A recursive skeletonization factorization based on strong admissibility

TL;DR

The paper introduces a new recursive skeletonization factorization method, RS-S, that efficiently approximates matrices from elliptic PDE discretizations by focusing on far-field interactions, reducing rank growth and computational complexity.

Contribution

It presents the strong recursive skeletonization (RS-S) and hybrid RS-WS methods, which improve efficiency and reduce storage costs for hierarchical matrix factorizations.

Findings

RS-S achieves linear complexity under certain rank assumptions.

RS-WS reduces storage costs compared to RS-S.

Numerical examples demonstrate the effectiveness of the methods.

Abstract

We introduce the strong recursive skeletonization factorization (RS-S), a new approximate matrix factorization based on recursive skeletonization for solving discretizations of linear integral equations associated with elliptic partial differential equations in two and three dimensions (and other matrices with similar hierarchical rank structure). Unlike previous skeletonization-based factorizations, RS-S uses a simple modification of skeletonization, strong skeletonization, which compresses only far-field interactions. This leads to an approximate factorization in the form of a product of many block unit-triangular matrices that may be used as a preconditioner or moderate-accuracy direct solver, with dramatically reduced rank growth. We further combine the strong skeletonization procedure with alternating near-field compression to obtain the hybrid recursive skeletonization factorization (RS-WS), a modification of RS-S that exhibits reduced storage cost in many settings. Under suitable rank assumptions both RS-S and RS-WS exhibit linear computational complexity, which we demonstrate with a number of numerical examples.