Efficient Solution of Advection Dominated PDE Constrained Optimization Problems

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The aim of this proposal is to develop, analyze and implement optimizationalgorithms that integrate multilevel iterative solvers, adaptive mesh refinement methods, and so-called `all-at-once' methods for the solution of optimization problems governed by advection dominated partial differential equations (PDEs). The presence of strong advection in the governing PDE creates many challenges forthe numerical solution of these optimization problems, beyond the challenges alreadyencountered in the numerical simulation of single advection dominated PDEs and beyondthe many difficulties in solving PDE constrained optimization problems with weak or no advection. One reason for the additional challenges arising in the optimization context is the presence of the so-called adjoint equation, which is also an advection dominated PDE, but with advection given by the negative of the advection in the governing equation. This can cause significant and perhaps unexpected differences in the behavior of discretization schemes and iterative solvers when they are extended from the application to single advection dominated PDEs to the solution of PDE constrained optimization problems. This research will analyze the sources of these differences, their impact on the quality of computed solutionand on the efficiency of solution algorithms. Another goal is to devise modifications of multilevel iterative solvers, adaptive mesh refinement methods, and so-called KKT solvers tomake them robust against the presence of advection.Many important real-life applications such as the shape optimization of technological devices, the optimal control of systems, and the identification of parameters in environmentalprocesses lead to optimization problems governed by systems of advection dominatedpartial differential equations. This research addresses mathematical and algorithmicissues that are crucial for the reliable and efficient solution of these problems. It will leadto a better theoretical understanding of the solution properties of these optimization problems as well as to new computational tools for their reliable and efficient solution. Furthermore, it will provide training opportunities for students in an important area of computational science.