Andreas A. Bock | Searcharxiv

arXiv2025

Connecting Kaporin's condition number and the Bregman log determinant divergence

This paper presents some theoretical results relating the Bregman log determinant matrix divergence to Kaporin's condition number. These can be viewed as nearness measures between a preconditioner and a given matrix, and we show under which conditions these two functions coincide. We also give examples of constraint sets over which it is equivalent to minimise these two objectives. We focus on preconditioners that are the sum of a positive definite and low-rank matrix, which were developed in a previous work. These were constructed as minimisers of the aforementioned divergence, and we show that they are only a constant scaling from also minimising Kaporin's condition number. We highlight connections to information geometry and comment on future directions.

math.NA↗

arXiv2024

A New Matrix Truncation Method for Improving Approximate Factorisation Preconditioners

In this experimental work, we present a general framework based on the Bregman log determinant divergence for preconditioning Hermitian positive definite linear systems. We explore this divergence as a measure of discrepancy between a preconditioner and a matrix. Given an approximate factorisation of a given matrix, the proposed framework informs the construction of a low-rank approximation of the typically indefinite factorisation error. The resulting preconditioner is therefore a sum of a Hermitian positive definite matrix given by an approximate factorisation plus a low-rank matrix. Notably, the low-rank term is not generally obtained as a truncated singular value decomposition (TSVD). This framework leads to a new truncation where principal directions are not based on the magnitude of the singular values, and we prove that such truncations are minimisers of the aforementioned divergence. We present several numerical examples showing that the proposed preconditioner can reduce the number of PCG iterations compared to a preconditioner constructed using a TSVD for the same rank. We also propose a heuristic to approximate the proposed preconditioner in the case where exact truncations cannot be computed explicitly (e.g. in a large-scale setting) and demonstrate its effectiveness over TSVD-based approaches.

math.NA↗