Pith. sign in

REVIEW 3 minor 28 references

A Nyström discretization with FFT-evaluated modal Green's functions solves acoustic scattering by many quasi-axisymmetric objects.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.3

2026-07-01 04:41 UTC pith:VYK6BIWH

load-bearing objection A targeted Nyström-FFT method for scattering from many quasi-axisymmetric bodies that looks workable and scales to 1000 objects.

arxiv 2606.31380 v1 pith:VYK6BIWH submitted 2026-06-30 math.NA cs.NA

A Spectral Solver for Acoustic Scattering by Multiple Quasi-Axisymmetric Structures

Jun Lai , Yuxin Li This is my paper
classification math.NA cs.NA
keywords acoustic scatteringquasi-axisymmetric objectsNyström discretizationboundary integral equationsfast Fourier transformmultiple scatteringspectral methodmodal Green's function
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper develops a fast algorithm for acoustic scattering problems involving multiple objects that are quasi-axisymmetric around an arbitrary curve. It discretizes the boundary integrals using a combination of Gauss-Legendre and trapezoidal quadrature, then accelerates the singular parts by recasting them as modal Green's functions evaluated with fast Fourier transforms. A convergence analysis is provided for smooth geometries, and the approach is demonstrated on problems with as many as 1000 scatterers. This matters for applications such as medical imaging and acoustic metamaterials where many such objects interact. The method couples individual scatterer solutions through inter-body interactions to handle the full multiple-scattering problem.

Core claim

We develop a fast and highly accurate algorithm for acoustic scattering by multiple quasi-axisymmetric objects, whose axis of rotation is an arbitrary curve. The method is based on a Nyström discretization that combines Gauss-Legendre quadrature with the trapezoidal rule. To treat the singular integrals that occur when target points are close to or coincide with source points, we reformulate them as evaluations of the modal Green's function and its derivatives, which are computed efficiently using the fast Fourier transform and convolution. The multiple scattering solver is then constructed by coupling the single scatterer discretizations through inter-body boundary integral interactions. We

What carries the argument

Nyström discretization of boundary integral equations combined with modal Green's functions computed via fast Fourier transform and convolution.

Load-bearing premise

The geometries are smooth and quasi-axisymmetric so that singular integrals reduce to modal Green's functions computable by FFT.

What would settle it

A numerical experiment on a single smooth sphere with a known exact analytical scattering solution; if the computed error does not decrease at the expected rate as the number of quadrature points increases, the accuracy and convergence claims would fail.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The solver efficiently handles multiple scattering problems involving up to 1000 quasi-axisymmetric structures.
  • High accuracy is achieved through spectral convergence of the quadrature rules when the surfaces are smooth.
  • Inter-body boundary integral interactions are incorporated by direct coupling of the single-scatterer discretizations.
  • The method applies to acoustic problems arising in medical imaging, geophysical exploration, and acoustic metamaterials.

Where Pith is reading between the lines

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

  • The modal reduction via FFT could be adapted to electromagnetic scattering or other linear wave problems that share the same axisymmetry.
  • For collections of objects that are not perfectly quasi-axisymmetric, the same framework might still provide a good preconditioner or initial approximation.
  • The convolution structure suggests that the method could be combined with fast multipole or tree-based accelerators for even larger numbers of scatterers.
  • Extending the solver to broadband or time-domain problems would require repeating the frequency-domain solve at multiple frequencies while reusing the modal Green's function infrastructure.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The manuscript develops a Nyström discretization for acoustic scattering by multiple quasi-axisymmetric objects (axis of rotation an arbitrary curve), combining Gauss-Legendre quadrature with the trapezoidal rule. Singular integrals are reformulated as modal Green's functions and derivatives, computed via FFT and convolution. Single-body discretizations are coupled through inter-body boundary integral interactions to solve the multiple-scattering system. A convergence analysis is presented for smooth geometries, and numerical examples demonstrate efficiency and accuracy for problems with up to 1000 such structures.

Significance. If the convergence analysis and numerical results hold, the approach offers a computationally efficient spectral method for a practically relevant class of multiple-scattering problems in acoustics. The FFT reduction of singular integrals exploiting quasi-axisymmetry and the direct coupling strategy are strengths that could enable scaling to large numbers of bodies, with potential impact in applications such as medical imaging and acoustic metamaterials. The explicit convergence statement and large-scale tests add value.

minor comments (3)
  1. The description of the modal Green's function computation via FFT and convolution (mentioned in the abstract and method) would benefit from an explicit statement of the convolution theorem application and any aliasing controls, to make the implementation fully reproducible from the text.
  2. In the numerical examples section, convergence plots or tables should report observed rates alongside the theoretical predictions from the analysis, with explicit mention of the smoothness assumptions used in each test case.
  3. Notation for the inter-body interaction operators and the overall system matrix should be introduced with a clear diagram or equation block early in the multiple-scattering section to aid readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript, accurate summary of the method, and recommendation of minor revision. The significance assessment correctly identifies the strengths of the FFT-based modal Green's function approach and the scaling to 1000 bodies. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents a Nyström discretization combining Gauss-Legendre quadrature and the trapezoidal rule for quasi-axisymmetric scatterers, reformulates singular integrals as modal Green's functions evaluated via FFT and convolution, then couples the resulting single-body systems through inter-body boundary integrals. Convergence analysis is stated only for smooth geometries, with numerical validation on up to 1000 bodies. No derivation step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation, or ansatz imported from the authors' prior work; the central algorithm is a direct discretization and coupling construction whose correctness is checked externally by the supplied numerical examples.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Assessment based solely on abstract; no free parameters or invented entities are mentioned. The method rests on standard assumptions of boundary integral methods and smoothness for analysis.

axioms (1)
  • domain assumption Geometries are smooth
    Convergence analysis is stated to hold for smooth geometries.

pith-pipeline@v0.9.1-grok · 5688 in / 1254 out tokens · 57550 ms · 2026-07-01T04:41:00.483155+00:00 · methodology

0 comments
read the original abstract

Acoustic scattering arises in a wide range of applications, including medical imaging, geophysical exploration, acoustic metamaterials, etc. In this paper, we develop a fast and highly accurate algorithm for acoustic scattering by multiple quasi-axisymmetric objects, whose axis of rotation is an arbitrary curve. The method is based on a Nystr\"om discretization that combines Gauss-Legendre quadrature with the trapezoidal rule. To treat the singular integrals that occur when target points are close to or coincide with source points, we reformulate them as evaluations of the modal Green's function and its derivatives, which are computed efficiently using the fast Fourier transform and convolution. The multiple scattering solver is then constructed by coupling the single scatterer discretizations through inter-body boundary integral interactions. We also present a convergence analysis for scattering problems with smooth geometries. Numerical examples demonstrate the efficiency and accuracy of the proposed method for solving multiple scattering problems involving up to 1000 quasi-axisymmetric structures.

Figures

Figures reproduced from arXiv: 2606.31380 by Jun Lai, Yuxin Li.

Figure 1
Figure 1. Figure 1: Geometry of quasi-axisymmetric structures 2. Integral equation formulations Consider a bounded quasi-axisymmetric obstacle Ω ⊂ R 3 with a smooth boundary ∂Ω parame￾terized by ∂Ω := (r(s) cos θ + p(s), r(s) sin θ + q(s), z(s)) (1) where s ∈ [0, T], θ ∈ [0, 2π]. We refer to such a geometry as quasi-axisymmetric, as it can be gener￾ated by sweeping a circular profile of radius r(s) along the generating center… view at source ↗
Figure 2
Figure 2. Figure 2: Numerical results for the quasi-wave geometry with k = 10 and Nf = 40. Here u∞ is referred to as the far-field pattern u∞(θ, φ) = 1 4π Z ∂Ω e −iky· x |x| µ(y)ds(y), (46) where µ is the single-layer density obtained by solving the boundary integral equation. Self￾convergence tests are performed on u∞(θ, φ) using 200 equispaced sampling points in each of the azimuthal and polar directions on the unit sphere.… view at source ↗
Figure 3
Figure 3. Figure 3: Three quasi-axisymmetric geometries and the corresponding spectral convergence for k = 5 [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Two-body scattering configurations used in the convergence tests [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Far-field results for eight quasi-axisymmetric ellipsoids with k = 10 [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Regular cubic array with 1000 quasi-axisymmetric ellipsoids in the box [−1, 4] × [−1, 4] × [−1, 4]. The minimum surface-to-surface distance is greater than 0.05. achieve comparable accuracy. The timing results show that both the matrix assembly time Tmat and the linear solve time Tsolve increase substantially with the total number of unknowns. For the larger discretizations in this eight-body example, Tsol… view at source ↗
Figure 7
Figure 7. Figure 7: Illustration of a randomly configured multi-body scattering problem. Each scat￾terer is independently rotated by random angles θ1 ∼ U[0, 2π) and θ2 ∼ U[0, 2π), and dis￾placed by a random perturbation ε ∼ U[−εmax, εmax] 3 [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

  1. [1]

    National Bureau of Standards, Washington, DC (1964)

    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards, Washington, DC (1964)

    work page 1964

  2. [2]

    Bremer, J., Gimbutas, Z.: A Nystr¨ om method for weakly singular integral operators on surfaces. J. Comput. Phys. 231(14), 4885–4903 (2012)

    work page 2012

  3. [3]

    Bremer, J., Gimbutas, Z., Rokhlin, V.: A nonlinear optimization procedure for generalized Gaussian quadratures. SIAM J. Sci. Comput. 32(4), 1761–1788 (2010)

    work page 2010

  4. [4]

    ESAIM Math

    Chernov, A., von Petersdorff, T., Schwab, C.: Exponential convergence of hp quadrature for integral operators with Gevrey kernels. ESAIM Math. Model. Numer. Anal. 45(3), 387–422 (2011)

    work page 2011

  5. [5]

    and Kress, R.: Integral Equation Method in Scattering Theory

    Colton, D. and Kress, R.: Integral Equation Method in Scattering Theory. Wiley-Interscience, New York (1983)

    work page 1983

  6. [6]

    (eds.): Acoustic Metamaterials: Negative Refraction, Imaging, Lensing and Cloaking

    Craster, R.V., Guenneau, S. (eds.): Acoustic Metamaterials: Negative Refraction, Imaging, Lensing and Cloaking. Springer, Dordrecht (2013)

    work page 2013

  7. [7]

    Fleming, J.L., Wood, A.W., Wood, W.D.: Locally corrected Nystr¨ om method for EM scattering by bodies of revolution. J. Comput. Phys. 196(1), 41–52 (2004)

    work page 2004

  8. [8]

    Ganesh, M., Hawkins, S.C.: A high-order algorithm for multiple electromagnetic scattering in three dimensions. Numer. Algorithms 50(4), 469–510 (2009)

    work page 2009

  9. [9]

    Garritano, J., Kluger, Y., Rokhlin, V., Serkh, K.: On the efficient evaluation of the azimuthal Fourier components of the Green’s function for Helmholtz’s equation in cylindrical coordinates. J. Comput. Phys. 471, 11585 (2022)

    work page 2022

  10. [10]

    Gimbutas, Z., Greengard, L.: Fast multi-particle scattering: a hybrid solver for the Maxwell equations in mi- crostructured materials. J. Comput. Phys. 232, 22–32 (2013)

    work page 2013

  11. [11]

    V.: A fast algorithm for particle simulations

    Greengard, L., Rokhlin. V.: A fast algorithm for particle simulations. J. Comput. Phys. 73(2), 325-348, (1987)

    work page 1987

  12. [12]

    Helsing, J., Karlsson, A.: An explicit kernel-split panel-based Nystr¨ om scheme for integral equations on axially symmetric surfaces. J. Comput. Phys. 272, 686–703 (2014)

    work page 2014

  13. [13]

    SIAM Rev

    Kleinman, R.E., Roach, G.F.: Boundary integral equations for the three-dimensional Helmholtz equation. SIAM Rev. 16(2), 214–236 (1974)

    work page 1974

  14. [14]

    Springer, New York (2014)

    Kress, R.: Linear Integral Equations, 3rd edn. Springer, New York (2014)

    work page 2014

  15. [15]

    J., Dong, H.: A fast solver for elastic scattering from axisymmetric objects by boundary integral equations

    Lai. J., Dong, H.: A fast solver for elastic scattering from axisymmetric objects by boundary integral equations. Adv. Comput. Math. 48(20), (2022)

    work page 2022

  16. [16]

    Lai, J., Kobayashi, M., Barnett, A.: A fast and robust solver for the scattering from a layered periodic structure containing multi-particle inclusions. J. Comput. Phys. 298, 194–208 (2015)

    work page 2015

  17. [17]

    Lai, J., Kobayashi, M., Greengard, L.: A fast solver for multi-particle scattering in a layered medium. Opt. Express 22(17), 20481–20499 (2014)

    work page 2014

  18. [18]

    Lai, J., O’Neil, M.: An FFT-accelerated direct solver for electromagnetic scattering from penetrable axisymmetric objects. J. Comput. Phys. 390, 152–174 (2019)

    work page 2019

  19. [19]

    Inverse Probl

    Lai, J., Zhang, J.: Fast inverse elastic scattering of multiple particles in three dimensions. Inverse Probl. 38(10), 104002 (2022)

    work page 2022

  20. [20]

    Liu, Y., Barnett, A.H.: Efficient numerical solution of acoustic scattering from doubly-periodic arrays of axisym- metric objects. J. Comput. Phys. 324, 226–245 (2016)

    work page 2016

  21. [21]

    Malhotra D, Barnett A.H.: Efficient convergent boundary integral methods for slender bodies. J. Comput. Phys. 503, 112855 (2024)

    work page 2024

  22. [22]

    Cambridge University Press, Cambridge (2006)

    Martin, P.A.: Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles. Cambridge University Press, Cambridge (2006)

    work page 2006

  23. [23]

    IEEE Trans

    Medgyesi-Mitschang, L., Putnam, J.: Electromagnetic scattering from axially inhomogeneous bodies of revolution. IEEE Trans. Antennas Propag. 32(8), 797–806 (1984) 23

    work page 1984

  24. [24]

    IEEE Trans

    Morgan, M., Mei, K.: Finite-element computation of scattering by inhomogeneous penetrable bodies of revolution. IEEE Trans. Antennas Propag. 27(2), 202–214 (1979)

    work page 1979

  25. [25]

    Soenarko, B.: A boundary element formulation for radiation of acoustic waves from axisymmetric bodies with arbitrary boundary conditions. J. Acoust. Soc. Am. 93(2), 631–639 (1993)

    work page 1993

  26. [26]

    50(1), 67–87 (2008)

    Trefethen, L.N.: Is Gauss quadrature better than Clenshaw–Curtis? SIAM Rev. 50(1), 67–87 (2008)

    work page 2008

  27. [27]

    Young, P., Hao, S., Martinsson, P.G.: A high-order Nystr¨ om discretization scheme for boundary integral equations defined on rotationally symmetric surfaces. J. Comput. Phys. 231(11), 4142–4159 (2012)

    work page 2012

  28. [28]

    Zhang, S., Xia, C., Fang, N.: Broadband acoustic cloak for ultrasound waves. Phys. Rev. Lett. 106(2), 024301 (2011) School of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang 310027, China, and Cen- ter for Interdisciplinary Applied Mathematics, Zhejiang University, Hangzhou, Zhejiang 310027, China Email address:laijun6@zju.edu.cn School of ...

    work page 2011