In this thesis, we investigate optimization problems with partial differential equation (PDE) constraints. In particular we are concerned with the efficient numerical solution of optimum experimental design (OED) problems for parameter estimation (PE) with PDE models, among them sampling design problems. We consider two dimensional (2D) stationary diffusion advection reaction PDE models, including the challenging case of an advection dominated PDE.
For the simulation of the PDE boundary value problem, we utilize discontinuous Galerkin finite element methods and adaptive spatial grid refinement. We solve the optimization problems with derivative-based algorithms. For the optimization algorithms to converge fast and to converge to the ”true“ optimum, we need to provide accurate sensitivities. It is a challenge to evaluate the sensitivities, which correspond to the approximate solution of the primal PDE model and are in this sense consistent. In this thesis we develop efficient and accurate methods for sensitivity generation. We transfer the principle of internal numerical differentiation (IND) from ordinary differential equations (ODE)s to PDEs. That means, we incorporate the sensitivity generation in the solution process. The standard upwind discontinuous Galerkin method is not differentiable. Therefore, we propose a differentiable discontinuous Galerkin method and give a rigorous convergence analysis of it. We develop methods for structure exploitation of the primal and tangential discretization schemes to efficiently generate the sensitivities with automatic differentiation (AD). Furthermore, we establish methods for frozen adaptivity to generate consistent sensitivities. We are especially concerned with frozen spatial grid refinement and the adaptive step number of the linear solver.
We implement the developed methods in the software SeafaND-Optimizer, short for structure exploiting and frozen adaptivity numerical differentiation optimizer. It is a software for efficient and accurate simulation, PE and OED with PDE models. We perform numerical case studies for PE and OED problems with advection dominated 2D diffusion advection PDE models. With the structure exploiting techniques developed in this thesis, the example problems are solved with efficient memory usage. Due to the frozen adaptivity methods, we computed efficiently the consistent sensitivities. We test the PE algorithm with different noise levels. We perform a case study with different diffusion coefficients for sequential OED. Finally, we investigate, whether the developed methods are stable under mesh refinements
