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A convergent Born series for solving the inhomogeneous Helmholtz equation in arbitrarily large media

Gerwin Osnabrugge, Saroch Leedumrongwatthanakun, Ivo M. Vellekoop

arXiv:1601.05997v1physics.comp-ph

TL;DR

Large inhomogeneous Helmholtz systems remain computationally challenging. The paper modifies the Born series to converge for arbitrarily large structures and high scattering potential, reporting faster solutions than PSTD and FDTD while converging to the exact solution.

  • Problem

    Solving the inhomogeneous Helmholtz equation in large systems remains computationally challenging.

    Source: §1 · p. 1

  • Method

    A modified Born series is designed to converge to the exact Helmholtz solution for arbitrarily large structures and arbitrarily high scattering potential.

    Source: §1 · p. 2, §5 · p. 11

  • Results

    The method is several orders of magnitude faster than PSTD and FDTD, especially for low-contrast media, and converges to the exact solution unlike finite-difference methods.

    Source: §5 · p. 11

  • Takeaways & Limitations

    The method simulated phase conjugation in a 725-μm adipose-tissue sample in 15 minutes, compared with a reported week for a 240-μm medium in prior work.

    Source: §4.3 · p. 10

    The method is demonstrated for scalar waves in 1-D and 2-D; electromagnetic calculations require modification for vector waves.

    Source: §5 · p. 11

Abstract

from arXiv · show

We present a fast method for numerically solving the inhomogeneous Helmholtz equation. Our iterative method is based on the Born series, which we modified to achieve convergence for scattering media of arbitrary size and scattering strength. Compared to pseudospectral time-domain simulations, our modified Born approach is two orders of magnitude faster and nine orders of magnitude more accurate in benchmark tests in 1-dimensional and 2-dimensional systems.

1. Introduction

Solving the inhomogeneous Helmholtz equation in large systems remains computationally challenging, while existing time-domain methods trade speed against accuracy. The paper introduces a modified Born series intended to converge for arbitrarily large structures and high scattering potential.

1. Introduction

  • Large inhomogeneous Helmholtz systems remain computationally challenging despite methods such as Krylov subspace, multigrid, and domain decomposition approaches.Direct matrix inversion is described as ineffective for large inhomogeneous systems.Source: §1 · p. 1
  • Finite-difference time-domain methods approximate derivatives, creating a speed–accuracy trade-off because they are exact only as the step size tends to zero.Source: §1 · pp. 1–2
  • The traditional Born series diverges for larger structures or structures with high refractive-index contrast, despite its use for small, weakly scattering particles.Source: §1 · p. 2
  • The proposed modified Born series is reported to converge to the exact solution for arbitrarily large structures and arbitrarily high scattering potential, and to be several orders faster than PSTD.The paper compares accuracy and runtime with PSTD for 1-D and 2-D media.Source: §1 · p. 2

2. Algorithm

The algorithm rewrites the Helmholtz solution using a Green’s-function operator and a preconditioned Born iteration. Its interpretation links convergence and finite-distance field updates to the choice of parameters, including ε.

2. Algorithm

  • The field ψ, source S, and wavenumber k define the scattering potential V(r) = k(r)^2 − k^2, and the Green-function formulation expands recursively into the traditional Born series.The formulation uses G for convolution with the Green’s function, with V diagonal and S and ψ represented as vectors.Source: §2 · p. 2
  • Choosing a suitable preconditioner γ makes the iteration convergent even when the traditional series criterion ρ(GV) < 1 does not hold.The paper states that its γ and ε combination ensures ρ(M) < 1; k0 and ε may be chosen as positive real numbers without changing the solution.Source: §2 · pp. 2–3
  • The modified iteration converges to the exact Helmholtz solution for arbitrarily large media.It starts from ψ0 = γGS and is expressed as a series in powers of M.Source: §2 · p. 3

2.1. Interpretation

  • ε makes the homogeneous-medium Green’s function decay exponentially, localizing its energy; an equal gain in the scattering potential compensates the background loss.Even a homogeneous medium therefore has a nonzero scattering potential in this construction.Source: §2.1 · p. 3
  • In a homogeneous medium, each iteration moves a wave packet 2k0/ε, while the solution expands through finite-volume updates; Figure 1 shows solutions and contributions at iterations 20, 40, and 60.This pseudo-propagation is an interpretation of a time-independent solver, not time-resolved propagation.Source: §2.1 · pp. 3, 5, Figure 1 · p. 4
  • For D = 3, the method is reported faster than PSTD when λc/λm ≤ 2.2; in the tissue example it reaches 2.7 wavelengths per iteration, 21 times the speed of lowest-accuracy PSTD.Source: §2.1 · p. 5

3. Implementation

The implementation discretizes the field and potential on a regular two-dimensional grid and compares the method’s parameters with PSTD and FDTD. Its reported setup also highlights time-step and boundary choices that affect accuracy and runtime.

3. Implementation

  • The field and potential map are discretized on a regular 2-D grid with spacing Δx = λ/4.Source: §3 · p. 5
  • Table 1 compares key parameters for modified Born, PSTD, and FDTD, defining λm as the smallest wavelength, λc = 2π/k0, and L as system length.Source: Table 1 · p. 6, §3 · p. 6
  • PSTD should approach the modified Born solution as its time step tends to zero, but its accuracy depends on Δt.Source: §3 · p. 7
  • For high-accuracy simulations, the authors chose absorbing boundaries as a trade-off between layer thickness and accuracy.Residual boundary reflection or transmission can affect accuracy, and selecting a suitable absorbing condition is not trivial.Source: §3 · p. 7

4. Results

The homogeneous benchmark reports a large accuracy and speed advantage over PSTD, while the disordered-medium test compares the methods at the zero-step limit. A phase-conjugation experiment applies the method to a large adipose-tissue sample.

4.1. Accuracy and efficiency for homogeneous media

  • 2 orders of magnitude faster and at least 9 orders more accurate than PSTD designed for 1% accuracy; modified Born reaches E < 10^-11 at 0.5 iteration per propagated wavelength.Figure 2 plots relative error against iterations per wave cycle for modified Born and PSTD.Source: §4.1 · pp. 7, 9, Figure 2 · p. 8
  • 10^-11 relative error in this run came from absorbing-boundary reflection and leakage; thicker, smoother boundaries reduced it below 10^-17.Source: §4.1 · p. 9

4.2. Disordered inhomogeneous medium

  • Within 10^-13 at the PSTD zero-time-step limit, the modified Born solution matched PSTD before its accuracy saturated from rounding errors; runtimes were 8 seconds versus 20 hours at 10^4 iterations per wavelength.Figure 3 shows the steady-state field and relative difference as PSTD iterations vary.Source: §4.2 · p. 9, Figure 3 · p. 10

4.3. Phase conjugation numerical experiments in adipose tissue

  • 15 minutes simulated a 725 µm adipose-tissue sample, compared with one week for a prior simulation of a 240 µm-thick medium.Source: §4.3 · p. 10
  • Phase conjugating the field at the top boundary produced a second steady-state field that focused at the original point-source location despite continued scattering in deeper tissue.Source: §4.3 · p. 10

5. Conclusion

The modified Born series converges for arbitrarily large structures with arbitrarily high scattering potential. It is reported to be faster than PSTD and FDTD, with demonstrated results in 1D and 2D media.

5. Conclusion

  • In Figure 4, phase conjugation of the field at the top boundary focuses light at the original point-source location in an adipose tissue model.The field remains scattered as it propagates deeper into the tissue.Source: Figure 4 · p. 11
  • The method converges for arbitrarily large structures with arbitrarily high scattering potential.Unlike the original Born series, which diverges for larger structures or high refractive-index contrast, the modified series converges to the exact solution.Source: §5 · pp. 10–11
  • The method is several orders of magnitude faster than PSTD and FDTD, especially for media with low scattering contrast.It converges to the exact solution, whereas finite-difference methods are limited in accuracy.Source: §5 · p. 11
  • For time-dependent solutions, the method could be applied at each frequency of interest, with the time-dependent result calculated by Fourier transform.The authors state this approach would still be more accurate and potentially faster than PSTD.Source: §5 · p. 11
  • The method was demonstrated for 1D and 2D media; electromagnetic wave calculations require modification for vector waves.The authors expect similar accuracy and speed improvements for such vector-wave calculations.Source: §5 · p. 11
  • Future work could seek faster-converging choices of γ and ϵ, use advanced iteration methods, or split large media into domains for parallel processing.Source: §5 · p. 11

Appendix A. Proof of convergence

The proof establishes that the iteration operator has spectral radius at most one under the stated conditions. Convergence is guaranteed when absorption rules out the exceptional eigenvalue-one case.

Appendix A. Proof of convergence

  • The proof establishes ρ(M) ≤ 1, using the unitary operator and diagonal V properties to bound the inner product involving M.A sufficient convergence condition is ρ(M) < 1.Source: Appendix A · pp. 11–12
  • The bound follows by requiring Im∆(r) ≥ 0 and using ϵ ≥ |∆(r)|, which ensures the stated condition is fulfilled.Source: Appendix A · p. 12
  • An eigenvalue of 1 is possible only for an infinite non-absorbing medium, whose Helmholtz solution has infinite total energy and cannot be found by the method.Absorbing boundaries or any finite-size volume with non-zero absorption make the inequality strict and guarantee convergence.Source: Appendix A · p. 12

Appendix B. Boundary conditions

The implementation uses absorbing boundary layers to prevent waves from wrapping around the periodic boundaries associated with FFT convolution. The layer design balances residual reflection, absorption, and convergence speed.

Appendix B. Boundary conditions

  • FFT convolution gives periodic boundaries by default, so absorbing layers are added to prevent waves from wrapping around the simulation boundaries.The chosen layer is designed for zero reflectivity at normal incidence while bounding its scattering potential.Source: Appendix B · p. 13
  • Discretization leaves residual reflectivity because high spatial frequencies at the medium-to-absorber transition cannot be represented; smoothing the boundary reduces it.The idealized absorbing boundary has zero reflectivity.Source: Appendix B · p. 13
  • Higher polynomial order N makes the boundary smoother and lowers residual reflection, but takes longer to reach its maximum absorption coefficient.The profile uses the first N terms of the Taylor expansion of exp(αx), partially canceling exp(−αx) near the boundary.Source: Appendix B · p. 13
  • Increasing α strengthens absorption but increases scattering potential, raises ϵ, and slows convergence; the simulations used layers 25 wavelengths thick with N = 4 and maximum potential 0.2.These settings were reported to balance scattering potential, residual reflectivity, and residual transmission in high-accuracy simulations.Source: Appendix B · p. 13
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