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A new interpolated pseudodifferential preconditioner for the Helmholtz equation in heterogeneous media

Sebastian Acosta, Tahsin Khajah, Benjamin Palacios

TL;DR

The paper addresses the ill-conditioning of the Helmholtz equation in heterogeneous media at intermediate to high frequencies by introducing a pseudodifferential preconditioner whose symbol is interpolated with respect to the wave speed $c(x)$. This univariate interpolation enables FFT-based, matrix-free evaluation with near-linear complexity in the number of degrees of freedom, and the leading symbol $q_{-2}$ provides a close approximation to the inverse of the Helmholtz operator up to a $\mathcal{S}^{-1}$-order error. Theoretical results show spectral clustering of the preconditioned operator near 1, ensuring rapid GMRES convergence, while numerical experiments across circular inclusions and head phantoms demonstrate large reductions in condition numbers (often by 10+ orders of magnitude) and substantial improvements in iteration counts. The work further extends to absorbing layers via complexified wave speeds and discusses limitations and potential extensions to non-Cartesian grids and higher-order symbol terms, highlighting practical impact for ultrasound and related wave-propagation simulations.

Abstract

This paper introduces a new pseudodifferential preconditioner for the Helmholtz equation in variable media with absorption. The pseudodifferential operator is associated with the multiplicative inverse to the symbol of the Helmholtz operator. This approach is well-suited for the intermediate and high-frequency regimes. The main novel idea for the fast evaluation of the preconditioner is to interpolate its symbol, not as a function of the (high-dimensional) phase-space variables, but as a function of the wave speed itself. Since the wave speed is a real-valued function, this approach allows us to interpolate in a univariate setting even when the original problem is posed in a multidimensional physical space. As a result, the needed number of interpolation points is small, and the interpolation coefficients can be computed using the fast Fourier transform. The overall computational complexity is log-linear with respect to the degrees of freedom as inherited from the fast Fourier transform. We present some numerical experiments to illustrate the effectiveness of the preconditioner to solve the discrete Helmholtz equation using the GMRES iterative method. The implementation of an absorbing layer for scattering problems using a complex-valued wave speed is also developed. Limitations and possible extensions are also discussed.

A new interpolated pseudodifferential preconditioner for the Helmholtz equation in heterogeneous media

TL;DR

The paper addresses the ill-conditioning of the Helmholtz equation in heterogeneous media at intermediate to high frequencies by introducing a pseudodifferential preconditioner whose symbol is interpolated with respect to the wave speed . This univariate interpolation enables FFT-based, matrix-free evaluation with near-linear complexity in the number of degrees of freedom, and the leading symbol provides a close approximation to the inverse of the Helmholtz operator up to a -order error. Theoretical results show spectral clustering of the preconditioned operator near 1, ensuring rapid GMRES convergence, while numerical experiments across circular inclusions and head phantoms demonstrate large reductions in condition numbers (often by 10+ orders of magnitude) and substantial improvements in iteration counts. The work further extends to absorbing layers via complexified wave speeds and discusses limitations and potential extensions to non-Cartesian grids and higher-order symbol terms, highlighting practical impact for ultrasound and related wave-propagation simulations.

Abstract

This paper introduces a new pseudodifferential preconditioner for the Helmholtz equation in variable media with absorption. The pseudodifferential operator is associated with the multiplicative inverse to the symbol of the Helmholtz operator. This approach is well-suited for the intermediate and high-frequency regimes. The main novel idea for the fast evaluation of the preconditioner is to interpolate its symbol, not as a function of the (high-dimensional) phase-space variables, but as a function of the wave speed itself. Since the wave speed is a real-valued function, this approach allows us to interpolate in a univariate setting even when the original problem is posed in a multidimensional physical space. As a result, the needed number of interpolation points is small, and the interpolation coefficients can be computed using the fast Fourier transform. The overall computational complexity is log-linear with respect to the degrees of freedom as inherited from the fast Fourier transform. We present some numerical experiments to illustrate the effectiveness of the preconditioner to solve the discrete Helmholtz equation using the GMRES iterative method. The implementation of an absorbing layer for scattering problems using a complex-valued wave speed is also developed. Limitations and possible extensions are also discussed.
Paper Structure (19 sections, 5 theorems, 48 equations, 3 figures, 6 tables)

This paper contains 19 sections, 5 theorems, 48 equations, 3 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Pseudodifferential inversion of the Helmholtz equation
  4. Evaluation algorithm
  5. Algorithm
  6. Complexity estimates
  7. Accuracy and conditioning estimates
  8. Absorbing layer
  9. Numerical experiments
  10. Example 1: Circular inclusion
  11. Complexity:
  12. Frequency:
  13. Points per wavelength:
  14. Wave speed contrast:
  15. Wave speed smoothness:
  16. ...and 4 more sections

Key Result

Theorem 1

The preconditioned operator $\mathcal{E}_{\Omega} Q P : H^{s}(\Omega) \to H^{s}(\Omega)$ is Fredholm (being a compact perturbation of the identity), and its spectrum consists of $z=1$ and at most a countable set of eigenvalues with no point of accumulation except, possibly, $z = 1$.

Figures (3)

  • Figure 1: Amplitude of the solution $u$ (normalized in the $L^{\infty}$-norm) and relative residual for the first 100 iterations of the GMRES method applied to \ref{['Eqn.MainDiscrete']} and to \ref{['Eqn.MainDiscretePrecond']} for frequencies $\omega=400 \pi$, $\omega = 600 \pi$ and $\omega=800 \pi$, respectively. In all cases, the GMRES+Predonditioner method reaches residuals more than $7$ orders of magnitude smaller than the GMRES method at the $100$th iteration. The wave speed profile $c$ and source $f$ are defined by \ref{['Eqn.WS']} (using $\delta = 1$ and $\eta = 1/800$) and \ref{['Eqn.Source']}, respectively.
  • Figure 2: (a) Wave speed profile $c$ from the supplemental material of Aubry2022. (b) Function $\zeta$ for the absorbing layer given by \ref{['Eqn.Zeta']}. The skull contour is shown for reference. The complex-valued wave speed is defined by \ref{['Eqn.Complex_wave_speed']}.
  • Figure 3: Amplitude of the solution $u$ (log-scale) and relative residual for the first 100 iterations of the GMRES method applied to \ref{['Eqn.MainDiscrete']} and to \ref{['Eqn.MainDiscretePrecond']} for frequencies $\omega=400 \pi$, $\omega = 600 \pi$ and $\omega=800 \pi$, respectively.

Theorems & Definitions (7)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Theorem 2
  • proof
  • Theorem 3
  • proof