[Paper Review] An Overlapping Domain Decomposition Preconditioner for the Helmholtz equation

Wei Leng, Lili Ju|arXiv (Cornell University)|Aug 12, 2015

Electromagnetic Scattering and Analysis8 references3 citations

TL;DR

This paper proposes a one-step overlapping domain decomposition preconditioner for the 2D Helmholtz equation that enables fully parallel solution of high-frequency wave problems. By decomposing the domain in both x and y directions and solving local subproblems simultaneously with PML layers, the method achieves scalable performance with iteration counts proportional to the number of subdomains, solving problems with nearly a billion unknowns efficiently on massively parallel machines.

ABSTRACT

In this paper, based on the overlapping domain decomposition method (DDM) proposed in \cite{Leng2015}, an one step preconditioner is proposed to solve 2D high frequency Helmholtz equation. The computation domain is decomposed in both $x$ and $y$ directions, and the local solution on each subdomain is updated simultaneously in one iteration, thus there is no sweeping along certain directions. In these ways, the overlapping DDM is similar to the popular DDM for Poisson problem. The one step preconditioner simply take the restricted source on each subdomain, solve the local problems and summarize the local solutions on all subdomains including their PML area. The complexity of solving the problem with the preconditioner is $O(N n_{ ext{iter}})$, where $n_{ ext{iter}}$ is the number of iteration, and it is shown numerically that $n_{ ext{iter}}$ is proportional to the number of subdomains in one direction. 2D Helmholtz problem with nearly a billion unknowns are solved efficiently with the preconditioner on massively parallel machines.

Motivation & Objective

• To develop a scalable, parallel preconditioner for high-frequency 2D Helmholtz problems that avoids sequential sweeping.
• To enable efficient solution of large-scale Helmholtz problems with up to nearly a billion unknowns using overlapping domain decomposition.
• To maintain robust convergence independent of frequency and problem size through a one-step, non-sweeping iterative method.
• To demonstrate the effectiveness of the preconditioner on complex media, including layered and Marmousi models, using massive parallelism.

Proposed method

• The computational domain is decomposed into overlapping subdomains in both x and y directions, enabling fully parallel updates without directional sweeping.
• A one-step preconditioner is constructed by solving local Helmholtz problems on each subdomain with restricted source terms and including PML regions in the solution.
• The method uses uniaxial PML for absorbing boundary conditions, with complex coordinate stretching applied in both x and y directions.
• The global solution is assembled as the sum of local solutions, including contributions from overlapping PML layers, ensuring wave continuity and absorption.
• The iterative process updates all subdomains simultaneously in one step, avoiding the sequential sweep of traditional DDMs.
• The method is implemented on massively parallel machines, with local problems solved independently and asynchronously.

Experimental results

Research questions

• RQ1 Can a non-sweeping, overlapping domain decomposition method achieve robust convergence for high-frequency Helmholtz problems?
• RQ2 How does the number of iterations scale with problem size and number of subdomains in a parallel setting?
• RQ3 Does the preconditioner maintain efficiency and scalability on complex media such as the Marmousi model?
• RQ4 What is the impact of overlapping region size on convergence and computational cost?
• RQ5 Can the method efficiently solve Helmholtz problems with nearly a billion unknowns using parallel architectures?

Key findings

• The number of GMRES iterations scales linearly with the number of subdomains in one direction, indicating good scalability.
• For the constant medium, the number of equivalent sweeps remained around 7–10 across different problem sizes, showing frequency robustness.
• In the layered medium test, the number of equivalent sweeps was consistently around 9–11, even as the problem size increased to 19,200² unknowns.
• For the Marmousi model with 977 million unknowns, the method achieved convergence in 1,123 GMRES iterations using 54,150 × 18,050 grid points and 4,332 processors.
• The time cost per iteration was significantly reduced without extra overlapping region, though larger overlap increased local solve time and total cost.
• The method demonstrated weak scalability, maintaining nearly constant equivalent sweeping counts as problem size increased, confirming robustness for large-scale problems.