text stringlengths 128 2.05k |
|---|
[EQUATION] With this subtraction, the integrands [MATH] and [MATH] behave as [MATH] at small momenta, but the leading [MATH] term is linear in [MATH] and thus [MATH] -integrates to zero. The result of the [MATH] -integration is hence [MATH] which, together with the factor [MATH] yields a finite loop integral at [MATH] ... |
[EQUATION] where it can be handled analytically: after performing the integrations with a small infrared regulator [MATH] in the potential [MATH] as in eq. ( 43 ), we obtain |
[EQUATION] The calculation for [MATH] is identical, without the overall factor [MATH] in the numerator, [EQUATION] Note that the full Poisson sum appears to be finite when the regulator is removed, while the [MATH] term is [MATH] and thus really diverges in the infrared. As explained in the previous section, both numer... |
[MATH] at finite [MATH] gives a finite result, while the limit [MATH] at finite [MATH] yields the expected [MATH] divergence [MATH] . This indicates that the correct formulation (the one which is continuously connected to the [MATH] case) must retain a small, but finite IR cutoff [MATH] . Taking [MATH] too early will l... |
[MATH] which ensures a smooth limit [MATH] and also stabilizes the iteration as explained in the previous section. With the Fourier integrand going as [MATH] at [MATH] , simple dimensional analysis suggests that the terms in the Poisson sum decay as [MATH] which, together with the alternating sign, amounts to a poorly ... |
This type of slowly converging alternating series can, however, be handled quite efficiently using the [MATH] -algorithm Wynn ( 1962 1956 . The alternative would be to attempt one more subtraction of the [MATH] behaviour under the integrands [MATH] and [MATH] . This leads, however, to a rather formidable expression inv... |
Eq. ( 44 ) is the final form of the gap equation which we solve iteratively: we start with an arbitrary function [MATH] , either the |
[MATH] solution or a constant, [EQUATION] and use Anderson’s algorithm as a sequence accelerator as in the T=0 case, cf. fig. Once the system has been iterated to convergence, we can extract the chiral condensate from |
[EQUATION] which is the finite temperature extension of eq. ( 36 ). III.3 The finite temperature case: Matsubara formulation The poisson resummation technique described in the last section is convenient at low temperatures were only a few terms are required. In addition, the [MATH] limit is recovered from the lowest te... |
Though our numerical procedure mainly relies on the Poisson technique, we have also solved quark gap equation ( 24 ) in the Matsubara representation and compared it with the results of the Poisson formulation. This will provide an independent test for the accuracy of our numerics. |
For the Matsubara formulation, we employ the residual [MATH] symmetry of rotations about the [MATH] -axis to let the external momentum component in the plane perpendicular to the heat bath point into 1-direction, [MATH] For the loop integration, we use polar coordinates [MATH] and |
[MATH] . We can then express eq. ( 24 ) in these coordinates, scale all dimensionfull quantities in the units of eq. ( 29 ) and finally go over to the more stable quotient form. This gives |
[EQUATION] where [MATH] Because of the symmetry [MATH] , we can restrict the external Matsubara index to [MATH] . On the rhs of eq. ( 52 ), we have also combined the terms with index [MATH] and [MATH] , since they only differ in the sign of the frequency [MATH] . This ensures that only mass functions with Matsubara ind... |
[MATH] appear on both sides of eq. ( 52 ) and the coupled integral equation system closes. The chiral condensate can be expressed in the Matsubara formulation as |
[EQUATION] For a numerical evaluation, the potential [MATH] must be infrared regularized as in eq. ( 43 ), and the number of Matsubara frequencies included in the system must be restricted to [MATH] . The system ( 52 ) then resembles the [MATH] |
equation, however with an [MATH] -component solution [MATH] and a different integration measure. It is the latter property which makes the Matsubara formulation less convenient: for small regulators [MATH] , the potential [MATH] has a strong singularity at [MATH] if the external and loop frequency match, |
[MATH] . This singularity is only partially cancelled by the integration measure and the [MATH] term dominates the entire Matsubara sum by a relative factor [MATH] . As before, this singular factor is canceled between the numerator and denominator, and the remaining [MATH] contributions from the other terms [MATH] |
in the Matsubara sum carry the actual corrections to the mass functions. This means that the iteration progresses much slower than at [MATH] and, more problematic, the |
[MATH] terms from the Matsubara sums must be computed to a very high accuracy. In addition to the high accuracy demand, the Matsubara formulation has the property that all components [MATH] are coupled by the system 52 ), so that the index cutoff [MATH] must be fixed once and for all and cannot be adjusted dynamically. |
If the external index [MATH] approaches the cutoff [MATH] , there are only a few frequencies larger than the dominating contribution [MATH] , i.e. the Matsubara sum is truncated unsymmetrically and the higher frequencies |
[MATH] thus have a systematic bias. The only solution is to include a very large number of frequencies [MATH] , so that the inaccurate modes near the cutoff give such a small contribution that their combined error does not matter. Unfortunately, the computational effort of the system 52 ) scales strictly as [MATH] , so... |
Finally, it should also be mentioned that the convergence of the Matsubara system 52 ) is non-uniform in the frequency index, i.e. the lowest frequencies are stable after a relatively small iteration count, while the highest frequencies (which contribute the least) are the slowest to converge. In practice, we have to s... |
IV Results We split our result section in two parts: first we consider numerical details on the individual parts of our calculation to demonstrate that the fairly complicated process actually works as intended. In the second part, we discuss the final results for the mass function and the chiral condensate at different... |
IV.1 Details on the numerical method In the following, we present some typical results of intermediate steps in the calculation. Numerical issues appear predominantly in the earlier steps of the iteration and the eventual mass function has a similar shape to the initial zero-temperature solution, cf. below. For simplic... |
[MATH] and [MATH] , which is in the region where the mass function changes most quickly. It should also be noted that we generally combine terms with both signs |
[MATH] in all internal calculations, i.e. we symmetrize the [MATH] -integrand [EQUATION] To keep the formulas simple, we will not always indicate this symmetrization, which is implicitly understood. |
We begin with the integrand of the momentum integral omitting the Fourier cosine factor for clarity and combining terms from [MATH] and [MATH] |
[EQUATION] The integration here can be done very efficiently using Gauss-Chebychev integration, which automatically takes care of the square root factor in the denominator. In Fig. , we have plotted eq. ( 55 ) for two values of [MATH] close to the boundary, for both the numerator (left panel) and the denominator (right... |
Next, we consider the integrand of the [MATH] -integral after performing the Fourier momentum integration, [EQUATION] Note that the argument [MATH] in this function can be restricted to [MATH] |
due to the symmetrization of [MATH] . Besides the external momentum (which we have fixed to the same standard value as in the previous figures), this function now depends on both the temperature and the Poisson summation index. In the left panel of Fig. , we have plotted eq. ( 56 ) for a fixed temperature [MATH] and tw... |
[MATH] [EQUATION] In the right panel of Fig. , we show the [MATH] transformation eq. ( 57 ) of the plots in the left panel. The detailed structure at low [MATH] is now spread out and the resulting function can be accurately integrated using Gauss-Legendre with about [MATH] sampling points. (By contrast, more than [MATH... |
[MATH] for [MATH] , respectively. This demonstrates the relatively slow [MATH] decay of the Poisson sum, even at a small temperature of |
[MATH] . The integral of the transformed function in the right panel of Fig. agree, of course, with the corresponding integral in the left panel. The integrands in the denominator of the gap equation show a qualitatively similar behavior and are not plotted here for brevity. |
Finally, we check the convergence of the Poisson sums in the gap equation 44 ). We fix the external momentum again at our preferred value |
[MATH] and [MATH] , and plot the partial Poisson sums in eq. ( 44 ), including the prefactor [MATH] The sums are even in [MATH] , i.e. we can combine terms with [MATH] |
[EQUATION] Fig. presents the partial sums in the gap equation ( 44 ) as a function of the upper summation bound, for our preferred external momentum setup. The left panel shows the situation for [MATH] , while the right panel displays [MATH] . The [MATH] term dominates in all cases, while the [MATH] terms contribute wi... |
[MATH] terms would have to be summed in this case, in order to suppress the oscillations and predict the value of the infinite series to a relative accuracy of [MATH] . By contrast, the [MATH] -algorithm is able to reach the same accuracy from only the first [MATH] terms in the series. |
IV.2 Results In the left panel of Fig. , we show the mass function [MATH] for the two extreme directions [MATH] of the external momentum (longitudinal and perpendicular to the heat bath). As explained earlier, the closest analog of the [MATH] mass function |
[MATH] in our finite temperature formulation is [MATH] , when the external momentum points in the direction of the heat bath. This curve has indeed a very similar form to the |
[MATH] solution plotted for comparison, while the [MATH] curve shows some deviations. Also, the mass function is already considerably smaller than at [MATH] , even though the temperature [MATH] in this plot is still in the confined phase. In order to recover the [MATH] limit, we would thus have to go to very small temp... |
[MATH] . From eq. ( 51 ), this does not directly translate into a drop of the condensate, since the mass function appears in the numerator and denominator of the integrand, which is thus less sensitive to the mass function at small momenta. (The temperature also appears through the Fourier sum which gives the main temp... |
For these reasons, the intercept [MATH] of the mass function is not a good indicator for the phase transition, in particular since it is also gauge dependent. This is also the reason why it cannot be directly compared to the constitutent mass in conventional covariant gauges; instead, it represents a gauge-dependent |
mass parameter from the instantaneous part of the quark propagator in Coulomb gauge. In Ref. Campagnari and Reinhardt ( 2018 it was demonstrated that such a mass parameter in Coulomb gauge could be as low as [MATH] |
and still be compatible with the (much larger) constituent masses in Landau gauge. The gauge-invariant order parameter for the phase transition is the chiral condensate plotted in Fig. . This shows the expected behaviour, i.e. it is roughly constant for small temperatures, and drops quickly to very small values at a ch... |
The exact location of the phase transition temperature [MATH] depends on the order of the transition: If it is a strong cross-over, one usually defines |
[MATH] from the inflexion point, while a second order transition would have a jump discontinuity in the derivative at the temperature where a non-vanishing condensate appears for the first time as we cool from the deconfined phase. This value of the transition temperature is always larger than the inflexion point. Our ... |
[EQUATION] Both the order and the critical temperature differ from the result of our previous investigation in ref. Reinhardt and Vastag ( 2016 , where a broad crossover phase transition with pseudo-critical temperature [MATH] was obtained in the limit of a vanishing quark-gluon coupling. It should, however, be mention... |
In Fig. , we have also included data from a direct summation of the original Matsubara series ( 24 ) with the same cutoff [MATH] |
to take care of the infrared-singularity and the transformation to quotient form for better stability. This system scales quadratically with the number of included Matsubara frequencies, and up to [MATH] frequencies (i.e. 10,000 times the effort as compared to [MATH] ) were necessary to even reach the transition region... |
By contrast, the Poisson formulation, though much more expensive per iteration, has no infrared singularity and provides the correct results even at larger temperatures, where the increase in computational effort as compared to lower temperatures is still moderate. Combined with the physical transparency of the method,... |
Returning to the results for the chiral phase transition, it should be emphasized again that the absolute value of [MATH] as well as the size of the condensate depends on the overall scale set by the Coulomb string tension, cf. eq. ( 29 ). The cited values are for the preferred value |
[MATH] . In view of the uncertainties about the fundamental scale, it is better to cite our findings for the critical temperature as |
[EQUATION] For instance, a somewhat larger value [MATH] would reproduce the lattice results for the chiral condensate and push the transition temperature to around [MATH] |
A second-order chiral phase transition is the expected result for a system of two chiral quark flavors, as can be seen from the so-called Columbia diagram Karsch ( 2002 ); Fukushima and Hatsuda ( 2011 . This might be surprising since we consider only one single quark flavor within our model. One should, however, notice... |
While the order of the chiral phase transition is the same in both the canonical and the present approach to finite temperatures, the critical temperature |
[MATH] found in the numerical calculations of ref. Lo and Swanson ( 2010 for vanishing chemical potential [MATH] is significantly smaller than our result. There are several possible reasons for this discrepancy: On the one hand, the evaluation of the partition function [see eq. ( )] |
[EQUATION] where [MATH] is the QCD Hamiltonian on [MATH] , necessitates some approximations in the canonical approach. This concerns especially the treatment of the density operator |
[MATH] [Eq. ( )] of the grand canonical ensemble, where a quasi-particle approximation is required for the Hamiltonian [MATH] Note that such an approximation is not required in the present approach and no equations of motion for the quasi-particle energies hence emerge. On the other hand, however, the present approach ... |
correlator Greensite and Olejnik ( 2003 – but no contribution from the spatial gluons. Nevertheless, the value for the critical temperature eq. ( 60 ) is too small as compared to the one found in lattice simulations for physical quark masses, |
[MATH] Borsányi et al. 2010 ); *Bazavov2012 Note that also the gauge invariant chiral quark condensate (Fig. falls behind the value expected from phenomenology. The fact that the Adler–Davis model predicts significantly too small results, e.g. for the pion decay constant, is well-known and also obtained in the canonica... |
Unfortunately, the inclusion of the coupling to the transversal spatial gluons will drastically increase the numerical costs, which is already considerable in the present study. Although the coupling effects turned out to be small when the solution of the zero-temperature gap equation is used Reinhardt and Vastag ( 201... |
Finally, one should emphasize again that the Coulomb string tension could also be adjusted to [MATH] , which reproduces the phenomenological value of the quark condensate, [MATH] , but is somewhat larger than the lattice prediction. With this arrangement, one would find a critical temperature of [MATH] |
Summary and Conclusions In the present paper, we have revisited the alternative Hamiltonian approach to finite-temperature QCD of Ref. Reinhardt ( 2016 and solved the temperature-dependent equations of motion of the fermion sector numerically. In a first study, we have ignored the coupling of quarks and transverse spat... |
[MATH] symmetry requires a Poisson sum and three nested integrations per temperature, of which the Fourier quadrature, in particular, is fairly expensive. As a consequence, the final solution amounts to 100+ CPU hours per temperature. |
For zero bare quark masses, the results for the chiral quark condensate show the expected weak second-order phase transition with a critical temperature of |
[MATH] While this value is larger than the one found within the usual (canonical) approach to finite temperature Hamiltonian QCD Lo and Swanson ( 2010 , our findings for the critical temperature are definitely smaller than the result of lattice calculations using dynamical quarks, [MATH] |
Borsányi et al. 2010 ); *Bazavov2012 We suspect that the mismatch between our findings and the lattice results is related to the neglect of the quark-gluon coupling in the variational ansatz, which also leads to a value of the quark condensate which is too small. However, if the scale is adjusted to reproduce the physi... |
Acknowledgments This work was partially supported by Deutsche Forschungsgemeinschaft (DFG) under Contract No. DFG-Re856/9-2 and under contract DFG-Re856/10-1. |
# Source: arxiv 1806.04563 # Title: Benchmarking Evolutionary Algorithms For Single Objective Real-valued Constrained Optimization - A Critical Review # Sections: all # Downloaded: 2026-03-03T01:57:43.192891+00:00 |
Benchmarking Evolutionary Algorithms For Single Objective Real-valued Constrained Optimization – A Critical Review – Abstract Benchmarking plays an important role in the development of novel search algorithms as well as for the assessment and comparison of contemporary algorithmic ideas. This paper presents common prin... |
keywords: Benchmarking, Constrained Optimization, Evolutionary Algorithms, Continuous Optimization Introduction Representing a subclass of derivative-free, nature-inspired methods for optimization, Evolutionary Algorithms (EA) provide powerful optimization tools for, but not restricted to, black-box or simulation-based... |
Due to the lack of theoretical performance results for optimization tasks of notable complexity, the development and the performance comparison of EA widely rely on benchmarking. First and foremost, benchmarking experiments are established for performance evaluation and algorithm comparison on given problem classes. Id... |
Yet, benchmarks can also be used to experimentally provide insight into the working principles of an algorithm (although, usually purpose-built experiments have to be conducted in addition) and foster the development of algorithms for specific problem branches. Furthermore, benchmarks may qualify to verify theoretical ... |
Currently, there are basically two main developing lines for EA benchmarking, the test environments provided in the IEEE Congress on Evolutionary Computation (CEC) competitions and the Comparing Continuous Optimizer (COCO) benchmark suite. |
The COCO suite represents the most elaborated platform for benchmarking and comparing unconstrained continuous optimizers for numerical (non-linear) optimization. The COCO framework advanced from the Black-box Optimization Benchmarking (BBOB) 2009 benchmark set |
The platform provides tools to ease the process of quantifying and comparing the performance of optimization algorithms for single-objective noiseless and noisy problems, and for bi-objective noiseless problems, respectively. A particular strength of the COCO platform is the large number of algorithm results available ... |
Alternatively, the competitions that are organized on a yearly basis during the CEC aim at the comparison of state-of-the-art stochastic search algorithms. These competitions, among others, include single objective, large-scale, noisy, multi-objective, and constrained optimization problems, respectively. The CEC compet... |
Being commonly recognized as successful optimization strategies in the context of unconstrained optimization, the application of EA to constrained optimization problems has gained the attention of the research community in recent years. Constrained optimization tasks are concerned with searching for the optimal solutio... |
for a survey on commonly used constraint handling approaches in the context of nature-inspired algorithms. Taking into account constrained optimization problems, the theoretical background of EA is even less developed. Hence, usage of benchmarks for performance assessment and algorithm development is essential. |
Regarding EA benchmarks for constrained optimization, the CEC competitions on constrained real-parameter optimization (organized in 2006, 2010, and 2017) introduced specific constrained test environments. The constrained test functions included in the CEC 2006 benchmark definitions were collected from |
. The following competitions refined some benchmark definitions and introduced new problem instances. To this end, the test-case generator developed in |
was called on. The respective paper introduces a method to generate test problems with varying features, e.g. with respect to the problem size, the size of the feasible region, or the number and the type of the constraints. Benchmark problems created by the test-case generator are included into the CEC2010 and CEC2017 ... |
Only recently, the development of a COCO branch for constrained black-box optimization benchmark (BBOB-constrained) problems is near completion |
Although the BBOB-constrained suite is still under development, a review of the corresponding benchmark principles is provided in Sec. Including the unfinished BBOB-constrained suite into the considerations is reasonable for the following reason. Representing the most sophisticated framework for unconstrained benchmark... |
An overview of additional problem collections is available at Each of these test problem sets is useful for demonstrations of the applicability of interesting algorithmic ideas. However, the presented problems are mainly related to the field of mathematical programming. They are provided in different mathematical model... |
Similar concerns apply to many real-world problem applications that are present in the literature . They usually come with limited reproducibility and comparability of the results reported, e.g. due to unavailable data or implicit modeling assumptions. Consequently, these studies can rather be thought of as a demonstra... |
The main goal of the present paper is to provide a critical review of state-of-the-art benchmarking environments that can be used for assessing and comparing Evolutionary Algorithms in the context of constrained single objective real-valued optimization. To this end, existing benchmark principles for constrained optimi... |
This way, the article may raise awareness of recent benchmarking techniques as well as their corresponding strengths and their incapabilities. By suggesting room for improvements with respect to framework definitions, experimentation principles, and reporting styles, the present paper aims at stimulating the debate on ... |
Such a discussion seems necessary as the field of constrained optimization is spacious and the available benchmarking approaches are comparably scarce. Although some investigations exist |
, it is by no means conclusively determined which features are making a constrained optimization problem hard even for a single algorithm subclass. Constrained real-valued optimization problems may differ with respect to the following features (and their combinations), including but not necessarily restricted to, |
1. the number of constraints, 2. the type of the constraints (refer to Sec. ), 3. the analytical structure of objective function and constraints, e.g. |
(a) the conditioning of the problem (b) the modality of the objective function (c) the ruggedness of the objective function (d) the (non-)linearity of the constraints |
(e) the separability of the objective function and/or constraints (f) the number of global optima (inside the feasible region) 4. |
the size of the search space, 5. the relative size of the feasible region in the search space, 6. the connectedness of the feasible region, |
7. the orientation of the feasible region within the search space, 8. the location of the global optimum on the boundary or aside. |
As there certainly is no such thing as free lunch , and as the EA development for constraint optimization tasks will further rely on the availability of suitable benchmarks, the need for benchmark definitions that take into account consistent subgroups of conceivable problems is beyond dispute. The test-case generator ... |
or test problem collections like , can be regarded as a meaningful step towards creating well structured problem groups of distinct characteristics. However, these problems and their reported solutions are commonly not scalable with respect to the problem dimensionality. Further, the issue of proposing a well-defined b... |
The remainder of this paper is organized as follows: Section introduces the general real-valued constrained optimization problem ( COP ), particularly with regard to a classification of the constraint functions commonly used for benchmarking EA on black-box problems. Section presents benchmarking principles appropriate... |
Problem formulation The present paper focuses on continuous optimization problems. That is, both the objective function and the constraint functions are assumed to be real-valued functions. The objective function might either be represented as a reward or as a cost function While the former calls for maximization, a co... |
The constraint functions fall into even more classes. A detailed taxonomy of constraints is provided in The paper subdivides constraints into nine distinct constraint classes which rely on the categorization according to the following features. |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.