The research objective of this project is to develop a strong theoretical foundation on local properties of finite element solutions for advection- dominated optimal control problems. The advection-diffusion partial differential equations (PDEs) are known to be difficult to approximate because their solutions often exhibit discontinuities, layers, and shocks. The structure of the solutions to optimal control problems is even more complicated because of the coupling in the optimality system of the governing advection-dominated PDE with an advection-dominated adjoint PDE. It is known that convergence behavior of finite element methods applied to single advection-dominated PDEs can be very different from the convergence behavior of finite element methods applied to advection-dominated optimal control problems. Understanding the global and local convergence behavior is crucial for reliable and efficient solution of advection-dominated optimal control problems, especially in the presence of control and state constraints, and objective functions that depend on pointwise state information. This project intends to deepen our understanding for various problems and to help develop reliable numerical methods.
Mathematics proved to be extremely useful in modeling many real life problems coming from environment, technology, climate, and etc. However, many mathematical models require special parameters that can not be measured directly. Examples can be shapes in modeling technological devises, physical coefficients in environmental processes, controls in navigation and etc. and need to be estimated. Mathematically, estimation of such paramters often leads to optimization problems with constraints in the form of system partial differential equations (PDEs). Usually, such system of PDEs is well understood and there are many available numerical techniques to solve it. However, it does not immediately apply that the method that works well for the underlying system of PDEs will work for the constrained optimization problem. In our previous work, we showed such differences in the case of a simple model problem. In this proposal we intend to investigate more complicated model problems that cover a broader range of applications.
Many optimal control problems with fluid and chemical-reactive flows are advection-dominated. To guarantee good approaximation and avoid nonphysical phenomena, numerical methods usually require a stabilization of the advection term. This poses several interesting questions. First interesting question concern the order when approxamates the problem, so called optimize-then-discretize and discretize-then-optimize approaches. For some stabilization techniques those optimize-then-discretize and discretize-then-optimize approaches do not commute. In the case of model advection-diffusion problems we analyzed in details the two approaches and run various numerical simulations to compare them numerically. The analytical and the numerical evidence show several possible problems with discretize-then-optimize approach. The second interesting question we addressed in details is the choice to treatt boundary conditions. The main result show that although for a single equation there is a small difference in numerical solution in treatment of boundary conditions, this difference can be substantial in the context of optimal control problems. Thus in particular the effect from the boundary layers on numerical solution in the interior of the domain can be significantly smaller if the boundary conditions are treated weakly. We provide a detailed analysis of this statement, which involves a rather technical weighted technique. We also provide various numerical simulations to support analytical findings.