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

Part of: Basic linear algebra Algorithms - Computer Science Parabolic equations and systems

Published online by Cambridge University Press:  05 December 2016

Feng Wei Yang ,
Chandrasekhar Venkataraman ,
Vanessa Styles  and
Anotida Madzvamuse
Feng Wei Yang*
Affiliation:
Department of Mathematics, University of Sussex, UK
Chandrasekhar Venkataraman*
Affiliation:
School of Mathematics & Statistics, University of St Andrews, UK
Vanessa Styles*
Affiliation:
Department of Mathematics, University of Sussex, UK
Anotida Madzvamuse*
Affiliation:
Department of Mathematics, University of Sussex, UK
*
*Corresponding author. Email addresses:[email protected] (F. W. Yang), [email protected] (C. Venkataraman), [email protected] (V. Styles), [email protected] (A. Madzvamuse)
*Corresponding author. Email addresses:[email protected] (F. W. Yang), [email protected] (C. Venkataraman), [email protected] (V. Styles), [email protected] (A. Madzvamuse)
*Corresponding author. Email addresses:[email protected] (F. W. Yang), [email protected] (C. Venkataraman), [email protected] (V. Styles), [email protected] (A. Madzvamuse)
*Corresponding author. Email addresses:[email protected] (F. W. Yang), [email protected] (C. Venkataraman), [email protected] (V. Styles), [email protected] (A. Madzvamuse)
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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 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 problemis 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.

Keywords

Optimal controlgeometric evolution lawphase fieldmultigridparallelmesh adaptivitytwo-grid solution strategy

MSC classification

Secondary: 15A29: Inverse problems 35K58: Semilinear parabolic equations 68W10: Parallel algorithms
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 1 , January 2017 , pp. 65 - 92
Copyright
Copyright © Global-Science Press 2016 

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References

[1] Papadakis, N. and Mémin, E., Variational optimal control technique for the tracking of deformable objects, Computer Vision, IEEE 11th International Conference, 1-7, 2007.CrossRefGoogle Scholar
[2] Haußer, F., Rasche, S. and Voigt, A., The influence of electric fields on nanostructures-simulation and control, Mathematics and Computers in Simulation, 80(7), 1449–1457, 2010.CrossRefGoogle Scholar
[3] Haußer, F., Rasche, S. and Voigt, A., Control of nanostructures through electric fields and related free boundary problems, in: Constrained Optimization and Optimal Control for Partial Differential Equations, 561-572, 2012.CrossRefGoogle Scholar
[4] Hogea, C., Davatzikos, C. and Biros, G., An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects, Journal of mathematical biology, 56(6), 793–825, 2008 CrossRefGoogle ScholarPubMed
[5] Croft, W., Elliott, C.M., Ladds, G., Stinner, B., Venkataraman, C. and Weston, C., Parameter identification problems in the modelling of cell motility, Journal of Mathematical Biology, 71(2), 399–436, 2015.CrossRefGoogle ScholarPubMed
[6] Blazakis, K.N., Madzvamuse, A., Reyes-Aldasoro, C.C., Styles, V. and Venkataraman, C., Whole cell tracking through the optimal control of geometric evolution laws, Journal of Computational Physics, 297, 495–514, 2015.CrossRefGoogle Scholar
[7] Vierling, M., Parabolic optimal control problems on evolving surfaces subject to point-wise box constraints on the control–theory and numerical realization, Interfaces and Free Boundaries, 16(2), 137–173, 2014.CrossRefGoogle Scholar
[8] Tröltzsch, F., Optimal control of partial differential equations: theory, methods and applications, AMS Bookstore, 112, 2010.CrossRefGoogle Scholar
[9] Blowey, J. and Elliott, C., Curvature dependent phase boundary motion and parabolic double obstacle problems, In Degenerate Diffusions, 51, 52, 55, 19-60, 1993.CrossRefGoogle Scholar
[10] Hinze, M., Pinnau, R., Ulbrich, M. and Ulbrich, S., Optimization with PDE Constraints, Mathematical Modelling: Theory and Applications, 23, 2009.Google Scholar
[11] Emmerich, E., Stability and error of the variable two-step BDF for semilinear parabolic problems, Journal of Applied Mathematics and Computing, 19, 33–55, 2005.CrossRefGoogle Scholar
[12] Bollada, P., Goodyer, C., Jimack, P., Mullis, A. and Yang, F., Thermalsolute phase field three dimensional simulation of binary alloy solidification, Journal of Computational Physics, 287, 130–150, 2015.CrossRefGoogle Scholar
[13] Yang, F.W., Goodyer, C.E., Hubbard, M.E. and Jimack, P.K., Parallel implementation of an adaptive, multigrid solver for the implicit solution of nonlinear parabolic systems, with application to a multi-phase-field of tumour growth, Proceedings of the Fourth International Conference on Parallel, Distributed, Grid and Cloud Computing for Engineering, paper 39, editors: Ivanyi, P. and Topping, B.H.V., 2015.Google Scholar
[14] Yang, F.W., Goodyer, C.E., Hubbard, M.E. and Jimack, P.K., An Optimally Efficient Technique for the solution of systems of nonlinear parabolic partial differential equations, Advances in Engineering Software, doi:10.1016/j.advengsoft.2016.06.003, In Press, 2016.CrossRefGoogle Scholar
[15] Goodyer, C.E., Jimack, P.K., Mullis, A.M., Dong, H.B. and Xie, Y., On the Fully Implicit Solution of a Phase-Field Model for Binary Alloy Solidification in Three Dimensions, Advances in Applied Mathematics and Mechanics, 4, 665–684, 2012.CrossRefGoogle Scholar
[16] Deckelnick, K., Dziuk, G. and Elliott, C.M., Computation of geometric partial differential equations and mean curvature flow, Acta Numerica, 139–232, 2005.CrossRefGoogle Scholar
[17] Trottenberg, U., Oosterlee, C. and Schuller, A., Multigrid, Academic Press, 2001.Google Scholar
[18] Brandt, A., Multi-Level Adaptive Solutions to Boundary-Value Problems, Mathematics of Computation, 31, 333–390, 1977.CrossRefGoogle Scholar
[19] Briggs, W.L., Henson, V.E. and McCormick, S.F., A Multigrid Tutorial, Society for Industrial and Applied Mathematics, 2000.CrossRefGoogle Scholar