Nonoverlapping domain decomposition preconditioners for discontinuous Galerkin approximations of Hamilton--Jacobi--Bellman equations

We analyse a class of nonoverlapping domain decomposition preconditioners for nonsymmetric linear systems arising from discontinuous Galerkin finite element approximation of fully nonlinear Hamilton--Jacobi--Bellman (HJB) partial differential equations. These nonsymmetric linear systems are uniformly bounded and coercive with respect to a related symmetric bilinear form, that is associated to a matrix $\mathbf{A}$. In this work, we construct a nonoverlapping domain decomposition preconditioner $\mathbf{P}$, that is based on $\mathbf{A}$, and we then show that the effectiveness of the preconditioner for solving the} nonsymmetric problems can be studied in terms of the condition number $\kappa(\mathbf{P}^{-1}\mathbf{A})$. In particular, we establish the bound $\kappa(\mathbf{P}^{-1}\mathbf{A}) \lesssim 1+ p^6 H^3 /q^3 h^3$, where $H$ and $h$ are respectively the coarse and fine mesh sizes, and $q$ and $p$ are respectively the coarse and fine mesh polynomial degrees. This represents the first such result for this class of methods that explicitly accounts for the dependence of the condition number on $q$; our analysis is founded upon an original optimal order approximation result between fine and coarse discontinuous finite element spaces. Numerical experiments demonstrate the sharpness of this bound. Although the preconditioners are not robust with respect to the polynomial degree, our bounds quantify the effect of the coarse and fine space polynomial degrees. Furthermore, we show computationally that these methods are effective in practical applications to nonsymmetric, fully nonlinear HJB equations under $h$-refinement for moderate polynomial degrees

Seventh Copper Mountain Conference on Multigrid Methods

The Seventh Copper Mountain Conference on Multigrid Methods was held on 2-7 Apr. 1995 at Copper Mountain, Colorado. This book is a collection of many of the papers presented at the conference and so represents the conference proceedings. NASA Langley graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection shows its rapid trend to further diversity and depth

Optimal Control of anisotropic Allen–Cahn equations

This thesis is concerned with the solution of an optimal control problem governed by an anisotropic Allen-Cahn equation as a model for, e.g., crystal growth. The first part treats the analytical existence theory and first order optimality conditions of the in time continuous and of the time discretized versions. The state equation is discretized implicitly in time with piecewise constant functions. To this end, we consider a more general quasilinear parabolic equation, where the quasilinear term is strongly monotone and obeys a certain growth condition while the lower order term is potentially non-monotone. The existence of the control-to-state operator and its Lipschitz continuity is shown for the time discretized as well as for the time continuous problem. Then we present for both the existence of global minimizers as well as the convergence of a subsequence of time discrete optimal controls to a global minimizer of the time continuous problem. The results hold in arbitrary space dimensions. Under some further restrictions we are able to show Fréchet differentiability of the in time discretized problem and use this to rigorously set up the first order conditions. For this the anisotropies are required to be smooth enough, which in this thesis is achieved by a suitable regularization. Therefore, the convergence behavior of the optimal controls are studied for a sequence of (smooth) approximations of the former quasilinear term. In addition the simultaneous limit in the approximation and the time step size is considered. For a class covering a large variety of anisotropies we introduce a certain regularization and show the previously formulated requirements. Finally, we will show that the results cannot be straightforwardly transferred to a semi-implicit discretization scheme. In the second part a trust region Newton method is presented, that eventually is used to numerically solve the optimal control problem. Different ways of preconditioning the involved Steihaug-CG solver are discussed and the limits of existing approaches in the present case are worked out. Then, several aspects of the implementation are examined, like the solver for the appearing partial differential equations, parallelization and the utility of adaptive meshes in the context of the control problem. In the final part, various numerical results based on the previously mentioned choice of anisotropies are presented. These include convergence with respect to the regularization parameter, numerical evidence for mesh independent behavior and a thorough discussion of the simulation in several relevant settings. We concentrate on two choices for the anisotropies and in addition include the isotropic case for comparison. Among others, crystal formation and topology changes are addressed and we see that the algorithm is able to handle these. Furthermore, the behavior of various quantities over the course of the algorithm is investigated. Here we observe that the number of Steihaug steps, and therefore the execution time per trust region step, growths considerably towards the end of the algorithm. Finally, we look at the impact of some implementational aspects with respect to execution speed. We observe that the implicit and semi-implicit approaches perform comparably fast if the implementation is suitably optimized. We however conclude that the implicit approach is preferable since it is less sensitive with respect to the regularization and is supported by more theoretical results

Seventh Copper Mountain Conference on Multigrid Methods

The Seventh Copper Mountain Conference on Multigrid Methods was held on April 2-7, 1995 at Copper Mountain, Colorado. This book is a collection of many of the papers presented at the conference and so represents the conference proceedings. NASA Langley graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The vibrancy and diversity in this field are amply expressed in these important papers, and the collection clearly shows the continuing rapid growth of the use of multigrid acceleration techniques

Bifurcation analysis of the Topp model

In this paper, we study the 3-dimensional Topp model for the dynamicsof diabetes. We show that for suitable parameter values an equilibrium of this modelbifurcates through a Hopf-saddle-node bifurcation. Numerical analysis suggests thatnear this point Shilnikov homoclinic orbits exist. In addition, chaotic attractors arisethrough period doubling cascades of limit cycles.Keywords Dynamics of diabetes · Topp model · Reduced planar quartic Toppsystem · Singular point · Limit cycle · Hopf-saddle-node bifurcation · Perioddoubling bifurcation · Shilnikov homoclinic orbit · Chao

ISOGEOMETRIC OVERLAPPING ADDITIVE SCHWARZ PRECONDITIONERS IN COMPUTATIONAL ELECTROCARDIOLOGY

In this thesis we present and study overlapping additive Schwarz preconditioner for the isogeometric discretization of reaction-diffusion systems modeling the heart bioelectrical activity, known as the Bidomain and Monodomain models. The cardiac Bidomain model consists of a degenerate system of parabolic and elliptic PDE, whereas the simplified Monodomain model consists of a single parabolic equation. These models include intramural fiber rotation, anisotropic conductivity coefficients and are coupled through the reaction term with a system of ODEs, which models the ionic currents of the cellular membrane. The overlapping Schwarz preconditioner is applied with a PCG accelerator to solve the linear system arising at each time step from the isogeometric discretization in space and a semi-implicit adaptive method in time. A theoretical convergence rate analysis shows that the resulting solver is scalable, optimal in the ratio of subdomain/element size and the convergence rate improves with increasing overlap size. Numerical tests in three-dimensional ellipsoidal domains confirm the theoretical estimates and additionally show the robustness with respect to jump discontinuities of the orthotropic conductivity coefficients

Space-time Methods for Time-dependent Partial Differential Equations

Modern discretizations of time-dependent PDEs consider the full problem in the space-time cylinder and aim to overcome limitations of classical approaches such as the method of lines (first discretize in space and then solve the resulting ODE) and the Rothe method (first discretize in time and then solve the PDE). A main advantage of a holistic space-time method is the direct access to space-time adaptivity and to the backward problem (required for the dual problem in optimization or error control). Moreover, this allows for parallel solution strategies simultaneously in time and space. Several space-time concepts where proposed (different conforming and nonconforming space-time finite elements, the parareal method, wavefront relaxation etc.) but this topic has become a rapidly growing field in numerical analysis and scientific computing. In this workshop the focus is the development of adaptive and flexible space-time discretization methods for solving parabolic and hyperbolic space-time partial differential equations