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

TL;DR

The paper tackles saddle point systems from PDE-constrained optimization with Neumann boundary control, where the stiffness matrix is singular. It tackles this by regularizing the pure Neumann problem to form an extended, nonsingular system and then applying a row-permutation-based block triangular preconditioner built from a Schur-complement approximation. A spectral analysis shows eigenvalue 1 with high multiplicity and provides explicit bounds for the remaining eigenvalues that depend on the mesh size $h$ and regularization parameter $eta$, indicating favorable clustering. Numerical experiments on $oldsymbol{ ext{Ω}}=(0,1)^2$ confirm that the proposed preconditioner yields robust, efficient convergence for GMRES, outperforming competing strategies and showing practical viability for Neumann boundary control problems.

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.

Figures (2)

• Figure 1: Desired state of Example 1.
• Figure 2: Computed state (left) and control (right) of the GMRES($\widehat{\mathcal{P}}_{2}$) method for Example 1 with DoF=2306 and $\beta=10^{-6}$.

Theorems & Definitions (3)

• lemma thmcounterlemma
• proof
• proposition thmcounterproposition