Permuted preconditioning for extended saddle point problem arising from Neumann boundary control

TL;DR

This paper introduces a novel block preconditioner for extended saddle point problems from Neumann boundary control, improving computational efficiency through eigenvalue analysis and regularization techniques.

Contribution

A new block preconditioner for saddle point problems from Neumann boundary control, addressing singularity and enhancing solution efficiency.

Findings

Eigenvalue bounds for the preconditioned matrix

Numerical results demonstrate improved efficiency

Regularization handles singularity of the stiffness matrix

Abstract

In this paper, a new block preconditioner is proposed for the saddle point problem arising from the Neumann boundary control problem. In order to deal with the singularity of the stiffness matrix, the saddle point problem is first extended to a new one by a regularization of the pure Neumann problem. Then after row permutations of the extended saddle point problem, a new block triangular preconditioner is constructed based on an approximation of the Schur complement. We analyze the eigenvalue properties of the preconditioned matrix and provide eigenvalue bounds. Numerical results illustrate the efficiency of the proposed preconditioning method.