A technique for updating hierarchical skeletonization-based factorizations of integral operators

TL;DR

This paper introduces an efficient method to update hierarchical factorizations of integral operators after local geometric or coefficient perturbations, significantly reducing computational costs for iterative problems.

Contribution

It develops a novel approach to locally update existing factorizations with polylogarithmic complexity, enhancing efficiency in solving perturbed integral equations.

Findings

Achieves polylogarithmic complexity in updating factorizations

Demonstrates scalability on various 2D problems

Applicable to recursive skeletonization and hierarchical interpolative factorizations

Abstract

We present a method for updating certain hierarchical factorizations for solving linear integral equations with elliptic kernels. In particular, given a factorization corresponding to some initial geometry or material parameters, we can locally perturb the geometry or coefficients and update the initial factorization to reflect this change with asymptotic complexity that is polylogarithmic in the total number of unknowns and linear in the number of perturbed unknowns. We apply our method to the recursive skeletonization factorization and hierarchical interpolative factorization and demonstrate scaling results for a number of different 2D problem setups.