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 $\Vert r{\Vert }_{\infty }$, i.e. maximum of the absolute residuals. The second minimizes $\Vert r{\Vert }_1$, 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.

22 pages