Optimal control of anisotropic Allen-Cahn equations

TL;DR

This paper develops a numerical approach for optimal control problems governed by anisotropic Allen-Cahn equations, including theoretical analysis and a trust region Newton solver, demonstrating convergence and mesh independence.

Contribution

It introduces a framework for analyzing and solving anisotropic Allen-Cahn optimal control problems with regularization and convergence analysis, including a numerical solver.

Findings

Convergence of optimal controls for smooth approximations.

Mesh independent behavior demonstrated numerically.

Effective regularization for a wide class of anisotropies.

Abstract

This paper aims at solving an optimal control problem governed by an anisotropic Allen-Cahn equation numerically. Therefore we first prove the Fr\'echet differentiability of an in time discretized parabolic control problem under certain assumptions on the involved quasilinearity and formulate the first order necessary conditions. As a next step, since the anisotropies are in general not smooth enough, the convergence behavior of the optimal controls are studied for a sequence of (smooth) approximations of the former quasilinear term. In addition the simultaneous limit in the approximation and the time step size is considered. For a class covering a large variety of anisotropies we introduce a certain regularization and show the previously formulated requirements. Finally, a trust region Newton solver is applied to various anisotropies and configurations, and numerical evidence for mesh independent behavior and convergence with respect to regularization is presented.