Matrix-Free Parallel Scalable Multilevel Deflation Preconditioning for Heterogeneous Time-Harmonic Wave Problems
Jinqiang Chen, Vandana Dwarka, Cornelis Vuik
TL;DR
This work develops a matrix-free, parallel scalable multilevel deflation preconditioner (MADP) for heterogeneous time-harmonic Helmholtz problems, integrating Complex Shifted Laplacian preconditioning with higher-order deflation and Galerkin-based coarse discretization. By extending deflation to multiple levels and carefully tuning coarsest-level tolerances, complex shifts, and multigrid coupling, the authors achieve near wavenumber-independent convergence while maintaining strong and weak parallel scalability. The method significantly reduces memory usage through matrix-free implementations and demonstrates robust performance on large-scale model problems (including Marmousi), with detailed complexity and roofline analyses validating efficiency advantages over CSR-based approaches. The proposed MADP variants (notably MADP-v2/v3) offer practical configurations that balance convergence and computational cost, making the approach applicable to real-world large-scale wave propagation tasks in heterogeneous media.
Abstract
We present a matrix-free parallel scalable multilevel deflation preconditioned method for heterogeneous time-harmonic wave problems. Building on the higher-order deflation preconditioning proposed by Dwarka and Vuik (SIAM J. Sci. Comput. 42(2):A901-A928, 2020; J. Comput. Phys. 469:111327, 2022) for highly indefinite time-harmonic waves, we adapt these techniques for parallel implementation in the context of solving large-scale heterogeneous problems with minimal pollution error. Our proposed method integrates the Complex Shifted Laplacian preconditioner with deflation approaches. We employ higher-order deflation vectors and re-discretization schemes derived from the Galerkin coarsening approach for a matrix-free parallel implementation. We suggest a robust and efficient configuration of the matrix-free multilevel deflation method, which yields a close to wavenumber-independent convergence and good time efficiency. Numerical experiments demonstrate the effectiveness of our approach for increasingly complex model problems. The matrix-free implementation of the preconditioned Krylov subspace methods reduces memory consumption, and the parallel framework exhibits satisfactory parallel performance and weak parallel scalability. This work represents a significant step towards developing efficient, scalable, and parallel multilevel deflation preconditioning methods for large-scale real-world applications in wave propagation.