Krylov Iterative Methods for the Geometric Mean of Two Matrices Times a Vector

TL;DR

The paper tackles the efficient computation of the geometric mean $A\#B=A(A^{-1}B)^{1/2}$ applied to a vector for large, sparse SPD matrices. It develops and analyzes polynomial and rational Krylov-space methods, including generalized Lanczos, Arnoldi, and rational Arnoldi variants, to approximate $f(A)b$ with $f(z)=z^{-1/2}$ in the representation of the geometric mean. Key contributions include extended Krylov, generalized Leja points, and adaptive-pole rational Arnoldi strategies, all designed to exploit matrix sparsity and require only matrix-vector products and linear solves. The experiments show that rational Krylov approaches, particularly with Leja or adaptive poles, achieve fast convergence with modest subspace sizes and offer practical efficiency for domain decomposition and large-scale PDE-like problems.

Abstract

In this work, we are presenting an efficient way to compute the geometric mean of two positive definite matrices times a vector. For this purpose, we are inspecting the application of methods based on Krylov spaces to compute the square root of a matrix. These methods, using only matrix-vector products, are capable of producing a good approximation of the result with a small computational cost.

Figures (1)

• Figure 1: Convergence rate for the presented methods.