Multigrid Methods for Elliptic Optimal Control Problems

In this dissertation we study multigrid methods for linear-quadratic elliptic distributed optimal control problems. For optimal control problems constrained by general second order elliptic partial differential equations, we design and analyze a $P_1$ finite element method based on a saddle point formulation. We construct a $W$-cycle algorithm for the discrete problem and show that it is uniformly convergent in the energy norm for convex domains. Moreover, the contraction number decays at the optimal rate of $m^{-1}$, where $m$ is the number of smoothing steps. We also prove that the convergence is robust with respect to a regularization parameter. The robust convergence of $V$-cycle and $W$-cycle algorithms on general domains are demonstrated by numerical results. For optimal control problems constrained by symmetric second order elliptic partial differential equations together with pointwise constraints on the state variable, we design and analyze symmetric positive definite $P_1$ finite element methods based on a reformulation of the optimal control problem as a fourth order variational inequality. We develop a multigrid algorithm for the reduced systems that appear in a primal-dual active set method for the discrete variational inequalities. The performance of the algorithm is demonstrated by numerical results

Preconditioners for Krylov subspace methods: An overview

When simulating a mechanism from science or engineering, or an industrial process, one is frequently required to construct a mathematical model, and then resolve this model numerically. If accurate numerical solutions are necessary or desirable, this can involve solving large-scale systems of equations. One major class of solution methods is that of preconditioned iterative methods, involving preconditioners which are computationally cheap to apply while also capturing information contained in the linear system. In this article, we give a short survey of the field of preconditioning. We introduce a range of preconditioners for partial differential equations, followed by optimization problems, before discussing preconditioners constructed with less standard objectives in mind

Discretizations & Efficient Linear Solvers for Problems Related to Fluid Flow

Numerical solutions to fluid flow problems involve solving the linear systems arising from the discretization of the Stokes equation or a variant of it, which often have a saddle point structure and are difficult to solve. Geometric multigrid is a parallelizable method that can efficiently solve these linear systems especially for a large number of unknowns. We consider two approaches to solve these linear systems using geometric multigrid: First, we use a block preconditioner and apply geometric multigrid as in inner solver to the velocity block only. We develop deal.II tutorial step-56 to compare the use of geometric multigrid to other popular alternatives. This method is found to be competitive in serial computations in terms of performance and memory usage. Second, we design a special smoother to apply multigrid to the whole linear system. This smoother is analyzed as a Schwarz method using conforming and inf-sup stable discretization spaces. The resulting method is found to be competitive to a similar multigrid method using non-conforming finite elements that were studied by Kanschat and Mao. This approach has the potential to be superior to the first approach. Finally, expanding on the research done by Dannberg and Heister, we explore the analysis of a three-field Stokes formulation that is used to describe melt migration in the earth\u27s mantle. Multiple discretizations were studied to find the best one to use in the ASPECT software package. We also explore improvements to ASPECT\u27s linear solvers for this formulation utilizing block preconditioners and algebraic multigrid

Mini-Workshop: Adaptive Methods for Control Problems Constrained by Time-Dependent PDEs

Optimization problems constrained by time-dependent PDEs (Partial Differential Equations) are challenging from a computational point of view: even in the simplest case, one needs to solve a system of PDEs coupled globally in time and space for the unknown solutions (the state, the costate and the control of the system). Typical and practically relevant examples are the control of nonlinear heat equations as they appear in laser hardening or the thermic control of flow problems (Boussinesq equations). Specifically for PDEs with a long time horizon, conventional time-stepping methods require an enormous storage of the respective other variables. In contrast, adaptive methods aim at distributing the available degrees of freedom in an a-posteriori-fashion to capture singularities and are, therefore, most promising

Schnelle Löser für partielle Differentialgleichungen

The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch(Leipzig), Gabriel Wittum (Heidelberg) was held May 22nd - May 28th, 2005. This meeting was well attended by 47 participants with broad geographic representation from 9 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds

Nonsmooth Schur-Newton methods for nonsmooth saddle point problems.

We introduce and analyze nonsmooth Schur-Newton methods for a class of nonsmooth saddle point problems. The method is able to solve problems where the primal energy decomposes into a convex smooth part and a convex separable but nonsmooth part. The method is based on nonsmooth Newton techniques for an equivalent unconstrained dual problem. Using this we show that it is globally convergent even for inexact evaluation of the linear subproblems