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Accelerated Born series for high-frequency strong scattering by weakly non-radial media

Mirza Karamehmedović, Faouzi Triki

arXiv:2610.23428v1math.NAcs.NA

TL;DR

The paper addresses practical selection of a modified Born series parameter for high-frequency Helmholtz scattering, including strong scattering and weakly nonradial media. It estimates the operator spectrum through WKB-based propagation and a first-order correction, then uses the estimate in a minimax selection. The method converges in tested strong-scattering cases, with computational savings at comparable accuracy in selected comparisons.

  • Problem

    Selecting a useful modified Born series parameter is difficult because sufficient convergence conditions can depend on the scattering operator's spectral geometry.

    Source: §1 · p. 2

  • Method

    The method estimates eigenvalue evolution from a constant reference medium to a radial profile using WKB-based scalar equations, corrects for weak nonradiality, and selects a complex parameter through a minimax problem.

    Source: §1 · pp. 2–3, §2.1 · p. 6, §3 · p. 10

  • Results

    The minimax selection sufficed for convergence in numerical tests, including one-dimensional cases with operator spectral radius significantly greater than one, and MBS had lower setup-plus-solve time than two implemented competitors at tested demanding accuracies.

    Source: §3 · p. 11, §4 · p. 12

  • Takeaways & Limitations

    The approach provides a practical parameter-selection procedure validated against direct integral-equation and finite-difference computations for high-frequency scattering, including weakly nonradial media.

    Source: §5 · p. 13

    The stated scope assumes bounded, nonnegative, compactly supported media, with only weak departures from radial symmetry considered in dimensions 2 and 3.

    Source: §1 · pp. 1–2

Abstract

from arXiv · show

We present a practical method for accelerating the modified Born series for high-frequency Helmholtz scattering in one, two, and three dimensions. We consider bounded, nonnegative, compactly supported media, allowing small departures from radial symmetry in two and three dimensions. The method selects a single complex relaxation parameter using an approximation of the scattering operator's spectrum. Starting from precomputed eigenvalues for a constant reference medium, we estimate their evolution towards the prescribed radial profile. A leading-order Wentzel-Kramers-Brillouin (WKB) approximation reduces this spectral propagation to scalar nonlinear equations, followed by a first-order correction for the nonradial part of the medium. The estimated eigenvalues, together with a bound on the omitted spectrum, define a convex minimax problem for selecting the relaxation parameter. This spectral preparation is independent of the source and can therefore be reused for multiple scattering problems in the same medium. Numerical examples demonstrate convergence in strongly scattering cases where the scattering operator has spectral radius greater than one. Comparisons with direct integral-equation and finite-difference solvers assess accuracy, residual convergence, and computation time, and show computational savings at comparable accuracy in selected examples.

1 Introduction

The paper develops a modified Born series solver for Helmholtz scattering in dimensions 1–3 and a practical way to select its complex relaxation parameter. It estimates the scattering operator spectrum to improve convergence, with spectral preparation reusable across sources for a fixed medium.

1 Introduction

  • The method targets bounded, nonnegative, compactly supported media in dimensions 1–3, allowing weak departures from radial symmetry in dimensions 2 and 3.The scattering problem is formulated for a fixed positive frequency.Source: §1 · pp. 1–2
  • The proposed method estimates the operator spectrum and selects the parameter by minimizing the spectral radius of the modified operator, approximated in practice by a minimax problem.The aim is to improve worst-case asymptotic convergence and reduce the terms needed to reach a prescribed tolerance.Source: §1 · p. 3
  • Starting from precomputed eigenvalues for a constant reference medium, the method propagates them toward the radial profile using WKB-based scalar nonlinear equations and applies a first-order correction for weak nonradiality.The medium is represented as q(x) = qrad(|x|) + ηQ(x), with small η.Source: §1 · p. 3
  • For a fixed medium, the spectral-estimation steps are shared across sources, leaving source-specific series iterations and matrix-vector operations.The intensive computation is described as suited to GPU implementation.Source: §1 · pp. 3–4
  • The modified Born series uses a complex overrelaxation parameter, and prior work established convergence for some such parameters even in strong-scattering and high-frequency cases.The paper notes that earlier sufficient conditions can depend on spectral geometry, making parameter selection less practical.Source: §1 · pp. 3–4

2 WKB approximation of the spectral evolution

The section estimates how eigenvalues evolve from a constant reference medium to a radial profile, then corrects them for weak nonradiality. High-frequency WKB approximations reduce propagation to scalar equations in dimensions 1–3.

2 WKB approximation of the spectral evolution

  • The spectral workflow starts with eigenvalues for a constant reference medium, propagates selected branches to the radial profile, and then corrects for weak nonradiality.A finite representative set of reference eigenvalues can be computed numerically once for the parameters of interest.Source: §2 · p. 4, §2.1 · p. 6, §2.2 · p. 10

2.1 Spectral propagation

  • The propagation follows eigenvalue branches along a homotopy from the constant medium to the radial profile, using a high-frequency WKB approximation for the eigenfunctions.The initial eigenvalues are precomputed or estimated, and the resulting propagation is expressed through scalar nonlinear equations.Source: §2.1 · pp. 5–6

2.1.1 Case d = 1

  • In dimension 1, the leading-order WKB treatment uses two rays and yields a scalar equation that can be solved numerically for the endpoint eigenvalue.The derivation assumes no stationary WKB phases, a controlled imaginary part for the relevant branch, and negligible oscillatory terms after high-frequency averaging.Source: §2.1.1 · pp. 6–7

2.1.2 Case d = 2

  • In dimension 2, separation into angular modes and leading-order WKB approximations produce a scalar equation for the endpoint eigenvalue, solved numerically from the precomputed reference eigenvalue.The propagation assumes no exceptional points along the selected branch.Source: §2.1.2 · pp. 7–8

2.1.3 Case d = 3

  • In dimension 3, the analogous spherical-coordinate WKB treatment yields a scalar equation for the endpoint eigenvalue, solved numerically from the precomputed reference eigenvalue.As in dimension 2, the argument assumes no exceptional points along the selected branch.Source: §2.1.3 · p. 9

2.2 Spectral correction

  • For a weakly nonradial medium, a first-order perturbation corrects the propagated radial-profile eigenvalues to estimate the spectrum of the full medium.The correction is applied after propagation along the radial homotopy.Source: §2.2 · p. 10

3 Estimating the optimal preconditioner parameter

The section selects an overrelaxation parameter by minimizing a worst-case contraction bound over an estimated truncated spectrum, with a bound for omitted eigenvalues. It also uses residuals to guide when to stop summing the modified Born series.

3 Estimating the optimal preconditioner parameter

  • The minimax objective accounts for estimated eigenvalues and a bound on the omitted spectrum when selecting the overrelaxation parameter.The estimated spectrum is truncated, with omitted eigenvalues assumed to satisfy |λ| ≤ δ.Source: §3 · pp. 10–11
  • The minimax solution is not guaranteed to make the true operator's spectral radius less than one, but it sufficed for convergence in the numerical tests.The computations use a rectangular admissible set in the first quadrant.Source: §3 · p. 11
  • The optimization is formulated as a convex second-order cone program in the real variables u, v, r, s0, and s1.The estimated contraction metric takes the maximum over the estimated spectrum and the omitted-spectrum bound.Source: §3 · p. 11
  • For a chosen parameter and tolerance, the relative residual provides a stopping rule: terminate at the first index N for which r(γ, N) < ε.The residual can be precomputed for any γ and N.Source: §3 · p. 11
  • The parameter affects the stopping index: as γ approaches zero, the spectral radius of Mγ(k) approaches one, and the expected stopping index grows for fixed tolerance.The spectral radius of Mγ(k) must also be less than one for convergence.Source: §3 · p. 12

4 Numerical examples

Numerical examples test the accelerated modified Born series in one and two dimensions, including one-dimensional cases with scattering-operator spectral radii well above one. At the more demanding tested accuracy levels, it has lower setup-plus-solve time than the two implemented competing methods.

4 Numerical examples

  • One-dimensional examples have spectral radii of approximately 42.7, 11.5, and 9.5, and Figures 1–6 demonstrate applicability when the radius is significantly greater than one.The three media are q(x) = 5 + 5 sin(2πx), q(x) = 1 + x², and a piecewise constant contrast.Source: §4 · p. 12
  • Figures 7–8 support the approach in two dimensions for a weakly non-radial medium with η = 0.12 and ∥Q∥∞ = 1.The source is supported in a disk of radius 0.34 centered at (-0.36, 0.08).Source: §4 · p. 12
  • At the more demanding accuracy levels tested, MBS has lower setup-plus-solve time than both implemented competing methods.The reported timing includes setup and one solve, charging the full spectral setup to each MBS result.Source: §4 · pp. 12–13
  • The comparisons use MBS and direct LU with the same volume-potential discretization, while the independent competitor uses second-order finite differences.The experiments use k = 20π and 768 cells in one dimension, and k = 12 with a 48 × 48 radial–angular grid in two dimensions.Source: §4 · pp. 12–13

5 Conclusion

The conclusion presents an accelerated Born series method and a practical procedure for choosing its overrelaxation parameter. It reports effectiveness at high frequency for weak and strong scattering and validation against two solver types.

5 Conclusion

  • The method addresses Helmholtz scattering for bounded, nonnegative, compactly supported, weakly non-radial media.The stated scope includes one, two, and three dimensions.Source: §5 · p. 13
  • The authors provide a practical numerical procedure for selecting the overrelaxation parameter.Source: §5 · p. 13
  • The method is reported effective at high frequency for weak and strong scattering and validated against direct integral-equation and finite-difference computations.Source: §5 · p. 13
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