TY - JOUR
T1 - A Robust and Efficient Adaptive Multigrid Solver for the Optimal Control of Phase Field Formulations of Geometric Evolution Laws
AU - Feng Wei Yang, Chandrasekhar Venkataraman, Vanessa Styles & Anotida Madzvamuse
JO - Communications in Computational Physics
VL - 1
SP - 65
EP - 92
PY - 2018
DA - 2018/04
SN - 21
DO - http://doi.org/10.4208/cicp.240715.080716a
UR - https://global-sci.org/intro/article_detail/cicp/11232.html
KW - 
AB - <p style="text-align: justify;">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.</p>