Randomized sketching for Krylov approximations of large-scale matrix functions

TL;DR

This paper introduces randomized sketching techniques combined with integral representations to efficiently approximate matrix functions times vectors, enabling computations on large sparse matrices with limited storage.

Contribution

It proposes new sketched Krylov methods for matrix function approximation, analyzing convergence and providing bounds, which are novel in combining randomized sketching with Krylov subspace methods.

Findings

Sketched FOM approximant has a closed-form expression.

Convergence analyzed for Stieltjes functions of positive matrices.

Numerical experiments show the effectiveness of the methods.

Abstract

The computation of f(A)b, the action of a matrix function on a vector, is a task arising in many areas of scientific computing. In many applications, the matrix A is sparse but so large that only a rather small number of Krylov basis vectors can be stored. Here we discuss a new approach to overcome these limitations by randomized sketching combined with an integral representation of f(A)b. Two different approximations are introduced, one based on sketched FOM and another based on sketched GMRES approximation. The convergence of the latter method is analyzed for Stieltjes functions of positive real matrices. We also derive a closed form expression for the sketched FOM approximant and bound its distance to the full FOM approximant. Numerical experiments demonstrate the potential of the presented sketching approaches.