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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2c7ae148e41bc37fa12737c90411c472e9702cfc | subsection | 51 | 167 | Stage-1 Discretization. | Stage 3: Evaluate direct interactionsboxes b
For each conventional target t owned by b, add to \mathsf {P}^{\text{near}}_b(t)
the contribution due to the interactions from sources owned by boxes in
{1}{1}{U_{b}}{{1}{2}{V_{b}}{{1}{3}{W_{b}}{{1}{3close}{W^\mathrm {close}_{b}}{{1}{3far}{W^\mathrm {far}_{b}}{{1}{4}{X_{b}}{... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0... |
c1a5d2356367a08ff89146a8a13b8ef9355e6b27 | subsection | 52 | 167 | Stage-1 Discretization. | Stage 5(a): Evaluate direct interactions due to {3close}{1}{U_{b}}{{3close}{2}{V_{b}}{{3close}{3}{W_{b}}{{3close}{3close}{W^\mathrm {close}_{b}}{{3close}{3far}{W^\mathrm {far}_{b}}{{3close}{4}{X_{b}}{{3close}{4close}{X^\mathrm {close}_{b}}{{3close}{4far}{X^\mathrm {far}_{b}}{}}}}}}}}Repeat Stage 3 with {3close}{1}{U_{b... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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495f7f08039f41fa236183792ee55164f6773132 | subsection | 53 | 167 | Stage-1 Discretization. | Stage 5(b): Evaluate multipoles due to {3far}{1}{U_{b}}{{3far}{2}{V_{b}}{{3far}{3}{W_{b}}{{3far}{3close}{W^\mathrm {close}_{b}}{{3far}{3far}{W^\mathrm {far}_{b}}{{3far}{4}{X_{b}}{{3far}{4close}{X^\mathrm {close}_{b}}{{3far}{4far}{X^\mathrm {far}_{b}}{}}}}}}}}boxes b
For each conventional target t owned by b, evaluate t... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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bef891beafbcb24a51ab6f971e8b08411da03553 | subsection | 54 | 167 | Stage-1 Discretization. | Stage 6(a): Evaluate direct interactions due to {4close}{1}{U_{b}}{{4close}{2}{V_{b}}{{4close}{3}{W_{b}}{{4close}{3close}{W^\mathrm {close}_{b}}{{4close}{3far}{W^\mathrm {far}_{b}}{{4close}{4}{X_{b}}{{4close}{4close}{X^\mathrm {close}_{b}}{{4close}{4far}{X^\mathrm {far}_{b}}{}}}}}}}}Repeat Stage 3 with {4close}{1}{U_{b... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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-... |
e0ab3499c8ffa29cbe94a4442c5abb91a05d6db8 | subsection | 55 | 167 | Stage-1 Discretization. | Stage 8: Form local expansions at QBX centersboxes b
For each QBX center c owned by b, translate \mathsf {L}^{\text{far}}_b to
c, obtaining \mathsf {L}^{\text{qbx},\text{far}}_{c}.
Stage 9: Evaluate final potential at targetsboxes b
For each conventional target t owned by b, evaluate \mathsf {L}^{\text{far}}_b(t).
A... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.... |
ca0b10ec9737fb1da9868c29c6c58cfbf1fb0306 | subsection | 56 | 167 | Stage-1 Discretization. | Assuming the truth of
Hypotheses REF , REF , and REF , there exists a
constant M > 0 such that for every target point x \in \mathbb {R}^3, we
have
\left|
\mathcal {S}_{\mathrm {QBX}({p_\mathrm {qbx}},N)} \mu (x)
- \mathcal {G}_{p_\mathrm {fmm}}[\mathcal {S}_{\mathrm {QBX}({p_\mathrm {qbx}},N)}] \mu (x)
\right|
\le \fr... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.0184... |
c6b91a7ad84eeaaa825a3261274602c3ddbbc934 | subsection | 57 | 167 | Stage-1 Discretization. | The potential at c is the sum of the contributions
\mathsf {L}^{\text{qbx},\text{near}}_{c}(t), \mathsf {L}^{\text{qbx},{3}{1}{U_{}}{{3}{2}{V_{}}{{3}{3}{W_{}}{{3}{3close}{W^\mathrm {close}_{}}{{3}{3far}{W^\mathrm {far}_{}}{{3}{4}{X_{}}{{3}{4close}{X^\mathrm {close}_{}}{{3}{4far}{X^\mathrm {far}_{}}{}}}}}}}}}_{c}(t), an... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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... |
942fdf6e906eccd031ddfd454b64ca3719dc2914 | subsection | 58 | 167 | Stage-1 Discretization. | The potential due to \mathsf {L}^{\text{qbx},\text{far}}_{c}(t) arrives via an interaction of {4far}{1}{U_{b^{\prime }}}{{4far}{2}{V_{b^{\prime }}}{{4far}{3}{W_{b^{\prime }}}{{4far}{3close}{W^\mathrm {close}_{b^{\prime }}}{{4far}{3far}{W^\mathrm {far}_{b^{\prime }}}{{4far}{4}{X_{b^{\prime }}}{{4far}{4close}{X^\mathrm {... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.011025628075003624,
0.... |
c022d2d3194b9fd2d079f2e770c077cb7d447728 | subsection | 59 | 167 | Stage-1 Discretization. | Lastly, the contribution due to all {2}{1}{U_{b^{\prime }}}{{2}{2}{V_{b^{\prime }}}{{2}{3}{W_{b^{\prime }}}{{2}{3close}{W^\mathrm {close}_{b^{\prime }}}{{2}{3far}{W^\mathrm {far}_{b^{\prime }}}{{2}{4}{X_{b^{\prime }}}{{2}{4close}{X^\mathrm {close}_{b^{\prime }}}{{2}{4far}{X^\mathrm {far}_{b^{\prime }}}{}}}}}}}} interac... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0... |
6eaa9f1779f3b6aa6d4ba84c7b078b68e77b551c | subsection | 60 | 167 | Stage-1 Discretization. | The cost of the tree build phase (Stage 1) and the
evaluation phase of the algorithm (Stages 2–9) are treated
separately. Under broadly applicable assumptions, the evaluation phase can be
shown to run in time that is proportional to the number of
particles. Nevertheless, the proportionality constant is affected by the ... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.007221753243356943,
... |
1e5b98e750ffc3801d16ef5a8a5b5a49e6a8a702 | subsection | 61 | 167 | Stage-1 Discretization. | Numerical Experiments
We use a family of smooth `urchin' test geometries \gamma _k
given analytically in spherical coordinates
(r_k, \theta ,\phi ) by prescribing r_k as a function of (\theta ,\phi ), where
r_k(\theta ,\phi ) &= 0.2 + \frac{\mathop {\mathrm {Re}}Y_{k}^{\lfloor k/2\rfloor }(\theta ,\phi ) - m_k}{M_k-m... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.01... |
64275f2709e78d72dde594d77487f3e822e53444 | subsection | 62 | 167 | Stage-1 Discretization. | We use the residual in this identity as a measure for the accuracy
that our scheme achieves in the evaluation of layer potential evaluations.
The achieved accuracy in Green's formula is predictive of the accuracy
one might achieve in the solution of boundary value problems. Data to support
this assertion (in two dimens... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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... |
ac447bd946a99572073fc182e64e891248a54907 | subsection | 63 | 167 | Stage-1 Discretization. | For our chosen value of t_f, Theorem REF roughly establishes
\Vert u\Vert _\infty {(3/4)}^{{p_\mathrm {fmm}}+1} as a bound on the absolute error incurred by
acceleration, neglecting a number of other factors given in the precise
statement of the theorem. We show {(3/4)}^{{p_\mathrm {fmm}}+1} in the left column of the
t... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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... |
d7357a5bc6081517dc18bbbd3be515dfec55d304 | subsection | 64 | 167 | Stage-1 Discretization. | We solve an exterior Dirichlet boundary value problem
\left( \triangle + k^2 \right) u &= 0 & \quad & \text{in } \mathbb {R}^3 \setminus \Omega , \\
u &= f & \quad & \text{on } \partial \Omega , \\
\lim _{r\rightarrow \infty } r \left(\frac{\partial }{\partial r} - ik \right) u
&= 0
where \Omega \subset \mathbb {R}^... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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253933a573ffa4dd307fd543910f54827b9ffaa2 | subsection | 65 | 167 | Stage-1 Discretization. | For brevity, we let p={p_\mathrm {fmm}} and q={p_\mathrm {qbx}}.Note that the rows shown do not add up to the shown total. Thelatter includes minor contributions to the overall cost (such as theupward and downward passes) that we have omitted.]It remains to examine both the computational cost and the scaling thereof th... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.01... |
4dcbaab36310c53f9fde330113ba545640c2bc89 | subsection | 66 | 167 | Stage-1 Discretization. | A further factor in the
large contribution of {3far}{1}{U_{}}{{3far}{2}{V_{}}{{3far}{3}{W_{}}{{3far}{3close}{W^\mathrm {close}_{}}{{3far}{3far}{W^\mathrm {far}_{}}{{3far}{4}{X_{}}{{3far}{4close}{X^\mathrm {close}_{}}{{3far}{4far}{X^\mathrm {far}_{}}{}}}}}}}} is the high cost of translations even
when the target (QBX) e... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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68df25107755de9b5103aac3375923efe1e85e3e | subsection | 67 | 167 | Stage-1 Discretization. | Conclusion
This paper introduces a fast algorithm for the accurate evaluation of layer
potentials in three dimensions using Quadrature by Expansion (QBX).
Our work builds on and extends the GIGAQBX algorithm in two
dimensions . Many features of the algorithm carry over broadly
unchanged from the two dimensional setting... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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90aa4b4d5c723fb628b47b95475812e1261d1376 | subsection | 68 | 167 | Stage-1 Discretization. | An area query proceeds by descending the tree towards the query center c until
the descent has reached a box whose size is commensurate with the size of the
query box \overline{B_\infty }(r, c). This box is referred to as the guiding
box. Specifically, the guiding box is the smallest box whose
1-near neighborhood conta... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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ab62011c9e35f8a00e5f4ee8f888feb2ecbc30b1 | subsection | 69 | 167 | Stage-1 Discretization. | Third, for each center c^{\prime }, 42 targets t are selected
from the sphere of radius R - |c^{\prime }| centered at c^{\prime }.
The points selected on the sphere are selected to be approximately
equispaced and included the poles of the sphere. The
points in the ball are selected from concentric spheres inside the b... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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12215e2334ffde7faf00b541ff739d0232dc72bf | subsection | 70 | 167 | Stage-1 Discretization. | As a result, we model the cost of forming
or evaluating a p-th order multipole/local expansion in spherical harmonics as
p^2 operations, which is correct to leading order.
The cost of translations of spherical harmonic expansions (multipole/local \rightarrow
local) may be modeled as follows. With a simple extension to... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.010251590050756931,
-0.005278348457068205,
-0.049793440848588943,
-0.00626994576305151,
0.020182818174362183,
0.014515458606183529,
0.05089182406663895,
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0.047108497470617294,
0.020548947155475616,
-0.031792134046554565,
0.02112865075469017,
-0.024957740679383278,
... |
2b93ac33a0eaeac34540306dc2630de4d525ad87 | subsection | 71 | 167 | Stage-1 Discretization. | This distance
is minimized when x is a box corner.
Since c is suspended, c cannot fit in any hypothetical child box of b_c,
which has radius |b_c| / 2. It follows from the previous observation that
r_c > \sqrt{3} |b_c| t_f / 2.
Regardless of where c is located in b_c, \overline{B_\infty }(c, 6|b_c|) is a
superset of ... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
0.004849905148148537,
0.011493245139718056,
-0.04511823505163193,
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0.01455353107303381,
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0.05244460701942444,
0.018422771245241165,
0.0006482121534645557,
0.0031575895845890045,
0.003749041585251689,
0.01436... |
ec3d07bceb630d17071b0c6b315e2b378c162e80 | subsection | 72 | 167 | Stage-1 Discretization. | By
Proposition REF , there are at most N_C M_C pairs (s, c) \in U_\mathrm {small} such that c is a suspended center. It follows that
|U_\mathrm {small}| \le N_C M_C + 27 N_S {n_{\mathrm {max}}}.
Lemma 12 (List 2 complexity) The amount of work done in Stage 4 (translation of multipole to local
expansions) is at most 875... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.00671670725569129,
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0.059683725237846375,
0.02688208594918251,
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0.031459059566259384,
0.009565873071551323,
0.... |
0cc853d17952e8df62b9c38bd359f25165102f79 | subsection | 73 | 167 | Stage-1 Discretization. | For t_f < \sqrt{3} - 1, \mathsf {TCR}(b) is contained strictly inside the
1-neighborhood of b. If the tree is level-restricted, this implies that if
b is a leaf box, any box in {3close}{1}{U_{b}}{{3close}{2}{V_{b}}{{3close}{3}{W_{b}}{{3close}{3close}{W^\mathrm {close}_{b}}{{3close}{3far}{W^\mathrm {far}_{b}}{{3close}{4... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.008378023281693459,
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-0.01038478035479784,
0.013246127404272556,
... |
51ba5886314c18ef2cb7e04569abe94a00b93dc7 | subsection | 74 | 167 | Stage-1 Discretization. | Lemma 15 (List 4 complexity) The cost of all Stage 6 interactions (evaluation of the potential due to List
4 close and far) is at most
375 N_B {n_{\mathrm {max}}}{p_\mathrm {fmm}}^2+ 250 N_C {n_{\mathrm {max}}}{p_\mathrm {qbx}}^2.
First, we show |X_b| \le 125. Every box in X_b is a leaf that is
either a 2-colleague ... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
0.0199420228600502,
0.0017203236930072308,
-0.0199420228600502,
-0.010207507759332657,
0.0028818282298743725,
0.02181873843073845,
0.04617028310894966,
0.006103144027292728,
0.050350937992334366,
-0.005180043168365955,
0.027616726234555244,
0.034421730786561966,
0.0015334149356931448,
0.00... |
69dad1eee75c2eefd4c6acc8af6878dc66d876bc | subsection | 75 | 167 | Stage-1 Discretization. | Recall that {4close}{1}{U_{b}}{{4close}{2}{V_{b}}{{4close}{3}{W_{b}}{{4close}{3close}{W^\mathrm {close}_{b}}{{4close}{3far}{W^\mathrm {far}_{b}}{{4close}{4}{X_{b}}{{4close}{4close}{X^\mathrm {close}_{b}}{{4close}{4far}{X^\mathrm {far}_{b}}{}}}}}}}} must be a subset of the List 4's of the
ancestors of b. If b^{\prime } ... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.04373500123620033,
0.021669121459126472,
-0.04059145227074623,
-0.028017258271574974,
0.006157387513667345,
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0.0447421558201313,
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0.006344321649521589,
0.0014945208095014095,
-0.01229188498109579,
-0.02908545359969139,
0... |
544f349f99f15f79fd05ed90d24c6f5460d5c5e8 | subsection | 76 | 167 | Stage-1 Discretization. | Finally, |{4far}{1}{U_{b}}{{4far}{2}{V_{b}}{{4far}{3}{W_{b}}{{4far}{3close}{W^\mathrm {close}_{b}}{{4far}{3far}{W^\mathrm {far}_{b}}{{4far}{4}{X_{b}}{{4far}{4close}{X^\mathrm {close}_{b}}{{4far}{4far}{X^\mathrm {far}_{b}}{}}}}}}}}| \le 375 follows since, by definition,
{4far}{1}{U_{b}}{{4far}{2}{V_{b}}{{4far}{3}{W_{b}}... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.0018861964344978333,
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0.016203362494707108,
0.011290478520095348,
0.052180320024490356,
0.01896495185792446,
0.024976368993520737,
... |
ae8765b57d4805da54363fe88b0cf2122f9ee9a6 | subsection | 77 | 167 | Error Estimates for FMM Translations | In , error estimates were presented for the GIGAQBX FMM that
applied to the 2D Laplace kernel with complex Taylor expansions. In this
section, we present their analogs in three dimensions. We restrict our
attention to the spherical harmonic approximation of the three-dimensional Laplace
potential (REF ).
This section l... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.030661024153232574,
0.024147463962435722,
0.000963877362664789,
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0.04567119851708412,
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0.013911867514252663,
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0.04435933381319046,
-0.0016026487573981285,
... |
476b03300dfd79b4dc5b27e6864dda8ebdadd212 | subsection | 78 | 167 | Types of Translations. | First, there is a direct
correspondence in these error estimates with the evaluation scenarios for the
point FMM. For instance, see for the
local case (cf. Hypothesis REF ), for the multipole case (cf. Hypothesis REF ),
and for the multipole-to-local
case (cf. Hypothesis REF ). | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.013181351125240326,
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0.004943006671965122,
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0.01908549852669239,
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0.010435236617922783,
-0.024516703560948372,
... |
5dbb3d26b21d2ae359ad0fe15b893f7727272a77 | subsection | 79 | 167 | `Sized' Targets. | Second, the results in this section
empirically confirm that, for purposes of FMM accuracy, the local expansion that is to be
approximated behaves much like a `sized target'. What this means is that, for a
given evaluation scenario, the accuracy that is expected is similar to the
accuracy expected if the local expansio... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.013618767261505127,
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0.026962870731949806,
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0.011665599420666695,
-0.05224723741412163,
0.... |
1357455a3e7467e917df90f4378c73e87b86d441 | subsection | 80 | 167 | Accuracy Dependence on Intermediate Expansion Order. | Last,
the results suggest that the accuracy chiefly depends on the order of the
intermediate multipole/local expansions used and not the `final' order of the
local expansion (i.e. the QBX order). The results also
suggest it might be possible to find an estimate independent of the final
expansion order. With regards to ... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.01698090322315693,
-0.029094861820340157,
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0.013151426799595356,
-0.038172703236341476,
-0.0013483264483511448,
-0.0216952767223119... |
8b2a4c627838dfaf0bc5b453f67f8798d868b0ef | subsection | 81 | 167 | Analytical Preliminaries | Local expansions have already been introduced in
Section REF . Recall that the p-th order local expansion
L_p due to a source s \in \mathbb {R}^3 centered at c \in \mathbb {R}^3,
with coefficients \langle L^m_n \rangle given by (REF ) and
evaluated at a target t \in \mathbb {R}^3 takes the formL_p(t) = \sum _{n=0}^p \s... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.04370530694723129,
0.0083015663549304,
-0.03860838711261749,
0.0076072257943451405,
0.007008261978626251,
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0.011964022181928158,
0.004036331549286842,
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-0.008133703842759132,
-0.027529459446668625,
0.023928044363856316,
-0.031100353226065636,
0.00... |
716f578e78910c72c95768ed3e6dd9421fbb2b25 | subsection | 82 | 167 | Analytical Preliminaries | See for the explicit formula
for this operator.We similarly define the operators \mathrm {M2M}_{{p} \rightarrow {q}}^{{c} \rightarrow {c^{\prime }}} for
multipole-to-multipole translation and \mathrm {M2L}_{{p} \rightarrow {q}}^{{c} \rightarrow {c^{\prime }}} for
multipole-to-local translation. An explicit formula for... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.05449789762496948,
0.02232094667851925,
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0.0152493417263031,
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0.0006350713665597141,
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-0.013342220336198807,
-0.0028530533891171217,
... |
e6afd4fafbd4f00c06ef53c6456e14bb5fdd9d69 | subsection | 83 | 167 | Accuracy of GIGAQBX FMM Translations | Recall the difference between the notion of accuracy in a point FMM and the notion of
accuracy used in the GIGAQBX FMM. A point FMM computes the value of a
potential at a point x. The accuracy is measured by\text{Point FMM Accuracy} = \left| \Phi (x) - \tilde{\Phi }_p(x) \right|,where \tilde{\Phi }_p is the point FMM's... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.0036153621040284634,
-0.035207830369472504,
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0.009999450296163559,
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0.017237801104784012,
0.012608002871274948,
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0.010037587024271488,
-0.024132922291755676,... |
e00bdc6f5f86043fe279421054c11392b76858f8 | subsection | 84 | 167 | Accuracy of GIGAQBX FMM Translations | The corresponding error bound is scaled by \sum _{i=1}^m |q_i|. | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.025535451248288155,
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-0.033589642494916916,
0.012264338321983814,
0.006383862812072039,... |
113711c3819ee37ac10b544c478fca52eb26ff2c | subsection | 85 | 167 | Multipole Accuracy | Recall that a multipole expansion is an `outgoing' approximation to the field
due to a set of sources at any point sufficiently far away from the expansion
center. In this section, we consider the ability of a local expansion obtained through translation from a multipole
expansion to approximate the local expansion of ... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.0314660519361496,
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0.03912655636668205,
-0.04330778494477272,
0.01544... |
1422769fb56b4540e6db3cb0de0c2c2d52a756eb | subsection | 86 | 167 | Multipole Accuracy | It
can be shown that this expansion must converge, in the sense that \lim _{q \rightarrow \infty } E_M(q) = \mathcal {E}(t). Therefore, by
Proposition REF , we may expect that for
every \varepsilon > 0, it is the case that for all sufficiently large q thatE_{M}(q) \le (1 + \varepsilon ) \left((4\pi )^{-1}/(\rho - r) \... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.017735473811626434,
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... |
3a5b2cd08358ba1438b62240164d18a8272b3287 | subsection | 87 | 167 | Local Accuracy | Recall that the role of the local expansion is complementary to the multipole
expansion, since the local expansion represents the potential in a neighborhood
of an expansion center. This section considers the case where a local expansion
is formed due to a potential at one center, subsequently translated to a second ce... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.002488787053152919,
0.0017106834566220641,
0.054040051996707916,
-0.052453331649303436,
... |
422f4b89dd0137367e610fbf7e381c9ab7b6e960 | subsection | 88 | 167 | Local Accuracy | This motivates the following hypothesis about the
behavior of E_L(q) for all q.Hypothesis 2 (Source \rightarrow Local(p) \rightarrow Local(q))
For the situation described above, there exists a constant C > 0 independent
of p, q, \rho , r, s, c, and t such that the error in the
local-mediated approximation to the local... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.05090425908565521,
0.01727326773107052,
-0.020737076178193092,
0.002811483573168516,
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0.057160478085279465,
-0.028000393882393837,
... |
9ecec5ffc57925eceba19a024a80126d667460e2 | subsection | 89 | 167 | Multipole-to-Local Accuracy | In this section, we work with the same geometrical situation as given in
Section REF and illustrated in Figure REF .
We consider the
accuracy achieved when a multipole expansion is converted into a local expansion
which is then translated to a third center, and used to approximate the local expansion
of the potential a... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.024178240448236465,
0.03618346154689789,
-0.019678188487887383,
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0.013103538192808628,
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0.026359621435403824,
-0.03551226854324341,
0... |
9a2ea742e421f7fbc02683181199878f23375df1 | subsection | 90 | 167 | Multipole-to-Local Accuracy | By the
triangle inequality,| \mathcal {G}(s, t) - T_p^0({t}) | \le | \mathcal {G}(s, t) - L_p^{0}({t}) | + | L_p^{0}({t}) - T_p^0({t}) |.The quantity |\mathcal {G}(s, t) - L_p^{0}(t)| may be bounded by
Proposition REF . Since the quantity |
L_p^{0}(t) - T_p^0(t)| is the difference between the
local expansion of a multi... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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dbfaef53d03aea2d6ec74dbe8686b1c0348df6f1 | subsection | 91 | 167 | The GIGAQBX Algorithm in Three Dimensions | This section is concerned with the precise statement of the GIGAQBX algorithm
and its complexity and accuracy analysis. The algorithm is presented in
Sections REF and REF . The accuracy
and complexity analyses follow in Sections REF
and REF . | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.00826929323375225,
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0.03805095702409744,
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0.024853652343153954,
... |
419ff4490aa541b04591dfae86b4a1c5dbe09367 | subsection | 92 | 167 | Target Confinement Rule. | To prevent inaccurate contributions from entering the QBX local expansion, while
still maintaining the efficiency enabled by the use of a tree, the design of the
GIGAQBX algorithm adopts the point of view of QBX centers as `targets with
extent' that each have their own near-field. The realization of this idea is
that G... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.03597264736890793,
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-0.02750580199062824... |
a811584c9758b46f858462549ae31dfef6819705 | subsection | 93 | 167 | Particles Owned by Non-Leaf Boxes. | It follows from the previous paragraph that the GIGAQBX algorithm, unlike the point FMM, allows for
particles (specifically, QBX centers) to be `owned' by non-leaf `ancestor'
boxes. The most important implication of this design is that interaction lists
involving direct evaluations at particles (List 1 and List 3), as ... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
0.0010959355859085917,
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d1639a09d576fcd9302ffaed807d0e0feb13e41d | subsection | 94 | 167 | Two-Away Near Neighborhood. | To obtain a good convergence factor in two or three dimensions, it is convenient to
consider the `near-field' to consist of both a box's nearest
neighbors and also its second nearest neighbors.
This is not a new modification, having been present in the original
three-dimensional FMM . | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.03943246230483055,
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0.... |
6ce85e449c03e296b8ee73945ce1c79c3f334f08 | subsection | 95 | 167 | `Close' and `Far' Lists. | In order to actually obtain the accuracy guarantees provided by the target
confinement rule, we disallow certain box near-field interactions that are too
close to the target confinement region from using expansion mediation that would
otherwise be mediated by expansions in the point FMM. Specifically, the fields
associ... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.030494630336761475,
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0.016819361597299576,
-0.027258964255452156,... |
f9ae7ce10cf99425a4e11fdf533355f1b72ba3bf | subsection | 96 | 167 | Changes from the 2D Version of GIGAQBX. | Perhaps the most significant difference with the two-dimensional
version is the use of an \ell ^2 target confinement region to
confine the QBX expansion centers, whereas the previous version used an
\ell ^\infty (square) region. The use of the \ell ^2 region improves the
efficiency of the scheme, especially in three d... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.0... |
30c7c1c2b8d0f082fe5d8c4fe12331d43666075c | subsection | 97 | 167 | Computational Domain | The computational domain for the algorithm is an octree whose axis-aligned root
box contains all sources, targets, and expansion centers (`particles') as well
as the entirety of each expansion disk. Starting with the root box, the octree
is refined by repeated subdivision of boxes that contain more than {n_{\mathrm {ma... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.029923390597105026,
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0.0... |
a266d04484540bcdffad7103b3ab3be0d32354ec | subsection | 98 | 167 | Notation | For a box b in the octree, we will use |b| to denote the (\ell ^\infty )
radius of b. The target confinement region (`TCR', also \mathsf {TCR}(b)) of
a box b with center c is \overline{B_2}(\sqrt{3} |b|(1 + t_f), c), where
t_f is the target confinement factor (`TCF'),
where \sqrt{3} |b| is half the box diagonal.The k-n... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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-0.02808309905230999,
0.... |
d4166de66b3bea27aa80d05bfd9262c77204b336 | subsection | 99 | 167 | Notation | The following notation refers to `point'
potentials evaluated at a target not requiring QBX owned by a box b:
[(a)]\mathsf {P}^{\text{near}}_b(t) denotes the potential at a target point t due to all
sources in {1}{1}{U_{b}}{{1}{2}{V_{b}}{{1}{3}{W_{b}}{{1}{3close}{W^\mathrm {close}_{b}}{{1}{3far}{W^\mathrm {far}_{b}}{{1... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.000358784687705338,
0.02316056378185749,
0.00961... |
4e9cacbd4824f30324249120a3fa7d8da4e150c9 | subsection | 100 | 167 | Notation | The following notation refers to
potentials evaluated with QBX mediation:
[(a)]\mathsf {L}^{\text{qbx},\text{near}}_{c}(t) denotes the (QBX) local expansion of the potential at
the center c, evaluated at target t, due to all sources in {1}{1}{U_{b}}{{1}{2}{V_{b}}{{1}{3}{W_{b}}{{1}{3close}{W^\mathrm {close}_{b}}{{1}{3fa... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.015150493010878563,
0.025571225211024284,
0.00580540020018816,
0.008... |
c7fe045d041290d1912b57fe855b72f13e351839 | subsection | 101 | 167 | Notation | Lastly, given a box b, \mathsf {M}_b and \mathsf {L}^{\text{far}}_b refer respectively to the
multipole and local expansions associated with the box.
Algorithmic Parameters
The parameters to the algorithm are {p_\mathrm {fmm}}, the FMM order; {p_\mathrm {qbx}}, the QBX
order; {p_\mathrm {quad}}, the upsampled quadratur... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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-0.02847011387348175,
... |
f3189697d8659f4257328c2f7b4492e6533bce4a | subsection | 102 | 167 | Notation | Stage 3: Evaluate direct interactionsboxes b
For each conventional target t owned by b, add to \mathsf {P}^{\text{near}}_b(t)
the contribution due to the interactions from sources owned by boxes in
{1}{1}{U_{b}}{{1}{2}{V_{b}}{{1}{3}{W_{b}}{{1}{3close}{W^\mathrm {close}_{b}}{{1}{3far}{W^\mathrm {far}_{b}}{{1}{4}{X_{b}}{... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.012810200452804565,
0.0037785894237458706,
0.014099612832069397,
-0.015762878581881523,
0... |
b03b51b09cf857f2de25113a6acc017ea777bac9 | subsection | 103 | 167 | Notation | Stage 5(a): Evaluate direct interactions due to {3close}{1}{U_{b}}{{3close}{2}{V_{b}}{{3close}{3}{W_{b}}{{3close}{3close}{W^\mathrm {close}_{b}}{{3close}{3far}{W^\mathrm {far}_{b}}{{3close}{4}{X_{b}}{{3close}{4close}{X^\mathrm {close}_{b}}{{3close}{4far}{X^\mathrm {far}_{b}}{}}}}}}}}Repeat Stage 3 with {3close}{1}{U_{b... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.027601998299360275,
0.03304915875196457,
-0.... |
2e2364f190398b693528fb525dc77de522e8e0fd | subsection | 104 | 167 | Notation | Stage 5(b): Evaluate multipoles due to {3far}{1}{U_{b}}{{3far}{2}{V_{b}}{{3far}{3}{W_{b}}{{3far}{3close}{W^\mathrm {close}_{b}}{{3far}{3far}{W^\mathrm {far}_{b}}{{3far}{4}{X_{b}}{{3far}{4close}{X^\mathrm {close}_{b}}{{3far}{4far}{X^\mathrm {far}_{b}}{}}}}}}}}boxes b
For each conventional target t owned by b, evaluate t... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.00108829524833709,
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0.012826814316213131,
-0.018023161217570305,
0.0... |
8472d8d621e458ddbb0f537698163aaaa8ae66e2 | subsection | 105 | 167 | Notation | Stage 6(a): Evaluate direct interactions due to {4close}{1}{U_{b}}{{4close}{2}{V_{b}}{{4close}{3}{W_{b}}{{4close}{3close}{W^\mathrm {close}_{b}}{{4close}{3far}{W^\mathrm {far}_{b}}{{4close}{4}{X_{b}}{{4close}{4close}{X^\mathrm {close}_{b}}{{4close}{4far}{X^\mathrm {far}_{b}}{}}}}}}}}Repeat Stage 3 with {4close}{1}{U_{b... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.040061596781015396,
0.003173195756971836,
-... |
7682def736e09fe648a4aefe4c4d1c40ed48061e | subsection | 106 | 167 | Notation | Stage 8: Form local expansions at QBX centersboxes b
For each QBX center c owned by b, translate \mathsf {L}^{\text{far}}_b to
c, obtaining \mathsf {L}^{\text{qbx},\text{far}}_{c}.
Stage 9: Evaluate final potential at targetsboxes b
For each conventional target t owned by b, evaluate \mathsf {L}^{\text{far}}_b(t).
A... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.04242026433348656,
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0.007362509611994028,
-0.013893399387598038,
0.... |
0086c927fe4965fdd05e00c36984a4aa36174a92 | subsection | 107 | 167 | Notation | Assuming the truth of
Hypotheses REF , REF , and REF , there exists a
constant M > 0 such that for every target point x \in \mathbb {R}^3, we
have
\left|
\mathcal {S}_{\mathrm {QBX}({p_\mathrm {qbx}},N)} \mu (x)
- \mathcal {G}_{p_\mathrm {fmm}}[\mathcal {S}_{\mathrm {QBX}({p_\mathrm {qbx}},N)}] \mu (x)
\right|
\le \fr... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.009590471163392067,
-0.014984634704887867,
0.0184... |
fc5f133a135c1128120edd50de7b3944e7f6fa71 | subsection | 108 | 167 | Notation | The potential at c is the sum of the contributions
\mathsf {L}^{\text{qbx},\text{near}}_{c}(t), \mathsf {L}^{\text{qbx},{3}{1}{U_{}}{{3}{2}{V_{}}{{3}{3}{W_{}}{{3}{3close}{W^\mathrm {close}_{}}{{3}{3far}{W^\mathrm {far}_{}}{{3}{4}{X_{}}{{3}{4close}{X^\mathrm {close}_{}}{{3}{4far}{X^\mathrm {far}_{}}{}}}}}}}}}_{c}(t), an... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.03393419086933136,
0.025694774463772774,
... |
2a188f6370ada462df3d15a889b851f0d337f948 | subsection | 109 | 167 | Notation | The potential due to \mathsf {L}^{\text{qbx},\text{far}}_{c}(t) arrives via an interaction of {4far}{1}{U_{b^{\prime }}}{{4far}{2}{V_{b^{\prime }}}{{4far}{3}{W_{b^{\prime }}}{{4far}{3close}{W^\mathrm {close}_{b^{\prime }}}{{4far}{3far}{W^\mathrm {far}_{b^{\prime }}}{{4far}{4}{X_{b^{\prime }}}{{4far}{4close}{X^\mathrm {... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.03888345882296562,
0.011025628075003624,
0.... |
7b4efa9ae96a17e705b86037f7121b53d094f67d | subsection | 110 | 167 | Notation | Lastly, the contribution due to all {2}{1}{U_{b^{\prime }}}{{2}{2}{V_{b^{\prime }}}{{2}{3}{W_{b^{\prime }}}{{2}{3close}{W^\mathrm {close}_{b^{\prime }}}{{2}{3far}{W^\mathrm {far}_{b^{\prime }}}{{2}{4}{X_{b^{\prime }}}{{2}{4close}{X^\mathrm {close}_{b^{\prime }}}{{2}{4far}{X^\mathrm {far}_{b^{\prime }}}{}}}}}}}} interac... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
0.0021321428939700127,
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0... |
9ef374da287906de83f5ec1cb6d3d51068b7135e | subsection | 111 | 167 | Notation | The cost of the tree build phase (Stage 1) and the
evaluation phase of the algorithm (Stages 2–9) are treated
separately. Under broadly applicable assumptions, the evaluation phase can be
shown to run in time that is proportional to the number of
particles. Nevertheless, the proportionality constant is affected by the ... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.008179312571883202,
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0.007221753243356943,
... |
a6f84a1b447c01c4a27e43db384f61b475bfadbd | subsection | 112 | 167 | Notation | Numerical Experiments
We use a family of smooth `urchin' test geometries \gamma _k
given analytically in spherical coordinates
(r_k, \theta ,\phi ) by prescribing r_k as a function of (\theta ,\phi ), where
r_k(\theta ,\phi ) &= 0.2 + \frac{\mathop {\mathrm {Re}}Y_{k}^{\lfloor k/2\rfloor }(\theta ,\phi ) - m_k}{M_k-m... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.03954697772860527,
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0.01... |
f05cecb9ba5158257f73e728ef6ed0a9bc5abbe9 | subsection | 113 | 167 | Notation | We use the residual in this identity as a measure for the accuracy
that our scheme achieves in the evaluation of layer potential evaluations.
The achieved accuracy in Green's formula is predictive of the accuracy
one might achieve in the solution of boundary value problems. Data to support
this assertion (in two dimens... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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... |
94ab3e12bf9009b3f241cacd591168a3244f5bcc | subsection | 114 | 167 | Notation | For our chosen value of t_f, Theorem REF roughly establishes
\Vert u\Vert _\infty {(3/4)}^{{p_\mathrm {fmm}}+1} as a bound on the absolute error incurred by
acceleration, neglecting a number of other factors given in the precise
statement of the theorem. We show {(3/4)}^{{p_\mathrm {fmm}}+1} in the left column of the
t... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.03356495872139931,
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0.012876738794147968,
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0.034632936120033264,
-0.0032325342763215303,
... |
d1f598f5523948013b94a46b983e8571a1d63ee7 | subsection | 115 | 167 | Notation | We solve an exterior Dirichlet boundary value problem
\left( \triangle + k^2 \right) u &= 0 & \quad & \text{in } \mathbb {R}^3 \setminus \Omega , \\
u &= f & \quad & \text{on } \partial \Omega , \\
\lim _{r\rightarrow \infty } r \left(\frac{\partial }{\partial r} - ik \right) u
&= 0
where \Omega \subset \mathbb {R}^... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.01584126614034176,
0.02766... |
11dead69841d736bca33f9fc131376987196cf4e | subsection | 116 | 167 | Notation | For brevity, we let p={p_\mathrm {fmm}} and q={p_\mathrm {qbx}}.Note that the rows shown do not add up to the shown total. Thelatter includes minor contributions to the overall cost (such as theupward and downward passes) that we have omitted.]It remains to examine both the computational cost and the scaling thereof th... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.03491761535406113,
0.029881421476602554,
0.01... |
01467273d24f0252f2adc189fa17277ea3698322 | subsection | 117 | 167 | Notation | A further factor in the
large contribution of {3far}{1}{U_{}}{{3far}{2}{V_{}}{{3far}{3}{W_{}}{{3far}{3close}{W^\mathrm {close}_{}}{{3far}{3far}{W^\mathrm {far}_{}}{{3far}{4}{X_{}}{{3far}{4close}{X^\mathrm {close}_{}}{{3far}{4far}{X^\mathrm {far}_{}}{}}}}}}}} is the high cost of translations even
when the target (QBX) e... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.027558013796806335,
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0.025711... |
900a950e6be508966b41204ae9896d65ebb476d4 | subsection | 118 | 167 | Notation | Conclusion
This paper introduces a fast algorithm for the accurate evaluation of layer
potentials in three dimensions using Quadrature by Expansion (QBX).
Our work builds on and extends the GIGAQBX algorithm in two
dimensions . Many features of the algorithm carry over broadly
unchanged from the two dimensional setting... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.038702718913555145,
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-0.01524606347084045... |
bc883abf01f2a9c21389da10bfb3001fc2ab7b8e | subsection | 119 | 167 | Notation | An area query proceeds by descending the tree towards the query center c until
the descent has reached a box whose size is commensurate with the size of the
query box \overline{B_\infty }(r, c). This box is referred to as the guiding
box. Specifically, the guiding box is the smallest box whose
1-near neighborhood conta... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.... |
7e86c6364182bd5bf5558b1265182a4205ecc81a | subsection | 120 | 167 | Notation | Third, for each center c^{\prime }, 42 targets t are selected
from the sphere of radius R - |c^{\prime }| centered at c^{\prime }.
The points selected on the sphere are selected to be approximately
equispaced and included the poles of the sphere. The
points in the ball are selected from concentric spheres inside the b... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.021895596757531166,
-0.010139111429452896,
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0.0047910925932228565,
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0.04440152645111084,
-0.03524657338857651,
0.0... |
242ab9cf203748b6bef4b2a8941c7b7d52142782 | subsection | 121 | 167 | Notation | As a result, we model the cost of forming
or evaluating a p-th order multipole/local expansion in spherical harmonics as
p^2 operations, which is correct to leading order.
The cost of translations of spherical harmonic expansions (multipole/local \rightarrow
local) may be modeled as follows. With a simple extension to... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.010251590050756931,
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0.020548947155475616,
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0.02112865075469017,
-0.024957740679383278,
... |
abd9919267dbf653165a26ebc95ab0efcf62ea8c | subsection | 122 | 167 | Notation | This distance
is minimized when x is a box corner.
Since c is suspended, c cannot fit in any hypothetical child box of b_c,
which has radius |b_c| / 2. It follows from the previous observation that
r_c > \sqrt{3} |b_c| t_f / 2.
Regardless of where c is located in b_c, \overline{B_\infty }(c, 6|b_c|) is a
superset of ... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
0.004849905148148537,
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0.0031575895845890045,
0.003749041585251689,
0.01436... |
c28af325399680fb831983402d50b63f451eef70 | subsection | 123 | 167 | Notation | By
Proposition REF , there are at most N_C M_C pairs (s, c) \in U_\mathrm {small} such that c is a suspended center. It follows that
|U_\mathrm {small}| \le N_C M_C + 27 N_S {n_{\mathrm {max}}}.
Lemma 12 (List 2 complexity) The amount of work done in Stage 4 (translation of multipole to local
expansions) is at most 875... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.00671670725569129,
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0.031459059566259384,
0.009565873071551323,
0.... |
ebf53a4c193f6e3729370c6e36c483178a3ad86c | subsection | 124 | 167 | Notation | For t_f < \sqrt{3} - 1, \mathsf {TCR}(b) is contained strictly inside the
1-neighborhood of b. If the tree is level-restricted, this implies that if
b is a leaf box, any box in {3close}{1}{U_{b}}{{3close}{2}{V_{b}}{{3close}{3}{W_{b}}{{3close}{3close}{W^\mathrm {close}_{b}}{{3close}{3far}{W^\mathrm {far}_{b}}{{3close}{4... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.05377805605530739,
0.03256594017148018,
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0.008378023281693459,
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-0.01038478035479784,
0.013246127404272556,
... |
e6f3058c9f33a5d307b5e05c214503ef99e8ebc5 | subsection | 125 | 167 | Notation | Lemma 15 (List 4 complexity) The cost of all Stage 6 interactions (evaluation of the potential due to List
4 close and far) is at most
375 N_B {n_{\mathrm {max}}}{p_\mathrm {fmm}}^2+ 250 N_C {n_{\mathrm {max}}}{p_\mathrm {qbx}}^2.
First, we show |X_b| \le 125. Every box in X_b is a leaf that is
either a 2-colleague ... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
0.0199420228600502,
0.0017203236930072308,
-0.0199420228600502,
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0.0028818282298743725,
0.02181873843073845,
0.04617028310894966,
0.006103144027292728,
0.050350937992334366,
-0.005180043168365955,
0.027616726234555244,
0.034421730786561966,
0.0015334149356931448,
0.00... |
d6b9628795dc4ee683f9fadd80389a0c78621490 | subsection | 126 | 167 | Notation | Recall that {4close}{1}{U_{b}}{{4close}{2}{V_{b}}{{4close}{3}{W_{b}}{{4close}{3close}{W^\mathrm {close}_{b}}{{4close}{3far}{W^\mathrm {far}_{b}}{{4close}{4}{X_{b}}{{4close}{4close}{X^\mathrm {close}_{b}}{{4close}{4far}{X^\mathrm {far}_{b}}{}}}}}}}} must be a subset of the List 4's of the
ancestors of b. If b^{\prime } ... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.04373500123620033,
0.021669121459126472,
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0.006157387513667345,
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0.006344321649521589,
0.0014945208095014095,
-0.01229188498109579,
-0.02908545359969139,
0... |
b5f0312a0a487167e8c78139bc676fadaff2813e | subsection | 127 | 167 | Notation | Finally, |{4far}{1}{U_{b}}{{4far}{2}{V_{b}}{{4far}{3}{W_{b}}{{4far}{3close}{W^\mathrm {close}_{b}}{{4far}{3far}{W^\mathrm {far}_{b}}{{4far}{4}{X_{b}}{{4far}{4close}{X^\mathrm {close}_{b}}{{4far}{4far}{X^\mathrm {far}_{b}}{}}}}}}}}| \le 375 follows since, by definition,
{4far}{1}{U_{b}}{{4far}{2}{V_{b}}{{4far}{3}{W_{b}}... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.016203362494707108,
0.011290478520095348,
0.052180320024490356,
0.01896495185792446,
0.024976368993520737,
... |
bdaf62b3c1766d26bdd9e29755fb298466d656f0 | subsection | 128 | 167 | Conventional Interaction Lists | The four conventional interaction lists in the FMM are defined in this section,
with two modifications to the standard definition.
First, non-leaf boxes are allowed as
target boxes. Thus, lists normally associated with only leaf boxes (Lists 1 and
3) may be associated with arbitrary boxes in the tree. Second, our defin... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.009436173364520073,
-0.005923353601247072,
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0.01185433566570282,
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0.0018241106299683452,
0.02946038730442524,
0.024395214393734932,
-0.008841168135404587,
0.0017192219384014606,
-0.01714835688471794,
0... |
45f44eca6e660e237adfb809025bbaf613d6ec96 | subsection | 129 | 167 | Conventional Interaction Lists | Unlike List 2,
the separation is insufficient for accurate multipole-to-local mediation. List
3 is not downward-propagating, and it is usually mediated with a
multipole-to-target interaction.Definition 4 (List 3, {3}{1}{U_{b}}{{3}{2}{V_{b}}{{3}{3}{W_{b}}{{3}{3close}{W^\mathrm {close}_{b}}{{3}{3far}{W^\mathrm {far}_{b}}... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.014317584224045277,
0.016949214041233063,
-0.03572912514209747,
-0.006239629350602627,
-0.0024619074538350105,
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0.05305973440408707,
0.016216933727264404,
-0.010579909197986126,
0.01829172484576702,
0.009306048974394798,
0.02312781848013401,
0.004489024169743061,
0.... |
9178af42bff530c6ab746e97fe1e1250a2630262 | subsection | 130 | 167 | Conventional Interaction Lists | Unlike List 3, List 4 is
downward-propagating, and one may form a local expansion of the field from the
source box that can be propagated to the descendants.Definition 5 (List 4, {4}{1}{U_{b}}{{4}{2}{V_{b}}{{4}{3}{W_{b}}{{4}{3close}{W^\mathrm {close}_{b}}{{4}{3far}{W^\mathrm {far}_{b}}{{4}{4}{X_{b}}{{4}{4close}{X^\math... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.0271066352725029,
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0.014537764713168144,
-0.001054083346389234,
-0.04746713861823082,
0.027915561571717262,
0.04850500449538231,
0.024527231231331825,
-0.0004025548987556249,
-0.00008406434062635526,
0.022451499477028847,
-0.005967733450233936,
... |
4be1b5d0a2af2550d96aa487ca71e50483739580 | subsection | 131 | 167 | Conventional Interaction Lists | For any d \in {4}{1}{U_{b}}{{4}{2}{V_{b}}{{4}{3}{W_{b}}{{4}{3close}{W^\mathrm {close}_{b}}{{4}{3far}{W^\mathrm {far}_{b}}{{4}{4}{X_{b}}{{4}{4close}{X^\mathrm {close}_{b}}{{4}{4far}{X^\mathrm {far}_{b}}{}}}}}}}}, either b \in {3}{1}{U_{d}}{{3}{2}{V_{d}}{{3}{3}{W_{d}}{{3}{3close}{W^\mathrm {close}_{d}}{{3}{3far}{W^\mathr... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.04907301440834999,
0.0073968106880784035,
-0.05438315495848656,
-0.028107117861509323,
-0.006229495629668236,
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0.0472114123404026,
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0.0072976271621882915,
-0.006389715243130922,
0.005893797148019075,
0.024734875187277794,
0... |
cd4db7fb7ddcf7173a9384d47cc6feb47f4aa514 | subsection | 132 | 167 | Close and Far Lists | Because of inadequate separation from the target confinement region, our algorithm
cannot make use of the interaction lists {3}{1}{U_{b}}{{3}{2}{V_{b}}{{3}{3}{W_{b}}{{3}{3close}{W^\mathrm {close}_{b}}{{3}{3far}{W^\mathrm {far}_{b}}{{3}{4}{X_{b}}{{3}{4close}{X^\mathrm {close}_{b}}{{3}{4far}{X^\mathrm {far}_{b}}{}}}}}}}}... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.008317598141729832,
0.004475478082895279,
-0.060466647148132324,
-0.029058177024126053,
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0.03024858422577381,
-0.0032946080900728703,
0.009233296848833561,
-0.010362658649682999,... |
4fc7a6bcf5d20bf3dfc825519938ff9bc02ac3ae | subsection | 133 | 167 | Close and Far Lists | The `far' list may be mediated via multipole-to-target
interaction.Definition 6 (List 3 close, {3close}{1}{U_{b}}{{3close}{2}{V_{b}}{{3close}{3}{W_{b}}{{3close}{3close}{W^\mathrm {close}_{b}}{{3close}{3far}{W^\mathrm {far}_{b}}{{3close}{4}{X_{b}}{{3close}{4close}{X^\mathrm {close}_{b}}{{3close}{4far}{X^\mathrm {far}_{b... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.04999151825904846,
0.013146426528692245,
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0.021928513422608376,
0.0024682930670678616,
0.0006447318592108786,
0.019364846870303154,
0.027803584933280945,
0.012299500405788422,
0.0036967170890420675,
... |
b398ffb903fca58dfadf46298be2ce53ac40c599 | subsection | 134 | 167 | Close and Far Lists | List 4 close is evaluated
directly only at the targets in the box, and is not
downward-propagating. List 4 far is downward-propagating. It consists of boxes
form which the TCR of the target box is adequately separated. By monotonicity of
`\prec ', this means that the TCR of the descendants is also adequately
separated ... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.06929225474596024,
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0.00473904050886631,
-0.005372439045459032,
0.... |
95c1186dd066749b05c1486dfa69c4de694bb6d7 | subsection | 135 | 167 | Close and Far Lists | A box d \in {4}{1}{U_{b}}{{4}{2}{V_{b}}{{4}{3}{W_{b}}{{4}{3close}{W^\mathrm {close}_{b}}{{4}{3far}{W^\mathrm {far}_{b}}{{4}{4}{X_{b}}{{4}{4close}{X^\mathrm {close}_{b}}{{4}{4far}{X^\mathrm {far}_{b}}{}}}}}}}} is in
List 4 far if \mathsf {TCR}(b) \prec d. Furthermore, if b has a parent, a
box d \in {4close}{1}{U_{\maths... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.027943525463342667,
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0.010278517380356789,
0.021991601213812828,
-0.014185422100126743,
0.011896220035851002,
... |
71cea1508625194565e9952a5fc180f48cd08e27 | subsection | 136 | 167 | Close and Far Lists | However, {4close}{1}{U_{b}}{{4close}{2}{V_{b}}{{4close}{3}{W_{b}}{{4close}{3close}{W^\mathrm {close}_{b}}{{4close}{3far}{W^\mathrm {far}_{b}}{{4close}{4}{X_{b}}{{4close}{4close}{X^\mathrm {close}_{b}}{{4close}{4far}{X^\mathrm {far}_{b}}{}}}}}}}} \cup {4far}{1}{U_{b}}{{4far}{2}{V_{b}}{{4far}{3}{W_{b}}{{4far}{3close}{W^\... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.005291417706757784,
0.0266592837870121,
0.0... |
e1a10e1b971903f28162d8a6f2402f88aa95a04f | subsection | 137 | 167 | Numerical Experiments | We use a family of smooth `urchin' test geometries \gamma _k
given analytically in spherical coordinates
(r_k, \theta ,\phi ) by prescribing r_k as a function of (\theta ,\phi ), wherer_k(\theta ,\phi ) &= 0.2 + \frac{\mathop {\mathrm {Re}}Y_{k}^{\lfloor k/2\rfloor }(\theta ,\phi ) - m_k}{M_k-m_k},\\
{M_k\\m_k} &= {{\m... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.037812843918800354,
0.0027733996976166964,
0.0704374834895134,
-0.003234997158870101,
0.00... |
b5f0bbdb37b83d818df9fc912009aa8c65ac0567 | subsection | 138 | 167 | Accuracy | table-format = 1.2e-1,
table-number-alignment = center,
table-sign-exponent = true,
scientific-notation = true,
round-mode = places,
round-precision = 2,
detect-weight = true,
mode = text,
[Table: \ell ^\infty error in Green's formula \mathcal {S}(\partial _nu)-\mathcal {D}(u)=u/2, scaled by 1/\Vert u\Vert _\infty , fo... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.06042005866765976,
0.0017927668523043394,
0.0399748869240284,
-0.009970835410058498,
0.0281... |
a307175a5d6581dadd16963f2fdbd3745e76ca64 | subsection | 139 | 167 | Accuracy | We choose the target confinement factor as
t_f=0.9.Table REF shows the results of these experiments for the
GIGAQBX FMM, scaled by the norm of the test function u and varying {p_\mathrm {qbx}}
across columns and {p_\mathrm {fmm}} across rows. We show table entries in bold if no
decrease in error is observed for at leas... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
0.006396611221134663,
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0.021131321787834167,
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0.05... |
b72e7175511102eb6b64f1c64473164c1476b00a | subsection | 140 | 167 | A BVP with Complex Geometry for the Helmholtz Equation | To support the assertion that our algorithm is broadly applicable and robust,
we demonstrate its use on a challenging, moderate-frequency boundary value
problem for the Helmholtz equation. While we have discussed a version of the
algorithm for the Laplace equation, a direct analog of our algorithm is
applicable for the... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.02212262526154518,
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0.07189090549945831,
0.027981307357549667,
0... |
71bdcc182e75676053234b2080ff295d92c33e3f | subsection | 141 | 167 | Cost and Scaling | It remains to examine both the computational cost and the scaling thereof that
the algorithm achieves on geometries of varying size. Rather than relying on
wall time (which is sensitive to machine details as well as varying levels of optimization
and code quality), we present an abstract operation count intended to
asy... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.026193423196673393,
0.02604087069630623,
0.... |
ed646a173b392fa2bf06e9fef6c1fb0bfa2cf5e5 | subsection | 142 | 167 | Cost and Scaling | A further factor in the
large contribution of {3far}{1}{U_{}}{{3far}{2}{V_{}}{{3far}{3}{W_{}}{{3far}{3close}{W^\mathrm {close}_{}}{{3far}{3far}{W^\mathrm {far}_{}}{{3far}{4}{X_{}}{{3far}{4close}{X^\mathrm {close}_{}}{{3far}{4far}{X^\mathrm {far}_{}}{}}}}}}}} is the high cost of translations even
when the target (QBX) e... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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9b9cd35ce32e80cdb6508180c1ab2d15085aafb5 | subsection | 143 | 167 | Cost Implications of the | Next, we seek to understand the impact of the change in the shape of the TCR,
which was box-shaped and defined by the \ell ^\infty -norm in the earlier
version of our algorithm , but which now is spherical and
measured by an \ell ^2-norm to better match the actual region of convergence of
the obtained local expansions.... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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... |
1bae3aaf23d1e45dd577650e525a3720b9a115ed | subsection | 144 | 167 | Conclusion | This paper introduces a fast algorithm for the accurate evaluation of layer
potentials in three dimensions using Quadrature by Expansion (QBX).Our work builds on and extends the GIGAQBX algorithm in two
dimensions . Many features of the algorithm carry over broadly
unchanged from the two dimensional setting. However, s... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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... |
805e25ad43164971e4eba38f4a1c2d53361fefbf | subsection | 145 | 167 | Area Queries | Area queries were introduced in in two dimensions.
We describe their (largely straightforward) three-dimensional generalization in this section. They form the core
mechanism on which the many of the geometric operations in this article are
performed. Given a center c and a radius r, the area query
computes the set of ... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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... |
7849ceed97fa6750fc8089465a90aeb2e81e1421 | subsection | 146 | 167 | Area Queries | Multipole and Multipole-to-Local Accuracy
We use the notation of
Section REF . Hypothesis REF pertains to the
accuracy of approximating a local expansion using an intermediate multipole
expansion.
As a numerical experiment, we test the truth of this hypothesis at selected
values of the parameters (R, r, \rho , p, q). F... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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... |
89efc2fcebbe422b29af4e61e45c9785e9539d9a | subsection | 147 | 167 | Area Queries | The
quantity E_L(q) is evaluated at t. The largest observed value
of E_L(q) is taken as an estimate of the upper bound on the quantity.
Detailed Complexity Analysis
This section provides the details of the complexity analysis from
Section REF , under the assumptions highlighted in
Section REF . In
Section REF and REF ... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.02921278215944767,
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... |
a4fd8434da175aa2febe9eed980c557d149f538f | subsection | 148 | 167 | Area Queries | To bound the number of algorithmic operations involving suspended QBX centers,
we introduce a parameter into the complexity analysis that corresponds to the
average size of a `neighborhood' of a suspended QBX center—in other words,
the average number of sources with which a QBX center must interact
directly. While this... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.00632476108148694,
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0.004337306134402752,
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0.008338918909430504,
... |
d25662999bd599cd91bb4b013e9abac862c2a437 | subsection | 149 | 167 | Area Queries | Each
source-center interaction costs {p_\mathrm {qbx}}^2 operations. The number of Stage 3
interactions is |U|, so the cost of Stage 3 is at most {p_\mathrm {qbx}}^2|U|. U
may be written as a disjoint union U = U_\mathrm {big} \cup U_\mathrm {small},
where U_\mathrm {big} contains all pairs (s, c) such that |b_s| \ge |... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.02736104652285576,
0.0016757305711507797,
0.02... |
2fa6be8ff9c659c58a99c0da52d69daa3ed08396 | subsection | 150 | 167 | Area Queries | If s interacts via List 3 close with a
leaf-settled center c, then c must be owned by a box that is a
2-colleague of either b_s or an ancestor of b_s. A box has at most 5^3
- 1 = 124 boxes that are 2-colleagues. Since each source-center interaction
costs {p_\mathrm {qbx}}^2 and there are at most {n_{\mathrm {max}}} lea... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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-0.004780129995197058,
0.032198160886764526,
0.01587018370628357,
0.021623125299811363,
0.015732845291495323,
-0.0010605555726215243,
0... |
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