Skip to main content
    • Aa
    • Aa

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

  • Feng Wei Yang (a1), Chandrasekhar Venkataraman (a2), Vanessa Styles (a1) and Anotida Madzvamuse (a1)

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.

Corresponding author
*Corresponding author. Email addresses: (F. W. Yang), (C. Venkataraman), (V. Styles), (A. Madzvamuse)
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[2] F. Haußer , S. Rasche and A. Voigt , The influence of electric fields on nanostructures-simulation and control, Mathematics and Computers in Simulation, 80(7), 14491457, 2010.

[4] C. Hogea , C. Davatzikos and G. Biros , An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects, Journal of mathematical biology, 56(6), 793825, 2008

[5] W. Croft , C.M. Elliott , G. Ladds , B. Stinner , C. Venkataraman and C. Weston , Parameter identification problems in the modelling of cell motility, Journal of Mathematical Biology, 71(2), 399436, 2015.

[6] K.N. Blazakis , A. Madzvamuse , C.C. Reyes-Aldasoro , V. Styles and C. Venkataraman , Whole cell tracking through the optimal control of geometric evolution laws, Journal of Computational Physics, 297, 495514, 2015.

[7] M. Vierling , 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), 137173, 2014.

[11] E. Emmerich , Stability and error of the variable two-step BDF for semilinear parabolic problems, Journal of Applied Mathematics and Computing, 19, 3355, 2005.

[12] P. Bollada , C. Goodyer , P. Jimack , A. Mullis and F. Yang , Thermalsolute phase field three dimensional simulation of binary alloy solidification, Journal of Computational Physics, 287, 130150, 2015.

[15] C.E. Goodyer , P.K. Jimack , A.M. Mullis , H.B. Dong and Y. Xie , On the Fully Implicit Solution of a Phase-Field Model for Binary Alloy Solidification in Three Dimensions, Advances in Applied Mathematics and Mechanics, 4, 665684, 2012.

[16] K. Deckelnick , G. Dziuk and C.M. Elliott , Computation of geometric partial differential equations and mean curvature flow, Acta Numerica, 139232, 2005.

[18] A. Brandt , Multi-Level Adaptive Solutions to Boundary-Value Problems, Mathematics of Computation, 31, 333390, 1977.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 46 *
Loading metrics...

Abstract views

Total abstract views: 157 *
Loading metrics...

* Views captured on Cambridge Core between 5th December 2016 - 28th July 2017. This data will be updated every 24 hours.