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

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 ∞, 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.