A Robust and Efficient Adaptive Multigrid Solver for the Optimal Control of Phase Field Formulations of Geometric Evolution Laws

DOI10.4208/CICP.240715.080716AzbMATH Open1373.49030arXiv1603.08572OpenAlexW3104765509WikidataQ58871734 ScholiaQ58871734MaRDI QIDQ5372590

Anotida Madzvamuse, Vanessa Styles, Feng-Wei Yang, Chandrasekhar Venkataraman

Publication date: 27 October 2017

Published in: Communications in Computational Physics (Search for Journal in Brave)

Abstract: We propose and investigate a novel solution strategy to efficiently and accurately compute approximate solutions to semilinear optimal control problems, focusing on the optimal control of phase field formulations of geometric evolution laws. The optimal control of geometric evolution laws arises in a number of applications in fields including material science, image processing, tumour growth anda cell motility. In the current work we focus on a phase field formulation of the optimal control problem, hence exploiting the well developed mathematical theory for the optimal control of semilinear parabolic partial differential equations. Approximation of the resulting optimal control problem is computationally challenging, requiring massive amounts of computational time and memory storage. The main focus of this work is to propose, derive, implement and test an efficient solution method for such problems. The solver for the discretised partial differential equations is based upon a geometric multigrid method incorporating advanced techniques to deal with the nonlinearities in the problem and utilising adaptive mesh refinement. An in-house two-grid solution strategy for the forward and adjoint problems, that significantly reduces memory requirements and CPU time, is proposed and investigated computationally. Furthermore, parallelisation as well as an adaptivestep gradient update for the control are employed to further improve efficiency. Along with a detailed description of our proposed solution method together with its implementation we present a number of computational results that demonstrate and evaluate our algorithms with respect to accuracy and efficiency. A highlight of the present work is simulation results on the optimal control in 3-D which would be computationally infeasible without the solution strategies proposed in the present work.

Full work available at URL: https://arxiv.org/abs/1603.08572

Mathematics Subject Classification ID

Inverse problems in linear algebra (15A29) Finite difference methods for initial value and initial-boundary value problems involving PDEs (65M06) Parallel numerical computation (65Y05) Discrete approximations in optimal control (49M25) Semilinear parabolic equations (35K58)

This page was built for publication: A Robust and Efficient Adaptive Multigrid Solver for the Optimal Control of Phase Field Formulations of Geometric Evolution Laws

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q5372590)