Janosch Rieger

Numerical fixed grid methods for differential inclusions

SummaryNumerical methods for initial value problems for differential inclusions usually require a discretization of time as well as of the set valued right hand side. In this paper, two numerical fixed grid methods for the approximation of the full solution set are proposed and analyzed. Convergence results are proved which show the combined influence of time and (phase) space discretization.

Shadowing and inverse shadowing in set-valued dynamical systems. Contractive case

We obtain several results on shadowing and inverse shadowing for set-valued dynamical systems that have a contractive property. Applications to

Approximation of reachable sets using optimal control and support vector machines

T

Shadowing and inverse shadowing in set-valued dynamical systems. Hyperbolic case

-flows of differential inclusions are discussed.

Discretizations of Linear Elliptic Partial Differential Inclusions

We propose and discuss a new computational method for the numerical approximation of reachable sets for nonlinear control systems. It is based on the support vector machine algorithm and represents the set approximation as a sublevel set of a function chosen in a reproducing kernel Hilbert space. In some sense, the method can be considered as an extension to the optimal control algorithm approach recently developed by Baier, Gerdts and Xausa. The convergence of the method is illustrated numerically for selected examples.

Robust boundary tracking for reachable sets of nonlinear differential inclusions

We introduce a new hyperbolicity condition for set-valued dynamical systems and show that this condition implies the shadowing and inverse shadowing properties.

Semi-Implicit Euler Schemes for Ordinary Differential Inclusions

Differential inclusions provide a suitable framework for modelling choice and uncertainty. In finite dimensions, the theory of ordinary differential inclusions and their numerical approximations is well-developed, whereas little is known for partial differential inclusions, which are the deterministic counterparts of stochastic partial differential equations. The aim of this article is to analyze strategies for the numerical approximation of the solution set of a linear elliptic partial differential inclusion. The geometry of its solution set is studied, numerical methods are proposed, and error estimates are provided.

Non-convex systems of sets for numerical analysis

The Euler scheme is, to date, the most important numerical method for ordinary differential inclusions because the use of the available higher-order methods is prohibited by their enormous complexity after spatial discretization. Therefore, it makes sense to reassess the Euler scheme and optimize its performance. In the present paper, a considerable reduction of the computational cost is achieved by setting up a numerical method that computes the boundaries instead of the complete reachable sets of the fully discretized Euler scheme from lower-dimensional data only. Rigorous proofs for the propriety of this method are given, and numerical examples illustrate the gain of computational efficiency as well as the robustness of the scheme against changes in the topology of the reachable sets.

Implementing Galerkin Finite Element Methods for Semilinear Elliptic Differential Inclusions

Two semi-implicit Euler schemes for differential inclusions are proposed and analyzed in depth. An error analysis shows that both semi-implicit schemes inherit favorable stability properties from the differential inclusion. Their performance is considerably better than that of the implicit Euler scheme, because instead of implicit inclusions only implicit equations have to be solved for computing their images. In addition, they are more robust with respect to spatial discretization than the implicit Euler scheme.

Semilinear parabolic differential inclusions with one-sided Lipschitz nonlinearities

The notion of a system of sets generated by a family of functionals is introduced. A generalization of the classical support function of convex subsets of