A robust and efficient adaptive multigrid solver for the optimal control of phase field formulations of geometric evolution laws
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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 and cell motility. Despite this, many open problems remain in the analysis and approximation of such problems. 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 adaptive-step 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 of phase field formulations of geometric evolution laws in 3-D which would be computationally infeasible without the solution strategies proposed in the present work.
Yang , F W , Venkataraman , C , Styles , V & Madzvamuse , A 2017 , ' A robust and efficient adaptive multigrid solver for the optimal control of phase field formulations of geometric evolution laws ' Communications in Computational Physics , vol 21 , no. 1 , pp. 65-92 . DOI: 10.4208/cicp.240715.080716a
Communications in Computational Physics
© 2016, Global Science Press. This work has been made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at www.cambridge.org / https://doi.org/10.4208/cicp.240715.080716a
All authors acknowledge support from the Leverhulme Trust Research Project Grant (RPG-2014-149). The work of CV, VS and AMwas partially supported by the Engineering and Physical Sciences Research Council, UK grant (EP/J016780/1). This work (AM) has also received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 642866. The work of CV is partially supported by an EPSRC Impact Accelerator Account award. The authors (FWY, CV, VS, AM) thank the Isaac Newton Institute for Mathematical Sciences for its hospitality during the programme (Coupling Geometric PDEs with Physics for Cell Morphology, Motility and Pattern Formation; EPSRC EP/K032208/1). AM was partially supported by Fellowships from the Simons Foundation.
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