Inexact inner-outer Golub-Kahan bidiagonalization method: A relaxation strategy

TL;DR

This paper investigates an inexact inner-outer Golub-Kahan method for saddle-point problems, focusing on how inner solver accuracy affects overall solution quality and proposing relaxation strategies to reduce computational cost.

Contribution

It introduces relaxation techniques to dynamically adjust inner solver tolerances, improving efficiency in solving large saddle-point systems.

Findings

Inner iteration accuracy significantly impacts overall solution precision.

Relaxation strategies effectively reduce total inner iterations.

Validated methods on Stokes and Poisson problems.

Abstract

We study an inexact inner-outer generalized Golub-Kahan algorithm for the solution of saddle-point problems with a two-times-two block structure. In each outer iteration, an inner system has to be solved which in theory has to be done exactly. Whenever the system is getting large, an inner exact solver is, however, no longer efficient or even feasible and iterative methods must be used. We focus this article on a numerical study showing the influence of the accuracy of an inner iterative solution on the accuracy of the solution of the block system. Emphasis is further given on reducing the computational cost, which is defined as the total number of inner iterations. We develop relaxation techniques intended to dynamically change the inner tolerance for each outer iteration to further minimize the total number of inner iterations. We illustrate our findings on a Stokes problem and validate them on a mixed formulation of the Poisson problem.