full-admissibility as Xad = Yt × U [5, 54, 55].
+2.2. High-Fidelity Discretization. In this work, the discretization of the optimality sistem (6)
+follows an one shot or all-at-once approach [25, 50, 51]. When we will consider Advection-Dominated
+OCP(µ)s, a stabilization technique will be also needed. Therefore, a SUPG method will be applied
+in a optimize-then-discretize approach, as we will see in Section 3.
+This means that firstly the
+optimality conditions are computed obtaining system (6) and then we discretize and stabilize it.
+Concerning numerical implementation, we use a FEM discretization for all three variables, where
+Th is a regular triangularization on Ω. Its elements K are triangles and the parameter h denotes the
+mesh size, i.e. the maximum diameter of an element of the chosen grid. In addition, we define
+Ωh := int
+� �
+K∈Th
+K
+�
+,
+as a quasi-uniform mesh for Ω. Considering Pr(K) as the space of polynomials of degree at most
+equal to r defined on K and defining
+XN ,r =
+�
+qN ∈ C(¯Ω) : qN
+|K ∈ Pr(K), ∀K ∈ Th
+�
+we set Y N = Y ∩ XN ,r, U N = U ∩ XN ,r and
+�
+QN �∗ = Q∗ ∩ XN ,r. In this work, the numerical
+implementation will always made by a P1-FEM approximation and the same triangulation Th for
+Y N , U N , and
+�
+QN �∗. A similar discussion can be made for time-dependent problem, as we will
+see in Section 3.2. This first discretization procedure will be indicated as the truth or high-fidelity
+approximation.
+From now on, Y, U, Q will be always Hilbert spaces and we will consider the Identity restricted to
+our observation domain Ωobs as the Observation function O for both steady and unsteady problems.
+Therefore, Z = Y for steady problems and Z = Y for unsteady ones are assumed. Our desired state
+will be denoted by yd and with the same symbol will also indicate its FEM discretization.
+3. SUPG stabilization for Advection-Dominated OCP(µ)s
+In this work we only deal with Advection-Diffusion equations.
+Definition 3.0.1 (Advection-Diffusion Equations). Let us take into account the following problem:
+(8)
+T(µ)y := −γ(µ)∆y + η(µ) · ∇y = f(µ) in Ω ⊂ R2,
+where suitable boundary conditions are applied on ∂Ω. In addition, we require that:
+• the diffusion coefficient γ : Ω → R is uniformly bounded, i.e. there exists γmax, γmin > 0 such
+that
+(9)
+P
+�
+ω ∈ A : γmin < γ(x; µ) < γmax ∀x ∈ Ω
+�
+= 1.
+• the advection field η : Ω → R2 belongs to (L∞(Ω))2 for a.e. µ ∈ P. More precisely, for a.e.
+µ the following inequality holds: 0 ≥ div η(x) ≥ −ϑ, ∀x ∈ Ω, with ϑ ∈ R+
+0 ;
+• f : Ω → R is an L2(Ω)-function for a.e. µ; in addition, f has bounded second moments with
+respect to the integral along A and Ω.
+With these hypotheses, Problem (8) is called Advection-Diffusion problem and the operator T(µ)y :=
+−γ(µ)∆y + η(µ) · ∇y is said the Advection-Diffusion operator.
+For more details regarding the well-posedness and theoretical results of Stochastic Advection-
+Diffusion OCP(µ)s, we refer to [14, 13].
+Definition 3.0.2 (P´eclet number and Advection-Dominated problem). Let us consider the FEM
+discretization related to an Advection-Diffusion problem and its regular triangulation Th. For any
+element K ∈ Th, the local P´eclet number is defined as [42, 38]:
+(10)
+PeK(x) := |η(x)|hK
+2γ(x)
+∀x ∈ K,
+
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+5
+where hK is the diameter of K. If PeK(x) > 1 ∀x ∈ K, ∀K ∈ Th, we say to study an Advection-
+Dominated problem.
+3.1. Setting for Stabilized Advection-Dominated OCP(µ)s - Steady case. Numerical in-
+stabilities can appear, when dealing with Advection-Dominated OCP(µ)s, i.e. when |η(µ)| ≫ γ. In
+order to adjust this unpleasant behaviour without modifying the mesh size, we use the Streamline
+upwind/Petrov Galerkin (SUPG) method [9, 27, 28, 42] in a optimize-then-discretize approach [19].
+This assures the strongly consistency of the optimality system [19]. For the sake of simplicity, we
+define our Advection-Dominated problem on H1
+0(Ω) and we do not indicate the parameter depen-
+dence. We denote with T ∗ the adjoint operator related to T. This last operator can be split into its
+symmetric and skew-symmetric parts as T = TS + TSS [42], where:
+(11)
+symmetric part: TSy = −γ∆y − 1
+2(div η)y,
+∀y ∈ H1
+0(Ω),
+skew-symmetric part: TSSy = η · ∇y + 1
+2(div η)y,
+∀y ∈ H1
+0(Ω).
+This two parts can be immediately recovered using the formulae:
+(12)
+TS = T + T ∗
+2
+,
+TSS = T − T ∗
+2
+.
+After having considered FEM spaces, the stabilization occurs in the bilinear and linear terms involved
+in the state and the adjoint equations. Instead, the gradient equation is left unstabilized [19]. We
+recall that we use distributed control.
+We defined the stabilized bilinear form as and cs, and the stabilized forcing term Fs as
+(13)
+as
+�
+yN , qN ; µ
+�
+:= a
+�
+yN , qN ; µ
+�
++
+�
+K∈Th
+δK
+�
+TyN , hK
+|η| TSSqN
+�
+K
+,
+yN , qN ∈ Y N ,
+cs
+�
+uN , qN ; µ
+�
+:= −
+�
+Ω
+uN qN −
+�
+K∈Th
+δK
+�
+uN , hK
+|η| TSSqN
+�
+K
+,
+uN ∈ U N , qN ∈ Y N ,
+Fs
+�
+qN ; µ
+�
+:= F
+�
+qN ; µ
+�
++
+�
+K∈Th
+δK
+�
+f(µ), hK
+|η| TSSqN
+�
+K
+,
+∀qN ∈ Y N .
+where δK is a local positive dimensionless parameter related to the element K ∈ Th, consequently
+it can be different for each triangle, and (·, ·)K is the inner scalar product in L2(K).
+In (13)
+a
+�
+yN , qN ; µ
+�
+=
+�
+TyN , qN �
+L2(Ω) and F
+�
+qN ; µ
+�
+=
+�
+f, qN �
+L2(Ω), where f collects all forcing and
+lifting terms of the problem.
+For the remaining conditions of the optimality system, we will always consider m and n form as
+the L2(Ωobs) and the L2(Ω) products for steady problems. The adjoint equation is an Advection-
+Dominated equation, too, where the advective term has opposite sign with respect to the state one:
+indeed, T ∗ = TS − TSS from (12). We use the next SUPG forms for zN ∈ Y N :
+(14)
+a∗
+s
+�
+zN , pN ; µ
+�
+:= a∗ �
+zN , pN ; µ
+�
++
+�
+K∈Th
+δa
+K
+�
+(TS − TSS)pN , hK
+|η| (−TSS) zN
+�
+K
+,
+�
+yN − yd, zN ; µ
+�
+s :=
+�
+Ωobs
+(yN − yd)zN dx +
+�
+K∈Th|Ωobs
+δa
+K
+�
+yN − yd, hK
+|η| (−TSS) zN
+�
+K
+,
+where a∗ is the adjoint form of a, δa
+K is the positive stabilization parameter of the stabilized adjoint
+equation.
+In our numerical experiments, we will always consider δK = δa
+K.
+Finally, the SUPG
+optimality system for a steady OCP(µ) reads as:
+(15)
+discretized adjoint equation:
+a∗
+s
+�
+zN , pN ; µ
+�
++
+�
+yN − yd, zN ; µ
+�
+s = 0, ∀zN ∈ Y N ,
+discretized gradient equation:
+c∗�
+vN , pN ; µ
+�
++ αn
+�
+uN , vN ; µ
+�
+= 0, ∀vN ∈ U N ,
+discretized state equation:
+as
+�
+yN , qN ; µ
+�
++ cs
+�
+uN , qN ; µ
+�
+= Fs(qN ; µ), ∀qN ∈
+�
+QN �∗ .
+
+6
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+We denote with Ks and KT
+s the stiffness matrices related to the stabilized forms as and a∗
+s,
+respectively, M is the not-stabilized mass matrix related to n, instead, Ms is the stabilized mass
+matrix related to m after stabilization, Cs is the matrix linked to stable form cs, the block CT refers
+to c, and fs is the vector that contains the coefficients of the stabilized force term as components.
+Moreover, we consider with the symbol y, u and p as the vectors of coefficients of yN , uN and pN ,
+expressed in terms of the nodal basis of Y N , U N , (QN )∗, respectively. Finally, the discretized block
+system related to (15) is:
+(16)
+�
+�
+Ms
+0
+KT
+s
+0
+αM
+CT
+Ks
+Cs
+0
+�
+�
+�
+�
+y
+u
+p
+�
+� =
+�
+�
+Msyd
+0
+fs
+�
+� .
+3.2. Setting for Stabilized Advection-Dominated OCP(µ)s - Unsteady case. We show the
+SUPG approach for time-dependent OCP(µ)s proposed in [63]. A classical implicit Euler discretiza-
+tion is applied to all forms including time-derivatives [3, 25, 50, 54, 55, 56]. We divide the time
+interval (0, T) in Nt sub-intervals of equal length ∆t := tj − tj−1, j ∈ {1, . . . , Nt}. Starting from
+this framework, a discretization along time is done, where each discrete instant of time is considered
+as a steady-state Advection-Dominated equation in a space-time approach [25, 50, 51, 54, 55, 56].
+In addition, the SUPG stabilization occurs for time-dependent forms, too. The general scheme is
+described as follows.
+Let us firstly define the discrete vectors y =
+�
+yT
+1 , . . . , yT
+Nt
+�T , u =
+�
+uT
+1 , . . . , uT
+Nt
+�T and p =
+�
+pT
+1 , . . . , pT
+Nt
+�T , where yi ∈ Y N , ui ∈ U N and pi ∈ (QN )∗ for 1 ≤ j ≤ Nt.
+Also here, yj, uj
+and pj indicate the column vectors containing the coefficients of the FEM discretization for state,
+control and adjoint, respectively (unlike the steady case, there are not denoted in bold style). This
+implies Ntot = 3 × Nt × N as the global dimension of the block system.
+We express all other
+terms in based of the respective nodal basis. The vector representing the initial condition for the
+state variable is y0 =
+�
+yT
+0 , 0T , . . . , 0T �T , where 0 is the zero vector in RN , yd =
+�
+yT
+d1, . . . , yT
+dNt
+�T
+is the vector including discrete time components of the discretized desired solution profile; instead,
+f s =
+�
+f T
+s1, . . . , f T
+sNt
+�T
+corresponds to the stabilized forcing term. We recall that Y, U, Q are Hilbert
+Spaces and, for the sake of simplicity, we assume Y N ≡ (QN )∗. So now we can see locally the time
+block discretization.
+• Adjoint equation: this equation is discretized backward in time using the forward Euler
+method, which is equal to a backward Euler with respect to time T − t, for t ∈ (0, T) [21].
+Firstly, we add a stabilized term to the form related to ∂tp and a∗ defined as:
+s∗ �
+zN , pN (t); µ
+�
+=
+�
+K∈Th
+δK
+�
+−∂tpN (t) + T ∗pN (t), −hK
+|η| TSSzN
+�
+K
+,
+where we define the form
+(17)
+m∗
+s
+�
+pN , zN ; µ
+�
+=
+�
+pN , zN �
+L2(Ω) −
+�
+K∈Th
+δK
+�
+pN , hK
+|η| TSSzN
+�
+K
+.
+Then, the time discretization is: for each j ∈ {Nt − 1, Nt − 2, ..., 1}, find pN
+j ∈ Y N
+s.t.
+(18)
+1
+∆tm∗
+s
+�
+pN
+j (µ) − pN
+j+1(µ), zN ; µ
+�
++ a∗
+s
+�
+zN , pN
+j (µ); µ
+�
+= −
+�
+yN
+j − ydj, zN ; µ
+�
+s
+∀zN ∈ Y N ,
+Considering M T
+s as the matrix inherent to m∗
+s, the block subsystem reads
+M T
+s pj = M T
+s pj+1 + ∆t
+�
+−M T
+s yj − KT
+s pj + M T
+s ydj
+�
+for j ∈ {Nt − 1, Nt − 2, . . . , 1} .
+
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+7
+Finally, we derive the following block system:
+�
+����
+M T
+s + ∆tKT
+s
+−M T
+s
+...
+...
+M T
+s + ∆tKT
+s
+−M T
+s
+M T
+s + ∆tKT
+s
+�
+����
+�
+��
+�
+AT
+s
+p +
+�
+�����
+∆tM T
+s y1
+...
+...
+∆tM T
+s yNt
+�
+�����
+=
+�
+�����
+∆tM T
+s yd1
+...
+...
+∆tM T
+s ydNt
+�
+�����
+.
+Setting the diagonal block matrix MT
+s ∈ RN ·Nt×RN ·Nt with diagonal entries [M T
+s , . . . , M T
+s ],
+the adjoint system to be solved is: ∆tMT
+s y + AT
+s p = ∆tMT
+s yd.
+• Gradient equation. We seek the solution of α∆tMuj+∆tCT pj = 0, ∀j ∈ {1, 2, . . . , Nt} ,
+which is equal to the following block system:
+(19)
+α∆t
+�
+����
+M
+M
+...
+...
+M
+�
+����
+�
+��
+�
+M
+�
+����
+u1
+u2
+...
+uNt
+�
+���� +∆t
+�
+����
+CT
+0
+· · ·
+CT
+...
+CT
+�
+����
+�
+��
+�
+CT
+�
+����
+p1
+p2
+...
+pNt
+�
+���� =
+�
+����
+0
+0
+...
+0
+�
+���� .
+More compactly, we solve α∆tMu+∆tCT p = 0.
+• State equation. A backward Euler method is used for a discretization forward in time. The
+stabilized term related to ∂ty and the bilinear form a is [29, 37, 58]:
+s
+�
+yN (t), qN ; µ
+�
+=
+�
+K∈Th
+δK
+�
+∂tyN (t) + TyN (t), hK
+|η| TSSqN
+�
+K
+,
+where yN (t) ∈ Y N for each t ∈ (0, T) and qN ∈ Y N . Defining the stabilized term ms as
+(20)
+ms
+�
+yN , qN ; µ
+�
+=
+�
+yN , qN �
+L2(Ω) +
+�
+K∈Th
+δK
+�
+yN , hK
+|η| TSSqN
+�
+K
+,
+then the backward Euler approach reads as: for each j ∈ {1, 2, · · · , Nt}, find yN
+j ∈ Y N s.t.
+(21)
+1
+∆tms
+�
+yN
+j (µ) − yN
+j−1(µ), qN ; µ
+�
++ as
+�
+yN
+j (µ), qN ; µ
+�
++ cs
+�
+uN
+j , qN ; µ
+�
+= Fs
+�
+qN ; µ
+�
+,
+given the initial condition yN
+0
+which satisfies
+�
+yN
+0 , qN �
+L2(Ω) =
+�
+y0, qN �
+L2(Ω) , ∀qN ∈ Y N .
+The matrix state equation to be solved becomes
+(22)
+Msyj + ∆tKsyj + ∆tCsuj = Msyj−1 + fsj∆t
+for j ∈ {1, 2, . . . , Nt} ,
+where the stabilized mass matrix of ms is Ms. Thus, we have
+�
+����
+Ms + ∆tKs
+0
+−Ms
+Ms + ∆tKs
+0
+...
+...
+0
+0
+−Ms
+Ms + ∆tKs
+�
+����
+�
+��
+�
+As
+y+∆t
+�
+��
+Cs
+0
+0
+...
+0
+0
+Cs
+�
+��
+�
+��
+�
+Cs
+u = Msy0 + ∆tf s,
+where Ms ∈ RN ·Nt×RN ·Nt is a block diagonal matrix which diagonal entries are [Ms, . . . , Ms].
+In a more compact notation, we have Asy+∆tCsu = Msy0 + ∆tf s.
+The final system considered and solved through an one shot approach is the following:
+(23)
+�
+�
+∆tMT
+s
+0
+AT
+s
+0
+α∆tM
+∆tCT
+As
+∆tCs
+0
+�
+�
+�
+�
+y
+u
+p
+�
+� =
+�
+�
+∆tMT
+s yd
+0
+Msy0 + ∆tf s
+�
+� .
+
+8
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+4. Weighted ROMs for random inputs advection-dominated OCP(µ)s
+Numerical simulations for OCP(µ)s can be very expensive in relation to computational time and
+storage. To overcome this problem, in this work we will consider ROMs [6, 24, 41, 40, 39]. We will
+study the case when the parameter µ can be affected by randomness, i.e. it can follow a particular
+probability distribution. That is the case of random inputs OCP(µ)s. In this scenario, a suitable
+modification of the ROMs, the wROMs [11, 15, 13, 16, 17, 18, 49, 59, 61, 60], takes into account
+the uncertainty quantification (UQ) of the problems and shows efficient results concerning errors
+and computational time. For the sake of notation, from now on we denote a generic probability
+distribution with the symbol ρ. ROM procedure is divided in two stages: an offline phase and an
+online phase.
+To exploit the potential of the ROMs setting, we assume an affine decomposition of the forms in
+(15) [24]. Therefore, Assumption 4.0.1 is required here.
+Assumption 4.0.1. We request that all the forms in (15) are affine in µ = (µ1, . . . , µp) ∈ P. More
+precisely, we request that [15, 13]:
+(1) the random diffusivity γ : Ω × P → R is of the form
+(24)
+γ(µ, x) = γ0(x) +
+p
+�
+k=1
+θγ
+k(µk)γk(x),
+with γk ∈ L∞(Ω), for k = 0, . . . , p and θγ
+k depending only on µk;
+(2) the random advection field η : Ω × P → R2 is of the form
+(25)
+η(µ, x) = η0(x) +
+p
+�
+k=1
+θη
+k(µk)ηk(x),
+with ηk ∈ (L∞(Ω))2, for k = 0, . . . , p and θη
+k depending only on µk;
+(3) the random forcing term f : Ω × P → R is of the form
+(26)
+f(µ, x) = f0(x) +
+p
+�
+k=1
+θf
+k(µk)fk(x),
+with fk ∈ L2(Ω), for k = 0, . . . , p and θf
+k depending only on µk.
+For example, Assumption 4.0.1 can be satisfied by truncating a Karhunen–Lo`eve expansion [47].
+4.1. Offline phase. The offline phase is the most expensive stage of the wROMs, which usually
+depends on N. However, this should be done only once. The aim of this procedure is to build reduced
+spaces Y N, U N and (QN)∗ that are good approximations of the high-fidelity ones and to compute
+all block matrix components that are µ-independent. Then, everything is memorized in order to be
+ready to be used in the online phase. The construction of the reduced basis is achieved through a
+modified version of the POD algorithm: the wPOD [11, 60, 61], described in Section 4.1.1. Here,
+we firstly compute high-fidelity evaluation of optimal solutions
+�
+yN (µ), uN (µ), pN (µ)
+�
+for different
+parameters µ, the so-called snapshots, to build the bases. Because of good performance presented in
+literature [30, 34, 53], this process will go through a partitioned approach, i.e. the wPOD is executed
+separately for all three variables. After this step, the three reduced spaces for state, control and
+adjoint are constructed as, respectively,
+(27)
+Y N = span {ξy
+n, n = 1, . . . , N},
+U N = span {ξu
+n, n = 1, . . . , N} ,
+(QN)∗ = span {ξp
+n, n = 1, . . . , N} .
+In order to ensure well-posedness for the reduced space approximation, we need to implement an
+enriched space for state and adjoint variables. This means to impose GN ≡ Y N ≡ (QN)∗, where
+GN = span {σn, n = 1, . . . , 2N} and {σn}2N
+n=1 = {ξy
+n}N
+n=1 ∪ {ξp
+n}N
+n=1 [20, 23, 30, 31, 35, 34]. This
+whole discussion holds true for parabolic problems in a space-time context, too. As a matter of fact,
+
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+9
+when dealing with time-dependent OCP(µ)s in a space-time approach, the time instances are not
+separated in the wPOD algorithm. Therefore, each snapshot carries all the time instances.
+4.1.1. Weighted Proper Orthogonal Decomposition. The peculiarity of wPOD is to take into account
+the probability distribution that characterizes µ to create reduced spaces with less number of basis
+with respect to the deterministic case without losing in accuracy [11, 60, 61]. We will notice that
+there will be different ways to consider randomness in the wPOD: the general idea is to suitably
+attribute a larger weight to those samples that are more significant according to the distribution of µ.
+From now we will refer to the POD algorithm based on the Monte-Carlo procedure in a deterministic
+context, i.e. when the distribution ρ is the uniform one, as Standard POD to distinguish it from
+the wPOD. As we will consider a partitioned approach, we show the procedure for the state space:
+adjoint and control variables will follow the same process.
+To consider stochasticity, wPOD needs to find the N-dimensional space Y N, with N ≪ N, such
+that it minimizes the following estimate:
+(28)
+E =
+��
+P
+inf
+ζy∈Y N ∥yN (µ) − ζy∥2
+Y ρ(µ)dµ.
+Let us consider a set of Ntrain ordered parameters µ1, . . . , µNtrain ∈ PNtrain, where PNtrain ∈ P is a
+discretization of P called the training set and its cardinality is |PNtrain| = Ntrain. One can choose
+Ntrain so that PNtrain is a good approximation of P. We can relate µ1, . . . , µNtrain to the ordered
+snapshots yN (µ1) , . . . , yN �
+µNtrain
+�
+. Considering w : P → R+ a weight function, a discretization
+of problem (28) is meant to find the N-dimensional space Y N which minimize the quantity
+(29)
+1
+Ntrain
+Ntrain
+�
+k=1
+w (µk)
+��yN (µk) − yN (µk)
+��2
+Y .
+One could think that the natural choice can be w(µ) = ρ(µ) and in an UQ context this means
+to just discretize the expectation of the square error
+(30)
+E
+���yN − yN��2
+Y
+�
+:=
+�
+P
+��yN (µ) − yN(µ)
+��2 ρ(µ)dµ,
+which is the argument of the square root in (28). However, this is not the unique choice in this
+scenario: therefore it will be interesting to understand which method is better to approximate (30).
+Here we illustrate different techniques that we use in the numerical tests in Section 5 to approximate
+(30). Considering the training set PNtrain =
+�
+µ1, . . . , µNtrain
+�
+, which can be composed by the nodes
+of the chosen quadrature formula that approximates (30), we indicate with ω = (ω1, . . . , ωNtrain) the
+standard weights of a chosen quadrature rule, with ρ1, . . . , ρNtrain the values of the density ρ in the
+nodes in PNtrain, and with w = (w1, . . . , wNtrain) the definitive weights used in wPOD algorithm. For
+a node µk, we have the correspondent quantities ωk, ρk, and wk. As a final result of this first step,
+the wPOD furnished the following sum to minimize
+(31)
+1
+Ntrain
+Ntrain
+�
+k=1
+wk
+��yN (µk) − yN (µk)
+��2
+Y ,
+which is achieved here through the following algorithms:
+• Weighted Monte-Carlo method, where µ1, . . . , µNtrain are Ntrain parameters extracted from
+the random variable µ according to its distribution ρ and ρi are the values of the density ρ in
+these points. For this approximation, we have PNtrain =
+�
+µ1, . . . , µNtrain
+�
+and wk = ρ(µk),
+for all k = 1, . . . , Ntrain;
+• Pseudo-Random method based on a Halton Sequence, where µ1, . . . , µNtrain are the nodes
+extracted by a sampling completely based on the Halton sequence [57] and ρk = ρ(µk). Also
+in this case, PNtrain =
+�
+µ1, . . . , µNtrain
+�
+and wk = ρk, for all k = 1, . . . , Ntrain;
+
+10
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+• Tensor product Gauss-Jacobi rule, where µ1, . . . , µNtrain are the nodes of the tensor product
+Gauss-Jacobi quadrature rule and ω1, . . . , ωNtrain are the correspondent quadrature weights.
+We can use this formula when the distribution is a Beta(αk, βk), as suitable Jacobi polyno-
+mials are orthogonal to this distribution [43]. As a matter of fact, simulations in Section 5
+will consider different Beta distributions for all components of µ. Therefore, we implement
+a Gauss-Jacobi formula using (αk, βk) as its parameters in each dimension [43], accordingly
+to the distribution of µ. For this approximation, we have PNtrain =
+�
+µ1, . . . , µNtrain
+�
+and
+wk = ωk, for all k = 1, . . . , Ntrain;
+• Tensor product Clenshaw-Curtis rule, where µ1, . . . , µNtrain are the nodes of the tensor prod-
+uct Clenshaw-Curtis quadrature rule and ω1, . . . , ωNtrain are the correspondent quadrature
+weights [57]. In this case we obtain PNtrain =
+�
+µ1, . . . , µNtrain
+�
+and wk = ρkωk, for all
+k = 1, . . . , Ntrain.
+In numerical tests of Section 5, we will respectively call as Weighted Monte-Carlo, Pseudo-
+Random, Gauss-Jacobi, and Clenshaw-Curtis wPOD algorithms the rules just specified. As it is
+know, tensor rule can be efficient, but their structure implies huge computational costs for elevate
+cardinality of the training set Ptrain or high-dimensional parameter space P. For this purpose, when
+we will use Clenshaw-Curtis or Gauss-Jacobi methods, we will consider sparse grid techniques based
+on a Smolyak algorithm, too [48, 62]: we will implement isotropic ones [36].
+Once chosen the rule (29) to approximate (30), the procedure to minimize (29) is described as
+follows. Let us define the deterministic correlation matrix of the snapshots of the state variable
+Dy ∈ RNtrain×Ntrain in the following way:
+(32)
+Dy
+kl :=
+1
+Ntrain
+�
+yN (µk) , yN (µl)
+�
+Y ,
+1 ≤ k, l ≤ Ntrain.
+Firstly, we define the weighted correlation matrix as
+(33)
+W y := W · Dy,
+where W := diag(w1, · · · , wNtrain) is the diagonal matrix whose elements are the weights of (29).
+The matrix W y is not symmetric in the usual matrix sense, but with respect to the scalar product
+induced by W y, hence W y is diagonalizable anyway [60]. Therefore, we seek the solution of the
+eigenvalue problem W ygy
+n = λy
+ngy
+n, 1 ≤ n ≤ Ntrain, where ∥gy
+n∥Y = 1, i.e. we pursue to find an
+eigenvalue λy
+n with the relative eigenvector of norm equal to one. We will indicate with (gy
+n)t the t-
+th component of the eigenvector gy
+n ∈ RNtrain. For the sake of simplicity, we rearrange the eigenvalues
+λy
+1, . . . , λy
+Ntrain in a decreasing layout. Then, let us look at the first N eigenvalue-eigenvector pairs
+(λy
+1, gy
+1), . . . , (gy
+N, λy
+N). The basis functions χy
+n for the state equation are constructed through the
+following relation:
+(34)
+ζy
+n =
+1
+√
+λy
+n
+Ntrain
+�
+t=1
+(gy
+n)t yN (µk) ,
+1 ≤ n ≤ N.
+In order to choose N, one can refer to same study of eigenvalues of W y [24, 39, 61]. At the end, our
+reduced spaces are built as (27) and, then, enriched spaces are constructed.
+We summarise all the wPOD procedure for OCP(µ)s in Algorithm 1.
+4.2. Online phase. In this stage, all operations have usually a N-independent cost. This process
+reflects to be computationally cheap and, therefore, it can be recalled multiple times using small
+machine resources. Firstly, we choose a parameter µ. We get all the µ-independent quantities and
+reduced spaces back from the storage. Immediately, we combined parameter independent part with
+the µ-dependent ones, that are rapidly calculated here. Then a Galerkin projector onto Y N, U N
+and (QN)∗ is performed, computing the reduced solution yN, uN and pN through a reduced block
+matrix system. As previously seen in Section 3, a stabilization is needed in the truth approximation.
+However, it could also not be the case for the online stage. This scenario lead to two possibilities:
+we do not use SUPG in the online phase, Offline-Only stabilization, or, on the contrary, stabilization
+occurs also here Offline-Online stabilization.
+
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+11
+Algorithm 1 Weighted POD algorithm for OCP(µ) problems through a partitioned approach
+Input: FEM spaces Y N , U N , and (QN )∗ parameter domain P, and Ntrain.
+Output: reduced spaces Y N, U N and (QN)∗.
+Considering the high-fidelity spaces Y N , U N and (QN )∗:
+1: Choose a quadrature rule (29) to approximate (30). This step defines a sample Ptrain ⊂ P and
+the respective weights w1, . . . , wNtrain. Define the matrix W := diag(w1, · · · , wNtrain) ;
+2: for all µ ∈ Ptrain do
+3:
+Solve the high-fidelity SUPG OCP(µ) system (15);
+4: end for
+5: Calculate the matrices Dy
+kl :=
+1
+Ntrain
+�
+yN (µk) , yN (µl)
+�
+Y , 1 ≤ k, l ≤ Ntrain and W y := W · Dy.
+Do the same for the control u and the adjoint p;
+6: Compute eigenvalues λy
+1, . . . , λy
+Ntrain
+and the corresponding orthonormalized eigenvectors
+gy
+1, . . . , gy
+Ntrain of W y. Do the same procedure for u and p variables;
+7: Fix N according to a certain criterion and construct Y N = span {ξy
+n, n = 1, . . . , N}, where
+ξy
+n =
+1
+√
+λy
+n
+�Ntrain
+t=1
+(gy
+n)t yN (µk). Do the same for u and p variables.
+8: Build the aggregated space GN = span
+�
+{ξy
+n}N
+n=1 ∪ {ξp
+n}N
+n=1
+�
+and set GN ≡ Y N ≡ (QN)∗.
+5. Numerical Results
+In this last part we illustrate numerical simulations concerning two Advection-Dominated OCP(µ)s
+under random inputs: the Graetz-Poiseuille Problem and the Propagating Front in a Square Prob-
+lem. In both experiments, the parameter µ will be a random vector and it will follow a prescribed
+probability density function that will be specified. The deterministic version of both experiments
+can be founded in [63].
+The Offline approximation will be always based on a P1−FEM, which means to consider a finite
+element method characterized by polynomials of degree less or equal than 1. In steady and unsteady
+simulations, the same stabilization parameter δK will be employed for both stabilization in the high-
+fidelity approximation and in the Online phase: namely in Offline-Online stabilization, δK is the
+same for both phases.
+For each simulation, relative errors between the FEM and the reduced solutions, i.e.
+(35)
+ey,N(µ) :=
+��yN (µ) − yN(µ)
+��
+Y
+∥yN (µ)∥Y
+, eu,N(µ) :=
+��uN (µ) − uN(µ)
+��
+U
+∥uN (µ)∥U
+, ep,N(µ) :=
+��pN (µ) − pN(µ)
+��
+Q∗
+∥pN (µ)∥Q∗
+,
+for the state, the control and the adjoint, respectively, will be shown. Due to the parametric nature
+of the problems, for each quantity in (35) a simple average is computed for µ distributed according
+to its probability density in a testing set Ptest ⊆ P of size Ntest, for every dimension N = 1, . . . , Nmax
+of the reduced space built through a chosen wPOD procedure. In every graph, the base-10 logarithm
+of these averages will be shown. When we will specify to use a POD procedure based on a Monte-
+Carlo sampling [57] of a uniform density distribution, we will talk about Standard POD. In order
+to compare the different wPOD possibilities, we use the same testing set for all of them: it will be
+taken using a Monte-Carlo method according to the distribution of µ. Obviously, the performance
+of the Standard POD will be based on a testing set of uniform density. The sum of the errors with
+respect to each discretized instant of time t will be taken into account in the unsteady versions.
+In order to compare the computational cost between the FEM solution with that of the reduced
+one for any possible dimension N, we use the speedup-index, i.e.
+(36)
+speedup-index = computational time of the high-fidelity solution
+computational time of the reduced solution
+,
+which will be calculated for any µ in the testing set. Again, we will shown its sample average
+for any dimension N. For each test case, we will use the same Ptest to compute relative errors and
+
+12
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+the speedup-index. The steady experiments are run using a machine with 16GB of RAM and Intel
+Core i7-7500U Dual Core, 2.7GHz for the CPU; whereas all parabolic simulations are computed
+considering 16GB of RAM and Intel Core i7 − 7700 Quad Core, 3.60GHz for the CPU.
+The code concerning steady experiments is implemented using the RBniCS library [2]; instead,
+the unsteady ones are provided using both RBniCS and multiphenics [1] libraries. These are python-
+based libraries, built on FEniCS [32].
+5.1. Numerical Tests for the Graetz-Poiseuille Problem. The Graetz-Poiseuille problem is
+an Advection-Diffusion problem that represents the heat conduction in a rectilinear pipe. Here the
+transfer of heat can be regulated through the walls of the duct, which can be held at certain fixed
+temperature [22, 37, 46, 59].
+Firstly, we present simulation concerning the stationary case, where a distributed control is em-
+ployed all over the whole domain. The parameter µ = (µ1, µ2) is composed by the diffusion compo-
+nent µ1 and the geometrical one µ2, which characterizes the length of the plate.
+Ωobs
+Ωobs
+Ωo
+Γo,1
+Γo,2
+Γo,3
+Γo,4
+Γo,5
+Γo,6
+(0,0)
+(1,0)
+(1+µ2,0)
+(1+µ2,0.2)
+(1+µ2,0.8)
+(1+µ2,1)
+(1,1)
+(0,1)
+Figure 1. Geometry of the Graetz-Poiseuille Problem.
+The problem is studied using (x0, x1) as spatial coordinates. Ωo is the domain observed for a
+certain value µ2 with boundary Γo.
+We deal with homogeneous Neumann boundary conditions
+(BC) on Γo,3 := {1 + µ2} × [0, 1] considering Figure 1. Instead, Dirichlet conditions are set on sides
+Γo,1 := [0, 1]×{0}, Γo,5 := [0, 1]×{1}, Γo,6 := {0}×[0, 1] by imposing y = 0 and Γo,2 := [1, 1+µ2]×{0}
+and Γo,4 := [1, 1 + µ2] × {1} by imposing y = 1.
+The formulation of the problem is the following: given µ ∈ P, find (y, u) ∈ ˜Y × U which solves
+min
+(y,u)
+1
+2
+�
+Ωobs(µ)
+(y(µ) − yd)2 dΩo(µ) + α
+2
+�
+Ωo(µ)
+u(µ)2 dΩo(µ),
+such that
+(37)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+− 1
+µ1
+∆y(µ) + 4x1(1 − x1)∂x0y(µ) = u(µ),
+in Ωo(µ),
+y(µ) = 0,
+on Γo,1(µ) ∪ Γo,5(µ) ∪ Γo,6(µ),
+y(µ) = 1,
+on Γo,2(µ) ∪ Γo,4(µ),
+∂y(µ)
+∂ν
+= 0,
+on Γo,3(µ),
+where ˜Y :=
+�
+v ∈ H1�
+Ωo
+�
+s.t. it satisfies the BC in (37)
+�
+and U = L2(Ωo). For the sake of clarity,
+a lifting function Ry ∈ H1(Ω) that fulfills the BC in (37) is used.
+Consequently, the variable
+¯y := y − Ry, with ¯y ∈ Y , is used, where
+Y :=
+�
+v ∈ H1
+0
+�
+Ω
+�
+s.t. ∂¯y
+∂ν = 0, on Γ3 and ¯y = 0 on Γ \ Γ3
+�
+.
+Furthermore, we settle Q := Y ∗ without any loss of generality. Therefore, the adjoint variable p
+is null on Γ \ Γ3. The observation domain is Ωobs := [1, 1 + µ2] × [0.8, 1] ∪ [1, 1 + µ2] × [0, 0.2] as
+illustrated in Figure 1. The value µ2 can change the domain under study. Having that the domain
+
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+13
+Ωo is µ-dependent itself, in the Offline Phase snapshots are based on different domains due to the
+sampling of the geometrical parameter components [24, 41, 45, 44]. To deal with the geometrical
+parametrization of the problem, we set a reference domain Ω and we build affine maps that transform
+Ω in Ωo for a defined µ. This procedure implies an automatic modification of some bilinear and
+linear forms involved in the weak formulation of Problem (37).
+We choose Ω = (0, 2) × (0, 1) as reference domain, that is the original one Ωo(µ) corresponding
+to µ2 = 1. We assume that µ2 is positive for the sake of simplicity. Considering Figure 1, we divide
+this into 2 subdomains, which are defined as Ω1 = (0, 1) × (0, 1) and Ω2 = (1, 2) × (0, 1). Then, we
+build two affine transformations:
+(38)
+V1(µ) : Ω1 → Ωo,1(µ) ⊂ R2,
+such that V1
+��� x
+y
+�
+; µ
+�
+:=
+� x
+y
+�
+,
+which is the identity map defined on the first subdomain Ω1 and V2(µ) : Ω2 → Ωo,2(µ) ⊂ R2 as
+(39)
+V2
+�� x
+y
+�
+; µ
+�
+=
+� µ2x
+y
+�
++
+� 1 − µ2
+0
+�
+= R2
+� x
+y
+�
++
+� 1 − µ2
+0
+�
+,
+where we have
+(40)
+R2 :=
+�
+µ2
+0
+0
+1
+�
+.
+Glueing together V1 and V2 for each µ ∈ P, we manage to build a one-to-one transformation
+V (µ) defined all over Ω. We denote the restrictions of Th to Ω1 and Ω2 with T 1
+h and T 2
+h , respec-
+tively. Therefore, we can express all the forms of the weak formulation under the effect of this
+transformation. For instance, after possible lifting, we have as = a + s and a∗
+s = a∗ + s∗ as
+(41)
+a
+�
+yN , qN ; µ
+�
+: =
+�
+Ω1
+1
+µ1
+∇yN · ∇qN + 4x1(1 − x1)∂x0yN qN
++
+�
+Ω2
+1
+µ1µ2
+∂x0yN ∂x0qN + µ2
+µ1
+∂x1yN ∂x1qN + 4x1(1 − x1)∂x0yN qN ,
+s
+�
+yN , qN ; µ
+�
+: =
+�
+K∈T 1
+h
+δKhK
+�
+K
+�
+4x1(1 − x1)∂x0yN �
+∂x0qN
++
+�
+K∈T 2
+h
+δK
+hK
+õ2
+�
+K
+�
+4x1(1 − x1)∂x0yN �
+∂x0qN ,
+a∗ �
+zN , pN ; µ
+�
+: =
+�
+Ω1
+1
+µ1
+∇pN · ∇zN − 4x1(1 − x1)∂x0pN zN
+−
+�
+Ω2
+1
+µ1µ2
+∂x0pN ∂x0zN − µ2
+µ1
+∂x1pN ∂x1zN − 4x1(1 − x1)∂x0pN zN ,
+s∗ �
+zN , pN ; µ
+�
+: =
+�
+K∈T 1
+h
+δKhK
+�
+K
+�
+4x1(1 − x1)∂x0pN �
+∂x0zN
++
+�
+K∈T 2
+h
+δK
+hK
+õ2
+�
+K
+�
+4x1(1 − x1)∂x0pN �
+∂x0zN ,
+for all yN , qN , zN , pN , ∈ Y N . In order to take into account the possible bad effect on stabilized
+forms due to a extension or shortening of our domain Ωo, we choose the stabilization parameter for
+K ∈ T 2
+h as δK
+hK
+õ2 , where õ2 =
+�
+| det(R2)| [35, 37, 58].
+For the FEM discretization, a quite coarse mesh of size h = 0.034 is used and the total dimension
+of the numerical problem is 13146. We take δK = 1.0 for all K ∈ Th. The parameter space is set
+as P :=
+�
+1, 105�
+×
+�
+0.5, 1.5
+�
+, from which we want to extract a training set Ptrain with cardinality
+Ntrain = 100. For the n bilinear form, we consider a penalization α = 0.01. Our aim is to minimize
+the L2-error between the state and the desired solution profile yd(x) = 1.0, function defined on Ωobs
+
+14
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+Figure 2. (top) FEM not stabilized and (bottom) FEM stabilized solution, y (right) and
+u (left), µ = (105, 1.5), h = 0.034, α = 0.01, δK = 1.0.
+of Figure 1. Each wPOD procedure is computed until a Nmax = 20 in a partitioned approach and
+then all algorithms are compared using a testing set Ptest of 100 elements in P.
+We suppose that µ follows a Beta(5, 3) distribution for both parameter µ1 and µ2, i.e.
+(42)
+µ1 ∼ 1 +
+�
+105 − 1
+�
+X1, where X1 ∼ Beta(5, 3),
+µ2 ∼ 0.5 +
+�
+1.5 − 0.5
+�
+X2, where X2 ∼ Beta(5, 3),
+where µ1 and µ2 are independent random variables. This implies that we consider more probable
+the parameters for which the Graetz-Poiseuille Problem has high values of the P´eclet number. In
+Figure 2, we highlight how the FEM solutions of the state and the control are for µ = (105, 1.5).
+The adjoint solution is not shown here because it is proportional to the control due to the gradient
+equation [19].
+From Figure 2, one can see that a stabilization is necessarily needed.
+We firstly exploit the
+Offline-Only stabilization procedure, which results regarding errors are shown in Figure 3. The
+performance is not good for any kind of wPOD. Moreover, the Standard POD does not perform
+good, either. Relative errors never drop under 10−2 for any variables, hence more stabilization is
+necessary in this case.
+In Figure 4 relative errors of the Offline-Online stabilization procedure are presented.
+Here
+the trend seems better than the Offline-Only one, because these quantities decay faster along the
+value of N. The wPOD Monte-Carlo is the best performer for all y, u, p variables, as a matter of
+fact, it reaches ey,16 = 2.13 · 10−7 for the state, for the adjoint ep,16 = 3.95 · 10−7 and the control
+eu,16 = 3.80·10−7. This procedure has a better performance of the Standard POD, which its accuracy
+is at least 100 times inferior of the wPOD Monte-Carlo after N > 11. Concerning other rules, it can
+be noticed that Smolyak grid techniques perform better than their tensor-rule counterparts, despite
+having a training set whose cardinality is similar, but less of 100: 93 and 91 for the Clenshaw-Curtis
+and Gauss-Jacobi sparse grids, respectively.
+In Figure 5 we visually compare the two possibilities of stabilization for the geometrical parametriza-
+tion of the Graetz-Pouiseuille problem for the wPOD Monte-Carlo.
+In Table 1 we compare the speedup-index for all wPOD algorithms. We see that computational
+values are all of the same order of magnitude. For the wPOD Monte-Carlo we calculate 87 reduced
+solutions in the time of a FEM one.
+Now we want to present the parabolic version of Problem (37). This unsteady problem has been
+studied without optimization in [37, 59] in a deterministic context and in [59] in a UQ one. Instead,
+the deterministic OCP(µ) Graetz Problem under boundary control without Advection-dominancy
+is studied in [54, 52] and the deterministicdistributed control configuration is analyzed in [63].
+
+1.2e+00
+0.8
+0.6
+0.4
+0.2
+-1.5e-021.0e+00
+0.5
+0
+-0.5
+-9.3e-011.2e+00
+0.8
+0.6
+0.4
+0.2
+-1.5e-021.0e+00
+-0.5
+- 0
+-0.5
+-9.3e-01STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+15
+Figure 3. Relative Errors for the Graetz-Poiseuille Problem - Offline-Only Stabiliza-
+tion; State (top-left), Control (top-right), Adjoint (bottom); Standard POD (blue), wPOD
+Monte-Carlo (orange), Gauss-Jacobi tensor rule (green), Gauss-Jacobi Smolyak grid (red),
+Clenshaw-Curtis tensor rule (cyan), Clenshaw-Curtis Smolyak grid (dark green), Pseudo-
+Random based on Halton numbers (pink).
+Speedup-index Graetz-Poiseuille Problem: Offline-Online Stabilization - µ1, µ2 ∼ Beta(5,3)
+N
+POD
+wPOD
+Gauss tensor
+Gauss Smolyak
+CC tensor
+CC Smolyak
+Ps. Random
+4
+113.0
+108.9
+110.1
+110.1
+106.5
+109.4
+112.0
+8
+108.4
+105.1
+104.9
+106.1
+102.1
+105.9
+107.4
+12
+103.3
+100.2
+99.9
+99.8
+99.1
+96.9
+101.7
+16
+97.2
+92.5
+95.1
+94.5
+92.6
+94.2
+96.9
+20
+90.5
+87.3
+87.0
+88.0
+85.8
+86.3
+89.7
+Table 1. Average Speedup-index of Offline-Online Stabilization for the Graetz-Poiseuille
+Problem under geometrical parametrization. From left to right: Standard POD, wPOD
+Monte-Carlo, Gauss-Jacobi tensor, Gauss-Jacobi Smolyak grid, Clenshaw-Curtis tensor,
+Clenshaw-Curtis Smolyak grid, Pseudo-Random based on Halton numbers.
+Recalling Figure 1, for a fixed T > 0 the unsteady Graetz-Poiseuille Problem is posed as follows:
+find (y, u) ∈ ˜Y × U which solves
+min
+(y,u)
+1
+2
+�
+Ωobs(µ)×(0,T )
+(y(µ) − yd)2 dΩ + α
+2
+�
+Ω(µ)×(0,T )
+u(µ)2 dΩ,
+such that
+
+FEM vs ROM averaged relative error - y (state)
+101
+Log-Error
+100
+Relative L
+StandardPOD
+WeightedPODMonte-Carlo
+Gaussjacobi-tensor
+Gausslacobi-Smolyak
+ClenshawCurtis  tensor
+10-1
+ClenshawCurtis+Smolyak
+PseudoRandom-Halton
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+NFEM vs ROM averaged relative error - u (control)
+101
+100
+Standard POD
+Weighted POD Monte-Carlo
+Gaussjacobi-tensor
+10-1
+Gaussjacobi- Smolyak
+ClenshawCurtis-tensor
+ClenshawCurtis  Smolyak
+PseudoRandom Halton
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+NFEM vs ROM averaged relative error - p (adjoint)
+102
+Log-Error
+101
+Relative
+StandardPOD
+WeightedPoDMonte-Carlo
+Gaussjacobi-tensor
+100
+Gaussjacobi--Smolyak
+ClenshawCurtis-tensor
+ClenshawCurtis Smolyak
+PseudoRandom Halton
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+N16
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+Figure 4. Relative Errors for the Graetz-Poiseuille Problem - Offline-Online Stabiliza-
+tion; State (top-left), Control (top-right), Adjoint (bottom); Standard POD (blue), wPOD
+Monte-Carlo (orange), Gauss-Jacobi tensor rule (green), Gauss-Jacobi Smolyak grid (red),
+Clenshaw-Curtis tensor rule (cyan), Clenshaw-Curtis Smolyak grid (dark green), Pseudo-
+Random based on Halton numbers (pink).
+Figure 5. (top) wPOD Monte-Carlo Offline-Only stabilized and (bottom) Offline-Online
+stabilized solution, y (right) and u (left), µ = (105, 1.5), h = 0.034, α = 0.01, Ntrain = 100,
+δK = 1.0, N = 20.
+(43)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+∂ty(µ) − 1
+µ1
+∆y(µ) + 4x1(1 − x1)∂x0y(µ) = u(µ),
+in Ω(µ) × (0, T),
+y(µ) = 0,
+on Γ1 ∪ Γ5 ∪ Γ6 × (0, T),
+y(µ) = 1,
+on Γ2(µ) ∪ Γ4(µ) × (0, T),
+∂y(µ)
+∂ν
+= 0,
+on Γ3(µ) × (0, T),
+y(µ)(0) = y0(x),
+in Ω(µ).
+
+FEM vs ROM averaged relative error - p (adjoint)
+101.4
+100
+10-1
+10-2
+10-3
+10
+Standard POD
+10-5
+Weighted POD Monte-Carlo
+Gaussjacobi+tensor
+GaussjacobiSmolyak
+ClenshawCurtis-tensor
+10-6
+ClenshawCurtis - Smolyak
+PseudoRandom - Halton
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+N1.2e+00
+1
+0.8
+0.6
+0.4
+ 0.2
+-1.5e-021.0e+00
+0.5
+一
+0
+-0.5
+-9.3e-011.2e+00
+0.8
+0.6
+0.4
+0.2
+-1.5e-021.0e+00
+0.5
+0
+-0.5
+-9.3e-01FEM vs ROM averaged relative error - y (state)
+10-1
+10-2
+10-3
+10
+10-5
+Standard POD
+Weighted PODMonte-Carlo
+Gaussjacobi +tensor
+10-6
+Gaussjacobi↓. Smolyak
+ClenshawCurtis -tensor
+ClenshawCurtis-Smolyak
+PseudoRandom-Halton
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+NFEM vs ROM averaged relative error - u (control)
+10-1
+10-3
+10
+StandardPOD
+10-5
+Weighted PODMonte-Carlo
+Gaussjacobi+tensor
+Gaussjacobi + Smolyak
+10-6
+ClenshawCurtis.-.tensor
+ClenshawCurtis - Smolyak
+PseudoRandom-Halton
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+NSTABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+17
+Figure 6. Relative Errors for the Parabolic Graetz-Poiseuille Problem - Offline-Only Sta-
+bilization; State (top-left), Control (top-right), Adjoint (bottom); Standard POD (blue),
+wPOD Monte-Carlo (orange), Gauss-Jacobi tensor rule (green), Gauss-Jacobi Smolyak
+grid (red), Clenshaw-Curtis tensor rule (cyan), Clenshaw-Curtis Smolyak grid (dark green),
+Pseudo-Random based on Halton numbers (pink).
+As made in the steady version, we firstly consider a lifting procedure. Simulations are run following
+the space-time setting proposed in Section 3.2 for a prearranged number of time-steps Nt.
+The initial condition is y0(x) = 0 for all x ∈ Ω referring to Figure 1 and we set T = 3.0. The
+penalization parameter is α = 0.01 and we want the state solution to be similar in the L2-norm to
+a desired solution profile yd(x, t) = 1.0, function defined for all t ∈ (0, 3.0) and for all x in Ωobs in
+Figure 1. Choosing Nt = 30, the time step is ∆t = 0.1. For the spatial discretization a quite coarse
+mesh of h = 0.038 is implemented: consequently the total high-fidelity dimension is Ntot = 314820
+and a single FEM space is characterized by N = 3498 for a fixed instant t . Again, δK = 1.0 for all
+K ∈ Th. We take P :=
+�
+1, 105�
+×
+�
+1, 3.0
+�
+and µ is determined by the probability distribution
+(44)
+µ1 ∼ 1 +
+�
+105 − 1
+�
+X1, where X1 ∼ Beta(5, 3),
+µ2 ∼ 1.0 +
+�
+3.0 − 1.0
+�
+X2, where X2 ∼ Beta(5, 3).
+We choose a training set Ptrain of cardinality Ntrain = 100 (with exception of sparse grids, which
+have similar cardinality) and we performed the wPOD algorithms with Nmax = 15.
+In Figure 6 we present relative errors related to Offline-Only stabilization. Also in the parabolic
+case this procedure does not perform well. Therefore an online stabilization is needed.
+As a matter of fact, one can see in Figure 7 that the trends for Offline-Online stabilization seems
+a lot better than the previous strategy. Besides the Clenshaw-Curtis quadrature rule, errors decrease
+along the dimension N. Again, the best performance is given by the wPOD Monte-Carlo, where the
+following values are reached for N = 14: ey,14 = 9.71·10−7,ep,14 = 9.21·10−7, and eu,14 = 2.64·10−7.
+
+FEM vs ROM averaged relative error - y (state)
+Standard POD
+WeightedPODMonte-Carlo
+Gaussjacobi-tensor
+GaussJacobi-Smolyak
+ClenshawCurtis -tensor
+101
+ClenshawCurtis--Smolyak
+PseudoRandom-Halton
+Relative Log-Error
+100
+10-1
+2
+4
+6
+8
+10
+12
+14
+NFEM vs ROM averaged relative error - u (control)
+StandardPOD
+WeightedPODMonte-Carlo
+Gaussjacobi-tensor
+101
+Gaussjacobi- Smolyak
+ClenshawCurtis-tensor
+ClenshawCurtis -Smolyak
+Relative Log-Error
+PseudoRandom - Halton
+100
+10-
+10-2
+2
+4
+6
+8
+10
+12
+14
+NFEM vs ROM averaged relative error - p (adjoint)
+Standard POD
+WeightedPODMonte-Carlo
+Gaussjacobi-tensor
+Gaussjacobi-Smolyak
+ClenshawCurtis -tensor
+ClenshawCurtis - Smolyak
+PseudoRandom-Halton
+Relative Log-Error
+101
+100
+2
+4
+6
+8
+10
+12
+14
+N18
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+Figure 7. Relative Errors for the Parabolic Graetz-Poiseuille Problem - Offline-Online
+Stabilization; State (top-left), Control (top-right), Adjoint (bottom); Standard POD
+(blue), wPOD Monte-Carlo (orange), Gauss-Jacobi tensor rule (green), Gauss-Jacobi
+Smolyak grid (red), Clenshaw-Curtis tensor rule (cyan), Clenshaw-Curtis Smolyak grid
+(dark green), Pseudo-Random based on Halton numbers (pink).
+Finally, in Table 2 we illustrate the performance of the speedup-index. All weighted algorithms
+performs similar: we compute an order of magnitude of 104 reduced solution in the time of a FEM
+one. This efficiency is given by the nature of the space-time procedure, where each snapshot carries
+all the time instances, and the reduction is very effective.
+Speedup-index Parabolic Graetz-Poiseuille Problem: Offline-Online Stab. - µ1, µ2 ∼ Beta(5,3)
+N
+POD
+wPOD
+Gauss tensor
+Gauss Smolyak
+CC tensor
+CC Smolyak
+Ps. Random
+3
+14299.7
+14571.0
+13970.7
+14013.4
+14524.6
+14578.1
+14106.2
+6
+14666.3
+15393.5
+14621.8
+14952.8
+15302.6
+15117.8
+14482.0
+9
+14245.6
+14803.1
+14125.6
+14546.5
+14756.7
+14608.5
+13986.9
+12
+13693.6
+14206.2
+13554.3
+13935.7
+14050.5
+14075.4
+13453.0
+15
+13090.9
+13606.4
+13055.8
+13455.1
+13548.6
+13544.0
+12875.2
+Table 2. Average Speedup-index of Offline-Online Stabilization for the Parabolic Graetz-
+Poiseuille Problem under geometrical parametrization. From left to right: Standard POD,
+wPOD Monte-Carlo, Gauss-Jacobi tensor, Gauss-Jacobi Smolyak grid, Clenshaw-Curtis
+tensor, Clenshaw-Curtis Smolyak grid, Pseudo-Random based on Halton numbers.
+5.2. Numerical Tests for Propagating Front in a Square Problem. Here we analyze an
+Advection-Dominated PDE problem illustrated without control in a deterministic and in a stochastic
+context in [37, 59] and in [59], respectively. A distributed control is applied all over the domain Ω,
+
+FEM vs ROM averaged relative error - y (state)
+Standard POD
+Weighted POD Monte-Carlo
+101
+Gaussjacobi- tensor
+GaussJacobi-Smolyak
+ClenshawCurtis-tensor
+100
+ClenshawCurtis-Smolyak
+PseudoRandom-Halton
+10-
+10-2
+10-3
+10
+10-5
+10-6
+2
+4
+6
+8
+10
+12
+14
+NFEM vs ROM averaged relative error - u (control)
+101
+100
+10
+10-2
+10-3
+10-
+.4
+Standard POD
+10-5
+Weighted POD Monte-Carlo
+Gaussjacobi-tensor
+Gaussjacobi-Smolyak
+10-6
+ClenshawCurtis.-.tensor
+ClenshawCurtis - Smolyak
+PseudoRandom -Halton
+2
+4
+6
+8
+10
+12
+14
+NFEM vs ROM averaged relative error - p (adioint)
+101
+10
+10-3
+Standard POD
+WeightedPODMonte-Cairlo
+Gaussjacobi-tensor
+10-5
+Gaussjacobi- Smolyak
+ClenshawCurtis - tensor
+ClenshawCurtis - Smolyak
+PseudoRandom-Halton
+2
+4
+6
+8
+10
+12
+14
+NSTABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+19
+which is the square (0, 1) × (0, 1), as shown under Cartesian coordinates (x0, x1) in Figure 8. The
+boundary is composed as follows: Γ1 := {0} × [0, 0.25], Γ2 := [0, 1] × {0}, Γ3 := {1} × [0, 1],
+Γ4 := [0, 1] × {1}, Γ5 := {0} × [0.25, 1]; instead Ωobs := [0.25, 1] × [0.75, 1].
+Γ1
+Γ2
+Γ3
+Γ4
+Γ5
+Ω
+Ωobs
+(0,0.25)
+(0,1)
+(1,0.75)
+(1,1)
+(0.25,1)
+(1,0)
+(0,0)
+Figure 8. Geometry of the Propagating Front in a Square Problem
+Given µ = (µ1, µ2), our aim is to solve the following OCP(µ) problem: find (y, u) ∈ ˜Y × U which
+solves
+min
+(y,u)
+1
+2
+�
+Ωobs
+(y(µ) − yd)2 dΩ + α
+2
+�
+Ω
+u(µ)2 dΩ,
+such that
+(45)
+�
+�
+�
+�
+�
+�
+�
+− 1
+µ1
+∆y(µ) + [cos µ2, sin µ2] · ∇y(µ) = u(µ),
+in Ω,
+y(µ) = 1,
+on Γ1 ∪ Γ2,
+y(µ) = 0,
+on Γ3 ∪ Γ4 ∪ Γ5.
+In this case, we have that the domain of definition of our state y is
+˜Y :=
+�
+v ∈ H1�
+Ω
+�
+s.t. BC in (45)
+�
+.
+Again, we define a lifting function Ry ∈ H1�
+Ω
+�
+such that satisfies BC in (45), applying a lifting
+procedure before the Lagrangian approach. We define ¯y := y − Ry, with ¯y ∈ Y and Y := H1
+0(Ω),
+U = L2(Ω) and Q := Y ∗, with p = 0 on ∂Ω.
+The mesh size h is equal to 0.025, which entails an overall dimension of the truth approximation
+of 12087.
+Consequently, we have N = 4029 for state, control and adjoint spaces.
+Concerning
+stabilization, δK = 1.0 for all K ∈ Th. The penalization parameter is α = 0.01 and we pursue the
+state solution to be similar in the L2-norm to yd(x) = 0.5, defined for all x in Ωobs of Figure 8. In
+our test cases, P :=
+�
+1, 4 · 104�
+×
+�
+0.9, 1.5
+�
+and µ follow the subsequent probability distribution:
+(46)
+µ1 ∼ 1 +
+�
+4 · 104 − 1
+�
+X1, where X1 ∼ Beta(10, 10),
+µ2 ∼ 0.9 +
+�
+1.5 − 0.9
+�
+X2, where X2 ∼ Beta(10, 10),
+where µ1 and µ2 are independent random variables. The training set Ptrain and the testing set
+Ptest have both cardinality equal to ntrain = 100, with exception of sparse grid samplings, whose
+cardinality is similar to 100. We apply a wPOD procedure for a Nmax = 50 dimension. In Figure 9,
+we show the performance of relative errors for the Offline-Only stabilization procedure. As in the
+Graetz-Poiseuille Problem, these trends are not acceptable, as no quantity drops under 10−1 for all
+state, control and adjoint variables. Therefore, a stabilization applied in the Online Phase is needed,
+too.
+In Figure 10 relative errors for Offline-Online Stabilization procedure are shown. Again, wPOD
+Monte-Carlo presents the best behaviour: in this case it reaches ey,50 = 5.03 · 10−7 for the state, for
+the adjoint ep,50 = 1.07·10−6, and the control eu,50 = 4.21·10−6. Moreover, the wPOD Monte-Carlo
+has an accuracy of nearly a factor of 100 better than a Standard POD in a deterministic context for
+N > 20. Also here, Smolyak grids perform better than their tensor counterpart: for instance, we
+obtain in this case it reaches ey,50 = 2.77 · 10−6 for the state, for the adjoint ep,50 = 5.80 · 10−6, and
+
+20
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+Figure 9. Relative Errors for the Propagating Front in a Problem - Offline-Only Stabiliza-
+tion; State (top-left), Control (top-right), Adjoint (bottom); Standard POD (blue), wPOD
+Monte-Carlo (orange), Gauss-Jacobi tensor rule (green), Gauss-Jacobi Smolyak grid (red),
+Clenshaw-Curtis tensor rule (cyan), Clenshaw-Curtis Smolyak grid (dark green), Pseudo-
+Random based on Halton numbers (pink).
+the control eu,50 = 1.02 · 10−5 for Gauss-Jacobi. Concerning the training set, we have Ntrain = 89
+and Ntrain = 93 for the Gauss-Jacobi and the Clenshaw-Curtis ones, respectively. In Figure 11
+we see a comparison between the FEM solution for the state and the adjoint without stabilization
+and the Offline-Online Stabilized wPOD Monte-Carlo reduced solution for these variables with
+µ = (2 · 104, 1.2).
+The values of the speedup-index for the Offline-Online stabilization for each type of wPOD are
+reported in Table 3. For N = 50 the wPOD Monte-Carlo is the best choice again with a computation
+of 50 reduced solutions in the time of a FEM one. All the other possibilities perform a little bit lower
+for N = 50; however, all weighted algorithms have similar performances concerning the speedup-
+index: an order of magnitude of 102 for the first 50 reduced basis.
+Numerical tests of the parabolic version of the Propagating Front in a Square Problem are here
+illustrated. For a fix T > 0 and a given µ ∈ P we have to find the pair (y, u) ∈ ˜Y × U which solves
+min
+(y,u)
+1
+2
+�
+Ωobs×(0,T )
+(y(µ) − yd)2 dΩ + α
+2
+�
+Ω×(0,T )
+u(µ)2 dΩ,
+such that
+(47)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+∂ty(µ) − 1
+µ1
+∆y(µ) + [cos µ2, sin µ2] · ∇y(µ) = u(µ),
+in Ω × (0, T),
+y(µ) = 1,
+on Γ1 ∪ Γ2 × (0, T),
+y(µ) = 0,
+on Γ3 ∪ Γ4 ∪ Γ5 × (0, T),
+y(µ)(0) = y0(x),
+in Ω,
+
+FEM vs ROM averaged relative error - y (state)
+Relative Log-Error
+StandardPOD
+100
+WeightedPoDMonte-Carlo
+GaussJacobi-tensor
+Gaussjacobi - Smolyak
+ClenshawCurtis-tensor
+ClenshawCurtis-Smolyak
+PseudoRandom-Halton
+10
+20
+30
+40
+50
+NFEM vs ROM averaged relative error - u (control)
+Relative Log-Error
+100
+Standard POD
+Weighted PODMonte-Carlo
+Gaussjacobi-tensor
+Gausslacobi-Smolyak
+ClenshawCurtis-tensor
+ClenshawCurtis-Smolyak
+PseudoRandom -Halton
+10
+20
+30
+40
+50
+NFEM vs ROM averaged relative error - p (adioint)
+Relative Log-Error
+101
+StandardPOD
+Weighted POD Monte-Carlo
+Gausslacobi-tensor
+Gaussjacobi- Smolyak
+ClenshawCurtis-tensor
+ClenshawCurtis-Smolyak
+PseudoRandom-Halton
+10
+20
+30
+40
+50STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+21
+Figure 10. Relative Errors for the Propagating Front in a Problem - Offline-Online Sta-
+bilization; State (top-left), Control (top-right), Adjoint (bottom); Standard POD (blue),
+wPOD Monte-Carlo (orange), Gauss-Jacobi tensor rule (green), Gauss-Jacobi Smolyak
+grid (red), Clenshaw-Curtis tensor rule (cyan), Clenshaw-Curtis Smolyak grid (dark green),
+Pseudo-Random based on Halton numbers (pink).
+Figure 11. FEM not stabilized and wPOD Monte-Carlo Offline-Online stabilized solu-
+tion for y (left) and for p (right), µ = (2 · 104, 1.2), h = 0.025 α = 0.01, Ntrain = 100,
+δK = 1.0, N = 50.
+where y0(x) = 0 for all x ∈ Ω in Figure 8. A final time T = 3.0 is set. Considering the time
+discretization, we chose a number of time steps equal to Nt = 30, then we have ∆t = 0.1. Instead,
+for the spatial approximation, the mesh size is set to h = 0.036, that implies an overall dimension
+of the space-time setting equal to Ntot = 174780.
+For a fixed instant t, a single FEM space is
+characterized by N = 1942.
+For the SUPG procedure, we impose δK = 1.0 for all K ∈ Th.
+Setting a penalization parameter α = 0.01, we try to achieve in a L2-mean a desired solution profile
+yd(x, t) = 0.5, defined for all t ∈ (0, 3) and x in Ωobs of Figure 8.
+P :=
+�
+1, 4 · 104�
+×
+�
+0.9, 1.5
+�
+, as in the steady version. We suppose that µ follows the probability
+distribution (46). Our training set has cardinality Ntrain = 100, with exception for Gauss-Jacobi and
+Clenshaw-Curtis Smolyak grids with Ntrain = 89 and Ntrain = 93, respectively, which are the number
+of nodes nearest to 100 for this kind of procedure. In Figure 12 and 13, we show a representative
+
+FEM vs ROM averaged relative error - y (state)
+10-1
+10-2
+10-3
+10
+4
+StandardPOD
+10-5
+WeightedPODMonte-Carlo
+GaussJacobi-tensor
+Gaussjaciobi-Smolyak
+ClenshawCurtis-tensor
+10-6
+ClenshawCurtis-Smolyak
+PseudoRandom-Halton
+10
+20
+30
+40
+50
+NFEM vs ROM averaged relative error - u (control)
+100
+Standard.POD
+Weighted POD Monte-Carlo
+GaussJacobi-tensor
+GaussJacobi - Smolyak
+ClenshawCurtis-tensor
+10-1
+ClenshawCurtis-Smolyak
+PseudoRandom -Halton
+10-2
+10-3
+10-5
+10
+20
+30
+40
+50
+NFEM ys ROM averaged relative error - p (adioint)
+100
+10-1
+10-2
+10-3
+StandardPOD
+WeightedPODMonte-Carlo
+10-5
+Gaussjacobi--tensor
+Gaussjacobi - Smolyak
+ClenshawCurtis-tensor
+ClenshawCurtis - Smolyak
+10-6
+PseudoRandom---Halton
+10
+20
+30
+40
+50
+N1.2e+00
+1.1
+0.9
+0.8
+0.7
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+-1.8e-029.7e-03
+0.008
+0.006
+0.004
+0.002
+0
+-0.002
+-0.004
+-0.006
+-8.4e-0322
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+Speedup-index Propagating front in a Square Problem: Offline-Online Stab. - µ1, µ2 ∼ Beta(10,10)
+N
+POD
+wPOD
+Gauss tensor
+Gauss Smolyak
+CC tensor
+CC Smolyak
+Ps. Random
+10
+151.3
+179.2
+175.0
+178.7
+181.5
+176.4
+173.9
+20
+123.3
+140.4
+139.8
+141.0
+140.9
+140.5
+143.6
+30
+88.5
+103.3
+102.6
+102.8
+100.6
+102.6
+104.3
+40
+61.6
+73.7
+73.2
+69.9
+68.6
+70.4
+70.2
+50
+43.4
+50.2
+49.0
+47.6
+46.8
+49.2
+48.2
+Table 3. Average Speedup-index of Offline-Online Stabilization for the Propagating
+Front in a Square Problem.
+From left to right: Standard POD, wPOD Monte-Carlo,
+Gauss-Jacobi tensor, Gauss-Jacobi Smolyak grid, Clenshaw-Curtis tensor, Clenshaw-
+Curtis Smolyak grid, Pseudo-Random based on Halton numbers.
+Figure 12. wPOD Monte-Carlo Offline-Online stabilized reduced solution of y, for t =
+0.1, t = 1.5, t = 3.0, µ = (2 · 104, 1.2), h = 0.036, α = 0.01, Ntrain = 100, δK = 1.0,
+N = 30.
+Figure 13. wPOD Monte-Carlo Offline-Online stabilized reduced solution of p, for t =
+0.1, t = 1.5, t = 3.0, µ = (2 · 104, 1.2), h = 0.036, α = 0.01, Ntrain = 100, δK = 1.0,
+N = 30.
+stabilized FEM solution for µ = (2 · 104, 1.2) for some instants of time of the state y and the adjoint
+p, respectively. We choose to perform all wPOD procedure with Nmax = 30.
+Let us move to the error analysis. In Figure 14 we illustrate the relative errors for the Offline-Only
+stabilization. The performance are not satisfactory here, too, where no quantity drops below the
+accuracy of 10−1 for all N.
+Instead, Offline-Online stabilization procedure performs well, as one can notice from Figure 15.
+Again, wPOD Monte-Carlo has the best behaviour, it reaches ey,30 = 1.12 · 10−7 for the state, for
+the adjoint ep,30 = 4.55 · 10−7 and the control eu,30 = 1.36 · 10−7. Also in this case, isotropic sparse
+grid techniques is a better choice than tensor rules, both for Gauss-Jacobi and Clenshaw-Curtis
+approximations.
+In Table 4 we compare the speedup-index for all the weighted algorithms: performance are similar
+for all N, for N = 30 we computed nearly 4000 Offline-Online stabilized reduced solutions in the
+time of a FEM one.
+
+1.2e+00
+1.1
+0.9
+0.8
+0.7
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+-1.8e-021.2e+00
+1.1
+0.9
+0.8
+0.7
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+-1.8e-021.2e+00
+0.9
+0.8
+0.7
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+-1.8e-029.7e-03
+0.008
+0.006
+0.004
+0.002
+0
+-0.002
+-0.004
+-0.006
+-8.4e-039.7e-03
+0.008
+0.006
+0.004
+0.002
+0
+-0.002
+-0.004
+-0.006
+-8.4e-039.7e-03
+0.008
+0.006
+0.004
+0.002
+0
+-0.002
+-0.004
+-0.006
+-8.4e-03STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+23
+Figure 14. Relative Errors for the Parabolic Propagating Front in a Problem - Offline-
+Only Stabilization; State (top-left), Control (top-right), Adjoint (bottom); Standard POD
+(blue), wPOD Monte-Carlo (orange), Gauss-Jacobi tensor rule (green), Gauss-Jacobi
+Smolyak grid (red), Clenshaw-Curtis tensor rule (cyan), Clenshaw-Curtis Smolyak grid
+(dark green), Pseudo-Random based on Halton numbers (pink).
+Speedup-index Parabolic Propagating front in a Square Problem: Offline-Online Stabilization
+N
+POD
+wPOD
+Gauss tensor
+Gauss Smolyak
+CC tensor
+CC Smolyak
+Ps. Random
+5
+6601.9
+6503.5
+6702.6
+6629.1
+6566.6
+6605.6
+6575.8
+10
+6275.9
+6208.0
+6336.0
+6277.9
+6204.4
+6293.4
+6212.4
+15
+5814.3
+5702.4
+5838.7
+5794.3
+5699.6
+5752.1
+5723.7
+20
+5327.9
+5190.4
+5329.8
+5270.3
+5277.2
+5235.9
+5197.6
+25
+4465.2
+4303.3
+4562.6
+4422.2
+4541.3
+4433.1
+4479.9
+30
+4061.5
+3959.5
+4140.3
+4026.3
+4100.3
+4035.6
+4043.8
+Table 4. Average Speedup-index of Offline-Online Stabilization for the Parabolic Propa-
+gating Front in a Square Problem. From left to right: Standard POD, wPOD Monte-Carlo,
+Gauss-Jacobi tensor, Gauss-Jacobi Smolyak grid, Clenshaw-Curtis tensor, Clenshaw-
+Curtis Smolyak grid, Pseudo-Random based on Halton numbers. µ1, µ2 ∼ Beta(10,10)
+6. Conclusions and Perspectives
+In this work, we illustrated some numerical tests concerning stabilized Parametrized Advection-
+Dominated OCPs with random parametric inputs in a ROM context. We deal with both steady and
+unsteady cases and we took advantage of the SUPG stabilization to overcome numerical issues due
+to high values of the P´eclet number. Two possibilities of stabilization were analyzed: when SUPG
+is only used occurs in the offline phase, Offline-Only stabilization, or when it is provided in both
+online and offline phases, Offline-Online stabilization.
+
+FEM vs ROM averaged relative error - y (state)
+StandardPOD
+Weighted POD Monte-Carlo
+100
+Gaussjacobi- tensor
+Gaussjacobi -Smolyak
+ClenshawCurtis-tensor
+ClenshawCurtis-Smolyak
+PseudoRandom-Halton
+5
+10
+15
+20
+25
+30
+NFEM vs ROM averaged relative error - u (control)
+Standard POD
+WeightedPODMonte-Carlo
+2 × 100
+Gaussjacobi-tensor
+GaussJacobi-Smolyak
+ClenshawCurtis-tensor
+ClenshawCurtis -Smolyak
+PseudoRandom-Halton
+100
+6×10-1
+5
+10
+15
+20
+25
+30
+NFEM vs ROM averaged relative error - p (adjoint)
+Standard POD
+Weighted PODMonte-Carlo
+Gaussjacobi-tensor
+GaussJacobi-Smolyak
+ClenshawCurtis-tensor
+ClenshawCurtis - Smolyak
+PseudoRandom-Halton
+Log-Error
+101
+Relative
+100
+5
+10
+15
+20
+25
+30
+N24
+STABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+Figure 15. Relative Errors for the Parabolic Propagating Front in a Problem - Offline-
+Online Stabilization; State (top-left), Control (top-right), Adjoint (bottom); Standard
+POD (blue), wPOD Monte-Carlo (orange), Gauss-Jacobi tensor rule (green), Gauss-Jacobi
+Smolyak grid (red), Clenshaw-Curtis tensor rule (cyan), Clenshaw-Curtis Smolyak grid
+(dark green), Pseudo-Random based on Halton numbers (pink).
+In order to deal with the uncertainty quantification caused by random inputs, we consider wROM.
+More precisely, we built our reduced bases using a wPOD in a partitioned approach, using different
+quadrature rules. We implemented wPOD Monte-Carlo, Gaussian quadrature formulae based on
+Jacobi polynomials in a tensor rule, approximation related to Clenshaw-Curtis tensor rule, Smolyak
+isotropic sparse grid approximation of the last two methods, quasi Monte-Carlo method as a Pseudo-
+Random rule defined on Halton numbers.
+We analyzed relative errors between the reduced and the high fidelity solutions and the speedup-
+index concerning the Graetz-Poiseuille and Propagating Front in a Square Problems, always under a
+distributed control. For the state, control and adjoint spaces we implemented a P1-FEM approxima-
+tion in a optimize-then-discretize framework as the truth solution. Concerning parabolic problems,
+a space-time approach is followed applying SUPG in a suitable way. In order to established which
+wPOD performs better, we compare them through the same testing set sampled by a Monte-Carlo
+method according to the probability distribution of the parameter.
+Offline-Only stabilization technique performed very poorly in terms of errors, this happened for
+all wROMs considered. Instead, in all the steady and unsteady experiments, the wROM technique
+performed excellently in an Offline-Online stabilization framework. For parabolic problems, the
+speedup-index features large values thanks to the space-time formulation. More precisely, wPOD
+Monte-Carlo technique was always the best performer for relative errors, instead, concerning compu-
+tational efficiency all methods seem equivalent. In addition, the efficiency of the wPOD Monte-Carlo
+
+FEM vs ROM averaged relative error - y (state)
+10-1
+10-2
+10-3
+10-5
+Standard POD
+WeightedPODMonte-Carlo
+10-6
+GaussJacobi-tensor
+Gaussjacobi- Smolyak
+ClenshawCurtis -tensor
+ClenshawCurtis - Smolyak
+10-7
+PseudoRandom-Halton
+5
+10
+15
+20
+25
+30
+NFEM vs ROM averaged relative error - u (control)
+10-1
+10-2
+10-3
+10-
+4
+10-5
+Standard POD
+WeightedPODMonte-Carlo
+Gaussjacobi-tensor
+10-6
+GaussJacobi-Smolyak
+ClenshawCurtis-tensor
+ClenshawCurtis-Smolyak
+10-7
+PseudoRandom-Halton
+5
+10
+15
+20
+25
+30
+NFEM vs ROM averaged relative error - p (adjoint)
+100
+10-
+10
+10-3
+10
+Standard POD
+10-5
+WeightedPODMonte-Carlo
+Gaussjacobi-tensor
+Gaussjacobi- Smolyak
+ClenshawCurtis -tensor
+10-6
+ClenshawCurtis -Smolyak
+PseudoRandom-Halton
+5
+10
+15
+20
+25
+30
+NSTABILIZED WEIGHTED ROMS FOR ADVECTION-DOMINATED OCPS WITH RANDOM INPUTS
+25
+is supported by the fact that after a small number of reduced basis it is nearly 100 times more accu-
+rate than a Standard POD in a deterministic context. Moreover, we notice that sparse grids perform
+better than relative tensor ones, although having a bit less number of quadrature nodes.
+Furthermore, in the Graetz-Poiseuille Problem we illustrate that under geometrical parametriza-
+tion affected by randomness, wROMs still have good performance, despite small fluctuations in the
+graph of relative errors.
+As a first perspective, it might be interesting to create a strongly-consistent stabilization technique
+that flattens all the fluctuations of geometrical parametrization in a UQ context. Moreover, we want
+to extend the study to boundary control. Finally, it might be interesting to study the performance
+of other stabilization techniques for the online phases, for instance, of the Online Vanishing Viscosity
+and the Online Rectification methods [4, 12, 33] combined with the SUPG technique in the offline
+phase or with the stabilization strategy used in [59].
+Acknowledgements
+We acknowledge the support by European Union Funding for Research and Innovation – Horizon
+2020 Program – in the framework of European Research Council Executive Agency: Consolidator
+Grant H2020 ERC CoG 2015 AROMA-CFD project 681447 “Advanced Reduced Order Methods
+with Applications in Computational Fluid Dynamics”. We also acknowledge the PRIN 2017 “Nu-
+merical Analysis for Full and Reduced Order Methods for the efficient and accurate solution of
+complex systems governed by Partial Differential Equations” (NA-FROM-PDEs) and the INDAM-
+GNCS project “Tecniche Numeriche Avanzate per Applicazioni Industriali”. The computations in
+this work have been performed with RBniCS [2] library, developed at SISSA mathLab, which is
+an implementation in FEniCS [32] of several reduced order modelling techniques; we acknowledge
+developers and contributors to both libraries.
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