Johannes Kepler Symposium on Mathematics

As part of the Johannes Kepler symposium on mathematics Dr. Stefan Takacs, Institute of Computational Mathematics, JKU Linz, will give a public talk (followed by a discussion) on Wed, Oct. 9, 2019 at 15:15 o'clock at HS 12 on the topic of "Robust multigrid solvers and related topics" . The organziers of the symposium,

O.Univ.-Prof. Dr. Ulrich Langer,
Univ.-Prof. Dr. Gerhard Larcher
A.Univ.-Prof. Dr. Jürgen Maaß, and
die ÖMG (Österreichische Mathematische Gesellschaft),

hereby cordially invite you.

Series B - Mathematical Colloquium:

The intention is to present new mathematical results for an audience interested in general mathematics.

Robust multigrid solvers and related topics

This presentation gives an overview on my scientific achievements after finishing my PhD project. The main concern of my scientific work is to provide iterative solvers for partial differential equations (PDEs) which are robust in model parameters (like a regularization parameter or a step size of a time-stepping approach) or in parameters of the discretization (like the grid size or a polynomial degree). Since the linear systems are large-scale and sparse, iterative solvers are a proper choice. Geometric multigrid solvers are of particular interest.

We are mainly concerned with multigrid solvers because they show robustness in the grid size. This means that the convergence rates are independent of the grid size or, equivalently, of the number of degrees of freedom. Consequently, the number of multigrid cycles required to reach a certain accuracy goal is independent of the number of unknowns as well. Since the computational costs for one multigrid cycle can be shown to grow only linearly with the number of unknowns, also the computational costs for the whole solution process grow only linearly, which can be seen as a first aspect of robustness.
A second aspect of robustness is linked to model parameters, like a regularization parameter of an optimal control problem or a step size of a time-stepping approach. We construct solvers whose convergence rates are independent of such model parameters and, still, of the grid size. Since the choice of the parameters does not affect the number of degrees of freedom or the sparsity pattern of the involved matrices, the computational costs are independent of the choice of the model parameters. Why are we interested in a robust solver? Because it allows us to choose the regularization parameter only according to the parameter choice rule appropriate for the underlying inverse problem. For similar reasons, robustness in the length of time-steps in a time-stepping approach or some cost or material parameters is of interest.

A third aspect of robustness is linked to the polynomial (spline) degree, which we consider in the context of Isogeometric Analysis. The dependence of the condition number on the spline degree is particularly severe since it grows exponentially with the spline degree. This being not enough, the spline degree also affects the sparsity pattern of the stiffness matrix. The number of non-zero entries of the stiffness matrix grows like $p^d$, where $p$ is the spline degree and $d$ is the spatial dimension. This means that the computational costs per multigrid cycle are not expected to be independent of the spline degree. To be able to talk about robustness, we have to have both robust convergence behavior and a well bounded computational complexity per multigrid cycle. In this framework, we first consider single-patch domains that are parameterized by one single geometry function. Then, the results are extended to domains that are composed of numerous patches.

We are interested in both numerical experiments and convergence theory. We will see results of numerical experiments that are performed using implementations of the proposed methods in the programming language C++. The solvers in the context of Isogeometric Analysis are implemented in and provided to the G+Smo library. The convergence analysis for the multigrid solvers is based on the splitting of the analysis into smoothing property and approximation property as introduced by Hackbusch, which allows to analyze the two-grid method and the W-cycle multigrid method. For the Poisson problem, the extension of the convergence analysis of the multigrid solvers to frameworks that also include the V-cycle, cf. the work by Braess and Hackbusch and by Bramble, seems to be possible but has not been worked out in detail. The analysis of the V-cycle multigrid solver for the saddle point problems discussed in the first part is certainly completely open.