A matrix-free parallel two-level deflation preconditioner for the two-dimensional Helmholtz problems

TL;DR

This work develops a matrix-free, parallel two-level deflation preconditioner coupled with the Complex Shifted Laplacian Preconditioner (CSLP) to solve 2D Helmholtz systems efficiently. By leveraging deflation vectors derived from multigrid prolongation and high-order extensions within a matrix-free framework, the method achieves improved convergence that is largely independent of the wavenumber for many test problems, while maintaining memory efficiency. The study compares multiple coarse-grid realizations—straight Galerkin, stencil-based Galerkin, and several re-discretization schemes (including high-order and compact forms)—and analyzes tolerance settings for the coarse-grid solver, showing that ReD-based approaches often offer favorable trade-offs between cost and convergence. Extensive parallel experiments demonstrate scalable weak and strong performance on large heterogeneous models (Wedge, Marmousi, Marmousi-like) in 2D, positioning the proposed method as a robust benchmark for matrix-free multilevel deflation preconditioners in Helmholtz applications. The results have practical implications for seismic, acoustic, and electromagnetic simulations requiring high-frequency Helmholtz solves in parallel environments.

Abstract

We propose a matrix-free parallel two-level-deflation preconditioner combined with the Complex Shifted Laplacian preconditioner(CSLP) for the two-dimensional Helmholtz problems. The Helmholtz equation is widely studied in seismic exploration, antennas, and medical imaging. It is one of the hardest problems to solve both in terms of accuracy and convergence, due to scalability issues of the numerical solvers. Motivated by the observation that for large wavenumbers, the eigenvalues of the CSLP-preconditioned system shift towards zero, deflation with multigrid vectors, and further high-order vectors were incorporated to obtain wave-number-independent convergence. For large-scale applications, high-performance parallel scalable methods are also indispensable. In our method, we consider the preconditioned Krylov subspace methods for solving the linear system obtained from finite-difference discretization. The CSLP preconditioner is approximated by one parallel geometric multigrid V-cycle. For the two-level deflation, the matrix-free Galerkin coarsening as well as high-order re-discretization approaches on the coarse grid are studied. The results of matrix-vector multiplications in Krylov subspace methods and the interpolation/restriction operators are implemented based on the finite-difference grids without constructing any coefficient matrix. These adjustments lead to direct improvements in terms of memory consumption. Numerical experiments of model problems show that wavenumber independence has been obtained for medium wavenumbers. The matrix-free parallel framework shows satisfactory weak and strong parallel scalability.

Figures (12)

• Figure 1: The velocity distribution of the wedge problem
• Figure 1: The allocation map of interpolation operator
• Figure 1: Tolerance of the coarse-grid solver for MP-2b with wavenumber $k=320$ and grid size $513 \times 513$. The present APD-preconditioned GMRES is employed for outer iteration.
• Figure 2: The velocity distribution of the marmousi problem
• Figure 2: Tolerance of the coarse-grid solver for non-constant wavenumber problems. The present APD-preconditioned GMRES is employed for outer iteration.

Theorems & Definitions (2)

• Remark 5.1: Stencil Operations
• Remark 6.1: FLOPs of sparse matrix-vector multiplication