2022 journal article

Overlapping Domain Decomposition Preconditioner for Integral Equations

SIAM Journal on Scientific Computing, 44(6), A3617–A3644.

By: C. Chen* & G. Biros*

topics (OpenAlex): Electromagnetic Scattering and Analysis; Electromagnetic Simulation and Numerical Methods; Numerical methods in engineering
TL;DR: This work introduces a new preconditioner based on a novel overlapping domain decomposition that can be combined efficiently with existing fast direct solvers and applies the recursive skeletonization to subproblems associated with every subdomain. (via Semantic Scholar)
Source: ORCID
Added: November 1, 2023

The discretization of certain integral equations, e.g., the first-kind Fredholm equation of Laplace's equation, leads to symmetric positive-definite linear systems, where the coefficient matrix is dense and often ill-conditioned. We introduce a new preconditioner based on a novel overlapping domain decomposition that can be combined efficiently with fast direct solvers. Empirically, we observe that the condition number of the preconditioned system is $O(1)$, independent of the problem size. Our domain decomposition is designed so that we can construct approximate factorizations of the subproblems efficiently. In particular, we apply the recursive skeletonization algorithm to subproblems associated with every subdomain. We present numerical results on problem sizes up to $16\,384^2$ in two dimensions (2D) and $256^3$ in three dimensions (3D), which were solved in less than 16 hours and three hours, respectively, on an Intel Xeon Platinum 8280M.