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arxiv: 2606.27134 · v1 · pith:OFS54GFSnew · submitted 2026-06-25 · 🧮 math.NA · cs.NA

Fast summation on rectangular cuboids with arbitrary periodicity in the DMK framework

David Krantz , Ludvig af Klinteberg , Anna-Karin Tornberg This is my paper

Pith reviewed 2026-06-26 03:43 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords fast summationDMKmixed periodicityrectangular cuboidsStokes potentialsEwald summationmultilevel kernel splittingoctree
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The pith

DMK fast summation extends to rectangular cuboids with mixed periodicity by cubical tiling of localized tree levels and Fourier treatment of the root contribution.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the Dual-space multilevel kernel-splitting framework from free-space and fully periodic cubes to rectangular cuboids that may be periodic in one, two, or three directions. Lower levels of the octree involve only localized interactions, so they can be evaluated on a cubical tiling of the domain with almost no change to the original algorithm. The smooth root-level far-field term is computed in Fourier space, using series in periodic directions and integrals in free directions, with truncated kernels to regularize singularities when periodicity is reduced. The resulting scheme is validated on electrostatic and Stokes potentials for all periodicity combinations and a wide range of aspect ratios, and the added cost of periodization remains small even for high-aspect-ratio boxes.

Core claim

By evaluating localized interactions below the root on a cubical tiling and handling the smooth root-level far-field in Fourier space with truncated kernels for regularization, DMK applies to mixed-periodicity problems on rectangular cuboids while preserving the efficiency of the free-space version.

What carries the argument

The periodization strategy that tiles localized lower-tree interactions on cubical domains and evaluates the root-level smooth contribution via Fourier series or integrals, regularized by truncated kernels when periodicity is incomplete.

If this is right

  • The method applies to electrostatic potential as well as Stokeslet, stresslet and rotlet kernels for any combination of periodicities.
  • For large-aspect-ratio cuboids the root-level sum can be accelerated by the fast Fourier transform.
  • The approach supplies a general framework for other non-oscillatory kernels whenever a kernel split exists.
  • Numerical cost remains comparable to the free-space DMK algorithm across the tested range of aspect ratios.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same tiling-plus-Fourier-root strategy could be tested on other multilevel summation schemes that already separate local and smooth contributions.
  • Channel or slab geometries common in confined-flow simulations become directly accessible without forcing artificial periodicity in all directions.
  • The regularization via truncated kernels may generalize to kernels whose Fourier transforms have different singularity structures.

Load-bearing premise

Interactions on all tree levels below the root are localized, allowing their evaluation with minimal modification on a cubical tiling of the domain.

What would settle it

A numerical run on a high-aspect-ratio cuboid with one periodic direction that shows the periodization overhead exceeding a small fraction of free-space DMK time, or that the truncated-kernel Fourier discretization fails to converge at the expected rate.

Figures

Figures reproduced from arXiv: 2606.27134 by Anna-Karin Tornberg, David Krantz, Ludvig af Klinteberg.

Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: A 2D sketch of how the truncation radius [PITH_FULL_IMAGE:figures/full_fig_p008_3_1.png] view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: a [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Cube tiling for a triply periodic cuboid. Panel (a) shows and exact tiling, while panel (b) shows [PITH_FULL_IMAGE:figures/full_fig_p010_4_1.png] view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: A sketch of the domain Ωext (outer red line) in the 1−2 plane for evaluation of u DP compact, as defined in (2.3) with D = 2 or 3 and L2 not an integer multiple of L1. Sources inside Ω in black, periodic copies of these sources in blue. Each of the red Mext cubes will be adaptively refined in an octree structure. 4.2. Evaluation of compact contribution. With the forest of octrees built on Ωext, where Ωex… view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: Depending on the domain, the structure factor [PITH_FULL_IMAGE:figures/full_fig_p012_4_3.png] view at source ↗
Figure 6.1
Figure 6.1. Figure 6.1: Panels (a), (b), and (c) show the total number of Fourier discretization points [PITH_FULL_IMAGE:figures/full_fig_p022_6_1.png] view at source ↗
Figure 6
Figure 6. Figure 6: a [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 6.2
Figure 6.2. Figure 6.2: Linear scaling of periodic DMK for the Laplace kernel on the unit cube. The figure shows the runtime [PITH_FULL_IMAGE:figures/full_fig_p023_6_2.png] view at source ↗
Figure 6
Figure 6. Figure 6: a [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 6.3
Figure 6.3. Figure 6.3: Normalized runtime for rectangular cells [PITH_FULL_IMAGE:figures/full_fig_p024_6_3.png] view at source ↗
Figure 6.4
Figure 6.4. Figure 6.4: Runtimes and self-normalized runtimes of the compact and far-field contributions, [PITH_FULL_IMAGE:figures/full_fig_p025_6_4.png] view at source ↗
Figure 6.5
Figure 6.5. Figure 6.5: Large-aspect-ratio version of [PITH_FULL_IMAGE:figures/full_fig_p025_6_5.png] view at source ↗
Figure 6.6
Figure 6.6. Figure 6.6: Panel (a) illustrates the singly periodic pipe geometry in the cell [PITH_FULL_IMAGE:figures/full_fig_p025_6_6.png] view at source ↗
read the original abstract

Dual-space multilevel kernel-splitting (DMK) is a fast summation framework that combines ideas from the fast multipole method, Ewald summation, and multilevel summation. Originally formulated for free-space problems, and later extended to fully periodic problems on a cube, it decomposes the kernel interaction into a smooth global contribution and a hierarchy of localized interactions evaluated on an octree. We extend DMK to problems on rectangular cuboids with periodic boundary conditions in one, two, or three coordinate directions. The periodization leverages the fact that interactions on all tree levels below the root are localized, allowing for their evaluation with minimal modification on a cubical tiling of the domain. The remaining smooth root-level far-field contribution is evaluated in Fourier space, with Fourier series in the periodic directions and Fourier integrals in the free directions. For reduced periodicity, truncated kernels are used to regularize singular and near-singular Fourier kernels, yielding rapidly convergent trapezoidal discretizations and a unified treatment of all periodicities. For large-aspect-ratio cuboids, the root-level sum can be accelerated using the fast Fourier transform. We validate the method for the electrostatic potential and Stokeslet, stresslet and rotlet potentials, for all periodicities and a wide range of aspect ratios. Numerical experiments show that the periodization adds only a small overhead to the original free-space DMK algorithm, also for high-aspect-ratio cuboids. The resulting method provides a framework for applying DMK to problems with mixed periodicity on rectangular cuboids, and extends naturally to other non-oscillatory kernels for which a kernel split is available.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper extends the dual-space multilevel kernel-splitting (DMK) framework, originally for free-space and fully periodic cubic domains, to rectangular cuboids with periodicity in one, two, or three directions. It exploits localization of sub-root tree interactions to evaluate them on a cubical tiling with minimal changes, while treating the smooth root-level far-field via Fourier series (periodic directions) and integrals (free directions), regularized by truncated kernels for reduced periodicity to enable convergent trapezoidal rules. The approach is claimed to apply to non-oscillatory kernels admitting a split (e.g., electrostatic potential, Stokeslet/stresslet/rotlet) and to incur only small overhead relative to free-space DMK, even at high aspect ratios; validation is asserted via numerical experiments across all periodicities and aspect ratios.

Significance. If the error analysis, convergence rates, and numerical results hold, the work supplies a practical, unified fast-summation tool for mixed-periodicity problems on cuboids that preserves DMK's multilevel structure and extends naturally to other kernels. The combination of established FMM/Ewald/multilevel ideas with a new handling of truncated Fourier kernels for arbitrary periodicity is a clear incremental advance in the field of fast summation methods.

major comments (2)
  1. [§3] §3 (periodization strategy) and the abstract claim that sub-root interactions are 'localized' and therefore evaluable on a cubical tiling with 'minimal modification'; however, the precise definition of the interaction cutoff radius relative to the cuboid aspect ratio and the resulting error bound when the tiling is non-cubic are not stated explicitly, which is load-bearing for the 'small overhead' claim across high-aspect-ratio cases.
  2. [§4] The abstract and §4 assert that truncated kernels yield 'rapidly convergent trapezoidal discretizations' for all periodicities, but no explicit truncation-error estimate or dependence on the truncation parameter appears; without this, it is unclear whether the unified treatment maintains the same convergence order as the fully periodic case.
minor comments (2)
  1. [§4] Notation for the truncated kernel (e.g., K_trunc) should be introduced once and used consistently; several paragraphs in §4 switch between K^trunc and K_tr without definition.
  2. [Numerical results] Figure captions for the numerical experiments should include the specific aspect ratios tested and the number of particles per run so that the 'wide range' claim can be assessed without consulting the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback. We address the two major comments point by point below. Both points identify places where explicit statements are missing from the current manuscript; we will add the requested definitions and estimates in the revision.

read point-by-point responses

  1. Referee: [§3] §3 (periodization strategy) and the abstract claim that sub-root interactions are 'localized' and therefore evaluable on a cubical tiling with 'minimal modification'; however, the precise definition of the interaction cutoff radius relative to the cuboid aspect ratio and the resulting error bound when the tiling is non-cubic are not stated explicitly, which is load-bearing for the 'small overhead' claim across high-aspect-ratio cases.

    Authors: We agree that the cutoff radius and the associated tiling error bound should be stated explicitly. The cutoff at every sub-root level is set by the finite support of the localized kernel component after the split; this support depends only on the octree level and the splitting parameter, remaining independent of the cuboid aspect ratio. Consequently the number of tiled neighbor interactions per source particle stays O(1) regardless of aspect ratio. We will insert a short paragraph in §3 that (i) defines the cutoff radius in terms of the octree depth and kernel support length and (ii) bounds the tiling error by the tail of the smooth kernel, which decays exponentially. This addition directly supports the small-overhead claim. revision: yes

  2. Referee: [§4] The abstract and §4 assert that truncated kernels yield 'rapidly convergent trapezoidal discretizations' for all periodicities, but no explicit truncation-error estimate or dependence on the truncation parameter appears; without this, it is unclear whether the unified treatment maintains the same convergence order as the fully periodic case.

    Authors: We acknowledge that an explicit truncation-error estimate is absent and would strengthen the justification for uniform convergence order. While the numerical experiments already illustrate rapid convergence for all periodicities, we will add a brief analytic paragraph in §4 deriving that the truncation error of the regularized Fourier kernels decays exponentially with the truncation radius. This error is independent of the trapezoidal discretization and therefore does not degrade the spectral accuracy already established for the fully periodic case. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper extends the existing DMK framework (free-space and fully periodic cube cases) by exploiting the localization of sub-root tree interactions to enable cubical tiling for periodization, then applies standard Fourier series/integrals with truncated kernels at the root level for arbitrary periodicity. No equation or claim reduces by construction to a fitted parameter, self-definition, or unverified self-citation chain; the central construction combines established FMM-style trees, Ewald-style Fourier treatment, and multilevel splitting without internal reduction to its own inputs. Numerical experiments serve as validation rather than definitional necessity. The derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Ledger constructed from abstract only; no explicit fitted parameters or new entities are named.

axioms (2)
  • domain assumption Interactions on tree levels below the root remain localized enough to be evaluated on a cubical tiling of the rectangular domain with minimal modification.
    Stated as the enabling fact for the periodization of all levels except the root.
  • domain assumption Truncated kernels regularize singular and near-singular Fourier kernels sufficiently to produce rapidly convergent trapezoidal discretizations.
    Invoked for the reduced-periodicity cases to unify the Fourier treatment.

pith-pipeline@v0.9.1-grok · 5828 in / 1358 out tokens · 70778 ms · 2026-06-26T03:43:33.251617+00:00 · methodology

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Reference graph

Works this paper leans on

288 extracted references · 186 canonical work pages · 1 internal anchor

  1. [1]

    A Method to Compute Periodic Sums , author =. J. Comput. Phys. , volume = 272, pages =. doi:10.1016/j.jcp.2014.04.039 , urldate =

    work page doi:10.1016/j.jcp.2014.04.039 2014

  2. [2]

    and Marple, Gary R

    Barnett, Alex H. and Marple, Gary R. and Veerapaneni, Shravan and Zhao, Lin , year = 2018, month = nov, journal =. A. doi:10.1002/cpa.21759 , urldate =

    work page doi:10.1002/cpa.21759 2018

  3. [3]

    Pressure Tensor for Electrostatic Interaction Calculated by Fast Multipole Method with Periodic Boundary Condition , author =. J. Comput. Chem. , volume =. doi:10.1002/jcc.25179 , urldate =

    work page doi:10.1002/jcc.25179

  4. [5]

    2605.30805 , archivePrefix=

    Tianyue Li and Dhairya Malhotra and Shravan Veerapaneni , year=. 2605.30805 , archivePrefix=

    Pith/arXiv arXiv

  5. [6]

    (2000), Calculation of atomic integration data

    Amisaki, Takashi , year = 2000, month = sep, journal =. Precise and Efficient. doi:10.1002/1096-987X(200009)21:12<1075::AID-JCC4>3.0.CO;2-L , langid =

    work page doi:10.1002/1096-987x(200009)21:12 2000

  6. [7]

    Accelerating Molecular Dynamics Simulations Using Fast

    Liang, Jiuyang and Lu, Libin and Barnett, Alex and Greengard, Leslie and Jiang, Shidong , year = 2026, month = may, journal =. Accelerating Molecular Dynamics Simulations Using Fast. doi:10.1038/s41467-026-73232-8 , urldate =

    work page doi:10.1038/s41467-026-73232-8 2026

  7. [8]

    Efficient numerical methods for non-local operators:

    B. Efficient numerical methods for non-local operators:. 2010 , publisher=

    2010

  8. [9]

    Hackbusch, Wolfgang and B. Appl. Numer. Math. , volume=. 2002 , publisher=

    2002

  9. [10]

    Computing , volume =

    Giebermann, Klaus , year =. Computing , volume =. doi:10.1007/s006070170005 , isbn =

    work page doi:10.1007/s006070170005

  10. [11]

    , year =

    Wang, Mu and Brady, John F. , year =. Spectral. J. Comput. Phys. , volume =

  11. [12]

    Parameter

    Nestler, Franziska , year =. Parameter. Front. Phys. , volume =

  12. [13]

    Wu, Bowei and Zhu, Hai and Barnett, Alex and Veerapaneni, Shravan , journal =

  13. [14]

    and Tornberg, Anna Karin , doi =

    Broms, Anna and Barnett, Alex H. and Tornberg, Anna Karin , doi =. J. Comput. Phys. , keywords =

  14. [15]

    Accelerating Fast

    Jiuyang Liang and Libin Lu and Alex Barnett and Leslie Greengard and Shidong Jiang , journal=. Accelerating Fast. 2025 , volume=

    2025

  15. [16]

    Fast Summation of. J. Comput. Phys. , volume =

  16. [17]

    2602.16591 , archivePrefix=

    Erik Boström and Anna-Karin Tornberg and Ludvig af Klinteberg , year=. 2602.16591 , archivePrefix=

    Pith/arXiv arXiv

  17. [18]

    A. SIAM J. Sci. Comput. , author =. 2019 , pages =

    2019

  18. [19]

    Bottom-up construction and 2:1 balance refinement of linear octrees in parallel , author=. SIAM J. Sci. Comput. , volume=. 2008 , publisher=

    2008

  19. [20]

    A new version of the adaptive fast

    Greengard, Leslie and Jiang, Shidong and Rachh, Manas and Wang, Jun , journal=. A new version of the adaptive fast. 2024 , publisher=

    2024

  20. [21]

    1972 , publisher=

    Fourier Series and Integrals , author=. 1972 , publisher=

    1972

  21. [22]

    Ethridge and L

    F. Ethridge and L. Greengard , title =. SIAM J. Sci. Comput. , year =

  22. [23]

    Dutt and V

    A. Dutt and V. Rokhlin , TITLE =. SIAM J. Sci. Comput. , VOLUME =. 1993 , PAGES =

    1993

  23. [24]

    Dutt and V

    A. Dutt and V. Rokhlin , TITLE =. Appl. Comput. Harmon. Anal. , VOLUME =. 1995 , PAGES =

    1995

  24. [25]

    and Nieslony, A

    Potts, Daniel and Steidl, G. and Nieslony, A. , title =. Numer. Math. , volume =. 2004 , pages =

    2004

  25. [26]

    Multilevel matrix multiplication and fast solution of integral equations , author=. J. Comput. Phys. , volume=. 1990 , publisher=

    1990

  26. [27]

    and Venner, C

    Brandt, A. and Venner, C. H. , title =. SIAM J. Sci. Comput. , volume =. 1998 , doi =

    1998

  27. [28]

    and Wu, Zhe and Phillips, James C

    Hardy, David J. and Wu, Zhe and Phillips, James C. and Stone, John E. and Skeel, Robert D. and Schulten, Klaus , title =. J. Chem. Theory Comput. , volume =. 2015 , doi =

    2015

  28. [29]

    and Zhang, Hao and Komatsu, Teruhisa S

    Morimoto, Gentaro and Koyama, Yohei M. and Zhang, Hao and Komatsu, Teruhisa S. and Ohno, Yousuke and Nishida, Keigo and Ohmura, Itta and Koyama, Hiroshi and Taiji, Makoto , year =. Hardware Acceleration of Tensor-Structured Multilevel Ewald Summation Method on. Proc. doi:10.1145/3458817.3476190 , isbn =

    work page doi:10.1145/3458817.3476190

  29. [30]

    A fast adaptive multipole algorithm in three dimensions , author=. J. Comput. Phys. , volume=. 1999 , publisher=

    1999

  30. [31]

    Malhotra , Title =

    D. Malhotra , Title =. 2022 , Howpublished =

    2022

  31. [32]

    Askham and Z

    T. Askham and Z. Gimbutas and L. Greengard and L. Lu and M. O'Neil and M. Rachh and V. Rokhlin , Title =. 2021 , Howpublished =

    2021

  32. [33]

    Gimbtuas and L

    Z. Gimbtuas and L. Greengard and L. Lu and J. Magland and D. Malhotra and M. O'Neil and M. Rachh and V. Rokhlin , title =

  33. [34]

    Malhotra , title =

    D. Malhotra , title =

  34. [35]

    Second kind integral equations for the first kind

    Jiang, Shidong and Ren, Bo and Tsuji, Paul and Ying, Lexing , journal=. Second kind integral equations for the first kind. 2011 , publisher=

    2011

  35. [36]

    1988 , publisher=

    The rapid evaluation of potential fields in particle systems , author=. 1988 , publisher=

    1988

  36. [37]

    Prolate spheroidal wave functions,

    Landau, Henry J and Pollak, Henry O , journal=. Prolate spheroidal wave functions,. 1961 , publisher=

    1961

  37. [38]

    Prolate spheroidal wave functions,

    Slepian, David and Pollak, Henry O , journal=. Prolate spheroidal wave functions,. 1961 , publisher=

    1961

  38. [39]

    Prolate spheroidal wave functions,

    Slepian, David , journal=. Prolate spheroidal wave functions,. 1978 , publisher=

    1978

  39. [40]

    Some comments on

    Slepian, David , journal=. Some comments on. 1983 , publisher=

    1983

  40. [41]

    The Journal of Physical Chemistry A , author =

    An. The Journal of Physical Chemistry A , author =. 2020 , pages =. doi:10.1021/acs.jpca.0c01684 , abstract =

    work page doi:10.1021/acs.jpca.0c01684 2020

  41. [42]

    A dual-space multilevel kernel-splitting framework for discrete and continuous convolution , author=. Comm. Pure Appl. Math. , volume=. 2025 , publisher=

    2025

  42. [43]

    doi:10.48550/arXiv.2408.10205 , abstract =

    Liu, Ziming and Ma, Pingchuan and Wang, Yixuan and Matusik, Wojciech and Tegmark, Max , month = aug, year =. doi:10.48550/arXiv.2408.10205 , abstract =

    work page doi:10.48550/arxiv.2408.10205

  43. [44]

    KAN: Kolmogorov-Arnold Networks

    Liu, Ziming and Wang, Yixuan and Vaidya, Sachin and Ruehle, Fabian and Halverson, James and Soljačić, Marin and Hou, Thomas Y. and Tegmark, Max , month = jun, year =. doi:10.48550/arXiv.2404.19756 , abstract =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2404.19756

  44. [45]

    SIAM Review , author =

    Exactness of. SIAM Review , author =. 2022 , pages =. doi:10.1137/20M1389522 , abstract =

    work page doi:10.1137/20m1389522 2022

  45. [46]

    Fast. J. Comput. Phys. , author =. 2023 , pages =. doi:10.1016/j.jcp.2023.112473 , language =

    work page doi:10.1016/j.jcp.2023.112473 2023

  46. [47]

    Optimal approximation using complex-valued neural networks , url =

    Geuchen, Paul and Voigtlaender, Felix , month = oct, year =. Optimal approximation using complex-valued neural networks , url =. doi:10.48550/arXiv.2303.16813 , abstract =

    work page doi:10.48550/arxiv.2303.16813

  47. [48]

    epubs.siam.org redirect -

  48. [49]

    Helsing, Johan and Jiang, Shidong , month = nov, year =. The

  49. [50]

    Recursive reduction quadrature for the evaluation of

    Jiang, Shidong and Zhu, Hai , month = nov, year =. Recursive reduction quadrature for the evaluation of

  50. [51]

    and Chuang, Isaac L

    Nielsen, Michael A. and Chuang, Isaac L. , year =. Quantum computation and quantum information , isbn =

  51. [52]

    Acta Numerica , author =

    Fit without fear: remarkable mathematical phenomena of deep learning through the prism of interpolation , volume =. Acta Numerica , author =. 2021 , pages =. doi:10.1017/S0962492921000039 , abstract =

    work page doi:10.1017/s0962492921000039 2021

  52. [53]

    Loss landscapes and optimization in over-parameterized non-linear systems and neural networks , volume =. Appl. Comput. Harmon. Anal. , author =. 2022 , pages =. doi:10.1016/j.acha.2021.12.009 , language =

    work page doi:10.1016/j.acha.2021.12.009 2022

  53. [54]

    Nature Machine Intelligence , author =

    Closed-form continuous-time neural networks , volume =. Nature Machine Intelligence , author =. 2022 , pages =. doi:10.1038/s42256-022-00556-7 , abstract =

    work page doi:10.1038/s42256-022-00556-7 2022

  54. [55]

    Hammad, M. M. , month = jul, year =. Comprehensive. doi:10.48550/arXiv.2407.19258 , abstract =

    work page doi:10.48550/arxiv.2407.19258

  55. [56]

    IEEE/CAA Journal of Automatica Sinica , author =

    Complex-. IEEE/CAA Journal of Automatica Sinica , author =. 2022 , pages =. doi:10.1109/JAS.2022.105743 , abstract =

    work page doi:10.1109/jas.2022.105743 2022

  56. [57]

    IEEE Transactions on Neural Networks and Learning Systems , author =

    Design of. IEEE Transactions on Neural Networks and Learning Systems , author =. 2024 , pages =. doi:10.1109/TNNLS.2022.3224779 , abstract =

    work page doi:10.1109/tnnls.2022.3224779 2024

  57. [58]

    , month = feb, year =

    Trabelsi, Chiheb and Bilaniuk, Olexa and Zhang, Ying and Serdyuk, Dmitriy and Subramanian, Sandeep and Santos, João Felipe and Mehri, Soroush and Rostamzadeh, Negar and Bengio, Yoshua and Pal, Christopher J. , month = feb, year =. Deep

  58. [59]

    High-order discretization of a stable time-domain integral equation for. J. Comput. Phys. , author =. 2020 , keywords =. doi:10.1016/j.jcp.2019.109047 , language =

    work page doi:10.1016/j.jcp.2019.109047 2020

  59. [60]

    SIAM Journal on Numerical Analysis , author =

    On the. SIAM Journal on Numerical Analysis , author =. 2022 , pages =. doi:10.1137/21M1456935 , abstract =

    work page doi:10.1137/21m1456935 2022

  60. [62]

    Fast, high-order numerical evaluation of volume potentials via polynomial density interpolation , volume =. J. Comput. Phys. , author =. 2024 , pages =. doi:10.1016/j.jcp.2024.113091 , language =

    work page doi:10.1016/j.jcp.2024.113091 2024

  61. [63]

    On quadrature for singular integral operators with complex symmetric quadratic forms , url =

    Hoskins, Jeremy and Rachh, Manas and Wu, Bowei , month = dec, year =. On quadrature for singular integral operators with complex symmetric quadratic forms , url =

  62. [64]

    SIAM Journal on Numerical Analysis , author =

    A. SIAM Journal on Numerical Analysis , author =. 2023 , pages =. doi:10.1137/22M1520372 , abstract =

    work page doi:10.1137/22m1520372 2023

  63. [66]

    Fast multipole methods for evaluation of layer potentials with locally-corrected quadratures , url =

    Greengard, Leslie and O'Neil, Michael and Rachh, Manas and Vico, Felipe , month = apr, year =. Fast multipole methods for evaluation of layer potentials with locally-corrected quadratures , url =. doi:10.1016/j.jcpx.2021.100092 , abstract =

    work page doi:10.1016/j.jcpx.2021.100092 2021

  64. [67]

    Computers & Mathematics with Applications , author =

    An efficient full-wave solver for eddy currents , volume =. Computers & Mathematics with Applications , author =. 2022 , pages =. doi:10.1016/j.camwa.2022.10.018 , language =

    work page doi:10.1016/j.camwa.2022.10.018 2022

  65. [68]

    Comparison of integral equations for the. Adv. Comput. Math. , author =. 2021 , pages =. doi:10.1007/s10444-021-09904-4 , abstract =

    work page doi:10.1007/s10444-021-09904-4 2021

  66. [69]

    Integral Equations and Operator Theory , author =

    Dirac. Integral Equations and Operator Theory , author =. 2021 , pages =. doi:10.1007/s00020-021-02657-1 , abstract =

    work page doi:10.1007/s00020-021-02657-1 2021

  67. [70]

    SIAM Journal on Numerical Analysis , author =

    Cut. SIAM Journal on Numerical Analysis , author =. 2024 , pages =. doi:10.1137/22M1542933 , abstract =

    work page doi:10.1137/22m1542933 2024

  68. [71]

    A. SIAM J. Sci. Comput. , author =. 2019 , keywords =. doi:10.1137/18M120885X , number =

    work page doi:10.1137/18m120885x 2019

  69. [72]

    A. H. Barnett and others , title =. 2018 , Howpublished =

    2018

  70. [73]

    Blackwell and L

    R. Blackwell and L. Greengard and S. Jiang and D. Malhotra , title =. 2025 , Howpublished =

    2025

  71. [74]

    af Klinteberg , title =

    L. af Klinteberg , title =. 2025 , Howpublished =

    2025

  72. [75]

    Fast and accurate integral equation methods with applications in microfluidics , url =

    af Klinteberg, Ludvig , year =. Fast and accurate integral equation methods with applications in microfluidics , url =

  73. [76]

    Physical Review E , author =

    Generalized approach to. Physical Review E , author =. 2007 , pages =. doi:10.1103/PhysRevE.75.056706 , language =

    work page doi:10.1103/physreve.75.056706 2007

  74. [77]

    Mathematics of Computation , author =

    Numerical solution of the. Mathematics of Computation , author =. 1968 , pages =. doi:10.1090/S0025-5718-1968-0242392-2 , abstract =

    work page doi:10.1090/s0025-5718-1968-0242392-2 1968

  75. [78]

    Acta Mechanica , author =

    Particle-scale computational approaches to model dry and saturated granular flows of non-. Acta Mechanica , author =. 2019 , pages =. doi:10.1007/s00707-019-02389-9 , abstract =

    work page doi:10.1007/s00707-019-02389-9 2019

  76. [79]

    A boundary integral equation approach to computing eigenvalues of the. Adv. Comput. Math. , author =. 2020 , pages =. doi:10.1007/s10444-020-09774-2 , abstract =

    work page doi:10.1007/s10444-020-09774-2 2020

  77. [80]

    Commun. Comput. Phys. , author =. 2015 , pages =. doi:10.4208/cicp.020215.150515sw , abstract =

    work page doi:10.4208/cicp.020215.150515sw 2015

  78. [81]

    International Journal of Multiphase Flow , author =

    On the application of immersed boundary, fictitious domain and body-conformal mesh methods to many particle multiphase flows , volume =. International Journal of Multiphase Flow , author =. 2012 , pages =. doi:10.1016/j.ijmultiphaseflow.2011.12.002 , abstract =

    work page doi:10.1016/j.ijmultiphaseflow.2011.12.002 2012

  79. [83]

    A highly accurate boundary integral equation method for surfactant-laden drops in. J. Comput. Phys. , author =. 2018 , pages =. doi:10.1016/j.jcp.2018.01.033 , language =

    work page doi:10.1016/j.jcp.2018.01.033 2018

  80. [84]

    Molecular Physics , author =

    Yukawa potentials in systems with partial periodic boundary conditions. Molecular Physics , author =. 2007 , pages =. doi:10.1080/00268970701481716 , language =

    work page doi:10.1080/00268970701481716 2007

Showing first 80 references.