| Abstract: |
| In this talk, I will present 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.
|
|