Double Saddle-Point Preconditioning for Krylov Methods in the Inexact Sequential Homotopy Method

TL;DR

This paper introduces a novel double saddle-point preconditioning technique for Krylov methods within the inexact sequential homotopy framework, enabling efficient, parallelized solutions of large-scale PDE-constrained optimization problems.

Contribution

It extends the sequential homotopy method to incorporate inexact solvers for saddle-point systems and proposes a new symmetric positive definite preconditioner based on double Schur complements.

Findings

Effective preconditioning for saddle-point systems in PDE-constrained optimization.

Parallelizable approach suitable for large 3D problems.

Numerical validation on a nonlinear elliptic PDE benchmark.

Abstract

We derive an extension of the sequential homotopy method that allows for the application of inexact solvers for the linear (double) saddle-point systems arising in the local semismooth Newton method for the homotopy subproblems. For the class of problems that exhibit (after suitable partitioning of the variables) a zero in the off-diagonal blocks of the Hessian of the Lagrangian, we propose and analyze an efficient, parallelizable, symmetric positive definite preconditioner based on a double Schur complement approach. For discretized optimal control problems with PDE constraints, this structure is often present with the canonical partitioning of the variables in states and controls. We conclude with numerical results for a badly conditioned and highly nonlinear benchmark optimization problem with elliptic partial differential equations and control bounds. The resulting method allows for the parallel solution of large 3D problems.