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We propose and investigate a novel solution strategy to efficiently andaccurately compute approximate solutions to semilinear optimal controlproblems, focusing on the optimal control of phase field formulations ofgeometric evolution laws. The optimal control of geometric evolution lawsarises in a number of applications in fields including material science, imageprocessing, tumour growth anda cell motility. In the current work we focus on aphase field formulation of the optimal control problem, hence exploiting thewell developed mathematical theory for the optimal control of semilinearparabolic partial differential equations. Approximation of the resultingoptimal control problem is computationally challenging, requiring massiveamounts of computational time and memory storage. The main focus of this workis to propose, derive, implement and test an efficient solution method for suchproblems. The solver for the discretised partial differential equations isbased upon a geometric multigrid method incorporating advanced techniques todeal with the nonlinearities in the problem and utilising adaptive meshrefinement. An in-house two-grid solution strategy for the forward and adjointproblems, that significantly reduces memory requirements and CPU time, isproposed and investigated computationally. Furthermore, parallelisation as wellas an adaptivestep gradient update for the control are employed to furtherimprove efficiency. Along with a detailed description of our proposed solutionmethod together with its implementation we present a number of computationalresults that demonstrate and evaluate our algorithms with respect to accuracyand efficiency. A highlight of the present work is simulation results on theoptimal control in 3-D which would be computationally infeasible without thesolution strategies proposed in the present work.

A robust and efficient adaptive multigrid solver for the optimal control of phase field formulations of geometric evolution laws