Overlapping Domain Decomposition Preconditioner for Integral Equations

TL;DR

This paper introduces a new overlapping domain decomposition preconditioner for integral equations, achieving near-constant condition numbers and efficient solutions for large dense systems in 2D and 3D.

Contribution

A novel overlapping domain decomposition preconditioner that combines with fast direct solvers and maintains an $O(1)$ condition number regardless of problem size.

Findings

Condition number of preconditioned system is $O(1)$, independent of size.

Numerical solutions for large problems up to $16,384^2$ in 2D and $256^3$ in 3D.

Problems solved efficiently within hours on high-performance hardware.

Abstract

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 2D and $256^3$ in 3D, which were solved in less than 16 hours and three hours, respectively, on an Intel Xeon Platinum 8280M.