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This thesis deals with the hp-finite element method (FEM) for linear quadratic optimal control problems. Here, a tracking type functional with control costs as regularization shall be minimized subject to an elliptic partial differential equation. In the presence of control constraints, the first order necessary conditions, which are typically used to find optimal solutions numerically, can be formulated as a semi-smooth projection formula. Consequently, optimal solutions may be non-smooth as well. The hp-discretization technique considers this fact and approximates rough functions on fine meshes while using higher order finite elements on domains where the solution is smooth.
The first main achievement of this thesis is the successful application of hp-FEM to two related problem classes: Neumann boundary and interface control problems. They are solved with an a-priori refinement strategy called boundary concentrated (bc) FEM and interface concentrated (ic) FEM, respectively. These strategies generate grids that are heavily refined towards the boundary or interface. We construct an elementwise interpolant that allows to prove algebraic decay of the approximation error for both techniques. Additionally, a detailed analysis of global and local regularity of solutions, which is critical for the speed of convergence, is included. Since the bc- and ic-FEM retain small polynomial degrees for elements touching the boundary and interface, respectively, we are able to deduce novel error estimates in the L2- and L∞-norm. The latter allows an a-priori strategy for updating the regularization parameter in the objective functional to solve bang-bang problems.
Furthermore, we apply the traditional idea of the hp-FEM, i.e., grading the mesh geometrically towards vertices of the domain, for solving optimal control problems (vc-FEM). In doing so, we obtain exponential convergence with respect to the number of unknowns. This is proved with a regularity result in countably normed spaces for the variables of the coupled optimality system.
The second main achievement of this thesis is the development of a fully adaptive hp-interior point method that can solve problems with distributed or Neumann control. The underlying barrier problem yields a non-linear optimality system, which poses a numerical challenge: the numerically stable evaluation of integrals over possibly singular functions in higher order elements. We successfully overcome this difficulty by monitoring the control variable at the integration points and enforcing feasibility in an additional smoothing step. In this work, we prove convergence of an interior point method with smoothing step and derive a-posteriori error estimators. The adaptive mesh refinement is based on the expansion of the solution in a Legendre series. The decay of the coefficients serves as an indicator for smoothness that guides between h- and p-refinement.
This thesis is concerned with applying the total variation (TV) regularizer to surfaces and different types of shape optimization problems. The resulting problems are challenging since they suffer from the non-differentiability of the TV-seminorm, but unlike most other priors it favors piecewise constant solutions, which results in piecewise flat geometries for shape optimization problems.The first part of this thesis deals with an analogue of the TV image reconstruction approach [Rudin, Osher, Fatemi (Physica D, 1992)] for images on smooth surfaces. A rigorous analytical framework is developed for this model and its Fenchel predual, which is a quadratic optimization problem with pointwise inequality constraints on the surface. A function space interior point method is proposed to solve it. Afterwards, a discrete variant (DTV) based on a nodal quadrature formula is defined for piecewise polynomial, globally discontinuous and continuous finite element functions on triangulated surface meshes. DTV has favorable properties, which include a convenient dual representation. Next, an analogue of the total variation prior for the normal vector field along the boundary of smooth shapes in 3D is introduced. Its analysis is based on a differential geometric setting in which the unit normal vector is viewed as an element of the two-dimensional sphere manifold. Shape calculus is used to characterize the relevant derivatives and an variant of the split Bregman method for manifold valued functions is proposed. This is followed by an extension of the total variation prior for the normal vector field for piecewise flat surfaces and the previous variant of split Bregman method is adapted. Numerical experiments confirm that the new prior favours polyhedral shapes.