Mixed precision sketching for least-squares problems and its application in GMRES-based iterative refinement

TL;DR

This paper analyzes a mixed precision algorithm for sketching-based preconditioners in least-squares problems, demonstrating how to set precisions to ensure effective preconditioning and convergence in GMRES-based iterative refinement.

Contribution

It provides a finite precision analysis of a mixed precision QR factorization algorithm and guides setting precisions for effective preconditioning in least-squares problems.

Findings

Mixed precision QR factorization can be effectively analyzed for stability.

Preconditioners can guarantee convergence of GMRES-based IR under certain conditions.

Numerical examples validate the theoretical analysis.

Abstract

Sketching-based preconditioners have been shown to accelerate the solution of dense least-squares problems with coefficient matrices having substantially more rows than columns. The cost of generating these preconditioners can be reduced by employing low precision floating-point formats for all or part of the computations. We perform finite precision analysis of a mixed precision algorithm that computes the $R$-factor of a QR factorization of the sketched coefficient matrix. Two precisions can be chosen and the analysis allows understanding how to set these precisions to exploit the potential benefits of low precision formats and still guarantee an effective preconditioner. If the nature of the least-squares problem requires a solution with a small forward error, then mixed precision iterative refinement (IR) may be needed. For ill-conditioned problems the GMRES-based IR approach can be used, but good preconditioner is crucial to ensure convergence. We theoretically show when the sketching-based preconditioner can guarantee that the GMRES-based IR reduces the relative forward error of the least-squares solution and the residual to the level of the working precision unit roundoff. Small numerical examples illustrate the analysis.