Krylov-Simplex method that minimizes the residual in $\ell_1$-norm or $\ell_\infty$-norm

TL;DR

This paper introduces two Krylov-Simplex iterative methods that efficiently minimize residuals in either the  or -norms by combining Krylov subspace techniques with simplex optimization, demonstrated through numerical experiments.

Contribution

The paper proposes novel Krylov-Simplex algorithms for residual minimization in  and -norms, integrating Krylov subspaces with simplex methods for improved residual control.

Findings

Effective residual minimization demonstrated in numerical experiments.

Methods efficiently solve small dense linear systems with rank-one updates.

Algorithms successfully minimize maximum and sum of residuals in test cases.

Abstract

The paper presents two variants of a Krylov-Simplex iterative method that combines Krylov and simplex iterations to minimize the residual $r = b-Ax$. The first method minimizes $\|r\|_\infty$, i.e. maximum of the absolute residuals. The second minimizes $\|r\|_1$, and finds the solution with the least absolute residuals. Both methods search for an optimal solution $x_k$ in a Krylov subspace which results in a small linear programming problem. A specialized simplex algorithm solves this projected problem and finds the optimal linear combination of Krylov basis vectors to approximate the solution. The resulting simplex algorithm requires the solution of a series of small dense linear systems that only differ by rank-one updates. The $QR$ factorization of these matrices is updated each iteration. We demonstrate the effectiveness of the methods with numerical experiments.