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

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 twogrid 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.

Parameter identification through mode isolation for reaction-diffusion systems on arbitrary geometries

We present a computational framework for isolating spatial patterns arising in the steady states of reaction-diffusion systems. Such systems have been used to model many nat- ural phenomena in areas such as developmental and cancer biology, cell motility and material science. In many of these applications, often one is interested in identifying parameters which will lead to a particular pattern for a given reaction-diffusion model. To attempt to answer this, we compute eigenpairs of the Laplacian on a variety of domains and use linear stability analysis to determine parameter values for the system that will lead to spatially inhomogeneous steady states whose patterns correspond to particular eigenfunctions. This method has previously been used on domains and surfaces where the eigenvalues and eigenfunctions are found analytically in closed form. Our contribution to this methodology is that we numerically compute eigenpairs on arbitrary domains and surfaces. Here we present examples and demonstrate that mode isolation is straightforward especially for low eigenvalues. Additionally, we show that in some cases the inhomogeneous steady state can be a linear combination of eigenfunctions. Finally,we show an example suggesting that pattern formation is robust on similar surfaces in cases that the surface either has or does not have a boundary.