A Robust and Efficient Adaptive Multigrid Solver for the Optimal Control of Phase Field Formulations of Geometric Evolution Laws (Communications in Computational Physics, 2018)
We propose and investigate a novel solution strategy to efficiently and accuratelycompute approximate solutions to semilinear optimal control problems, focusingon the optimal control of phase field formulations of geometric evolution laws.The optimal control of geometric evolution laws arises in a number of applicationsin fields including material science, image processing, tumour growth and cell motility.Despite this, many open problems remain in the analysis and approximation ofsuch problems. In the current work we focus on a phase field formulation of the optimalcontrol problem, hence exploiting the well developed mathematical theory forthe optimal control of semilinear parabolic partial differential equations. Approximationof the resulting optimal control problem is computationally challenging, requiringmassive amounts of computational time and memory storage. The main focus of thiswork is to propose, derive, implement and test an efficient solution method for suchproblems. The solver for the discretised partial differential equations is based upon ageometric multigrid method incorporating advanced techniques to deal with the nonlinearitiesin the problem and utilising adaptive mesh refinement. An in-house two-gridsolution strategy for the forward and adjoint problems, that significantly reducesmemory requirements and CPU time, is proposed and investigated computationally.Furthermore, parallelisation as well as an adaptive-step gradient update for the controlare employed to further improve efficiency. Along with a detailed description ofour proposed solution method together with its implementation, we present a numberof computational results that demonstrate and evaluate our algorithms with respect toaccuracy and efficiency. A highlight of the present work is simulation results on theoptimal control of phase field formulations of geometric evolution laws in 3-D whichwould be computationally infeasible without the solution strategies proposed in thepresent work.