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High-Resolution Solvers for 3D Helmholtz Scattering Problems Using PFFT and Eigenvector-Based Preconditioning

Yury Gryazin, Ron Gonzales, Xiaoye Sherry Li

TL;DR

This work develops high resolution solvers for 3D Helmholtz scattering with absorbing boundaries by coupling fourth and sixth order compact discretizations to two boundary aware preconditioners, EigT and PFFT, for GMRES. A key contribution is embedding a low order absorbing boundary model into the preconditioner to ensure fast direct solves and improved convergence, addressing boundary condition mismatches that limit prior FFT-based approaches. The paper provides rigorous 1D model analysis and extensive 3D numerical experiments showing that PFFT delivers near optimal $O(N\log N)$ performance on FFT friendly grids while EigT is superior for smaller, non FFT optimized grids, and that both methods dramatically outperform traditional FFT-GMRES schemes on large-scale problems. The results demonstrate near order-of-magnitude speedups and robust convergence across high order schemes and nonconstant coefficients, enabling efficient high-resolution 3D scattering simulations relevant to electromagnetics and acoustics. Practical impact includes scalable, accurate solvers for large 3D Helmholtz problems on midsize to large grids with potential for hybrid parallel implementations.

Abstract

This paper presents an efficient Krylov subspace iterative solver for the three-dimensional (3D) Helmholtz equation with non-constant coefficients and absorbing boundary conditions, combining high-resolution compact schemes with low-order preconditioners. To mitigate numerical dispersion and reduce pollution error, we employ fourth- and sixth-order compact finite-difference schemes, thereby significantly softening the strict points-per-wavelength requirement. The resulting large, ill-conditioned linear systems are solved using a preconditioned GMRES method. The key innovation lies in the construction of the preconditioner: we introduce two highly efficient direct solvers - one based on a low-dimensional eigenvector transformation (EigT) and another on a partial Fast Fourier Transform (PFFT) algorithm - both derived from a lower-order approximation of the original problem that incorporates the absorbing boundary conditions. The motivation and efficacy of this lower-order preconditioning strategy for high-resolution schemes are analyzed through model problems, providing insight into the convergence rate. The theoretical analysis is validated by a comprehensive set of numerical experiments, demonstrating the method's performance for realistic problem sizes and parameters.

High-Resolution Solvers for 3D Helmholtz Scattering Problems Using PFFT and Eigenvector-Based Preconditioning

TL;DR

This work develops high resolution solvers for 3D Helmholtz scattering with absorbing boundaries by coupling fourth and sixth order compact discretizations to two boundary aware preconditioners, EigT and PFFT, for GMRES. A key contribution is embedding a low order absorbing boundary model into the preconditioner to ensure fast direct solves and improved convergence, addressing boundary condition mismatches that limit prior FFT-based approaches. The paper provides rigorous 1D model analysis and extensive 3D numerical experiments showing that PFFT delivers near optimal performance on FFT friendly grids while EigT is superior for smaller, non FFT optimized grids, and that both methods dramatically outperform traditional FFT-GMRES schemes on large-scale problems. The results demonstrate near order-of-magnitude speedups and robust convergence across high order schemes and nonconstant coefficients, enabling efficient high-resolution 3D scattering simulations relevant to electromagnetics and acoustics. Practical impact includes scalable, accurate solvers for large 3D Helmholtz problems on midsize to large grids with potential for hybrid parallel implementations.

Abstract

This paper presents an efficient Krylov subspace iterative solver for the three-dimensional (3D) Helmholtz equation with non-constant coefficients and absorbing boundary conditions, combining high-resolution compact schemes with low-order preconditioners. To mitigate numerical dispersion and reduce pollution error, we employ fourth- and sixth-order compact finite-difference schemes, thereby significantly softening the strict points-per-wavelength requirement. The resulting large, ill-conditioned linear systems are solved using a preconditioned GMRES method. The key innovation lies in the construction of the preconditioner: we introduce two highly efficient direct solvers - one based on a low-dimensional eigenvector transformation (EigT) and another on a partial Fast Fourier Transform (PFFT) algorithm - both derived from a lower-order approximation of the original problem that incorporates the absorbing boundary conditions. The motivation and efficacy of this lower-order preconditioning strategy for high-resolution schemes are analyzed through model problems, providing insight into the convergence rate. The theoretical analysis is validated by a comprehensive set of numerical experiments, demonstrating the method's performance for realistic problem sizes and parameters.
Paper Structure (17 sections, 2 theorems, 52 equations, 2 figures, 6 tables)

This paper contains 17 sections, 2 theorems, 52 equations, 2 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Discretization
  3. Statement of the Forward Problem
  4. A Fourth-Order Approximation Compact Scheme
  5. A Sixth-Order Approximation Compact Scheme
  6. Model Problems
  7. One-Dimensional Problem
  8. Efficient Preconditioners: EigT and PFFT Methods
  9. Low-Order Approximation Direct Eigenvectors (EigT) Preconditioner
  10. Partial FFT (PFFT) Preconditioning Approach
  11. Numerical Results
  12. 3D Test Problem
  13. Transformation Methods Comparison
  14. Direct Solution by the Low-Order Preconditioners
  15. Preconditioned GMRES Solution for Higher-Order Schemes
  16. ...and 2 more sections

Key Result

Theorem 3.2

Suppose that (Preconditioned_System) is an $m$-th order preconditioned system. Then the iterations $U^{(n)}$ of the GMRES method applied to this system satisfy the convergence estimate: where $\kappa_2(V) = \| V^{-1} \|_2 \| V \|_2$ is the condition number of matrix $V$.

Figures (2)

  • Figure 1: Convergence history of the preconditioned PFFT- and FFT-GMRES methods.
  • Figure 2: Color plot of the real part of the wavefield amplitude $U$.

Theorems & Definitions (4)

  • Definition 3.1
  • Theorem 3.2
  • Theorem 3.3
  • proof