An inexact semismooth Newton-Krylov method for semilinear elliptic optimal control problem

TL;DR

This paper introduces an inexact semismooth Newton-Krylov method that combines GMRES and nonmonotonic line search to efficiently solve semilinear elliptic optimal control problems with proven superlinear convergence.

Contribution

The paper develops a novel inexact semismooth Newton method incorporating Krylov subspace techniques and adaptive line search for improved efficiency and convergence in elliptic optimal control problems.

Findings

Method achieves superlinear convergence near solutions.

Numerical experiments confirm accuracy and efficiency.

Algorithm effectively reduces computational overhead.

Abstract

An inexact semismooth Newton method has been proposed for solving semi-linear elliptic optimal control problems in this paper. This method incorporates the generalized minimal residual (GMRES) method, a type of Krylov subspace method, to solve the Newton equations and utilizes nonmonotonic line search to adjust the iteration step size. The original problem is reformulated into a nonlinear equation through variational inequality principles and discretized using a second-order finite difference scheme. By leveraging slanting differentiability, the algorithm constructs semismooth Newton directions and employs GMRES method to inexactly solve the Newton equations, significantly reducing computational overhead. A dynamic nonmonotonic line search strategy is introduced to adjust stepsizes adaptively, ensuring global convergence while overcoming local stagnation. Theoretical analysis demonstrates that the algorithm achieves superlinear convergence near optimal solutions when the residual control parameter $\eta_k$ approaches to 0. Numerical experiments validate the method's accuracy and efficiency in solving semilinear elliptic optimal control problems, corroborating theoretical insights.