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Recurrence is a fundamental characteristic of dynamical systems with complicated behavior.

Understanding the inner structure of recurrence is challenging, especially if the system has many

degrees of freedom and is subject to noise. The cycling signature is an algebraic topological notion

for identifying and classifying elementary recurrent motions — called cycling — and the transitions

between them. Statistics on these cycling motions can be computed from sampled trajectories

(time series data).

They provide a coarse global description of the structure of the recurrent behavior.

To construct localizable multiscale methods for nonlinear partial differential equations we consider a transfer operator that maps arbitrary admissible boundary data on the boundary of an oversampling domain to the respective (local) solution on the target subdomain; here the boundary of the latter must have a distance greater than zero from the boundary of the oversampling domain. Then, we try to approximate the set of all local solutions on the target subdomain. Interpreting the boundary data as some input parameter, we can view this set of local solutions as a set of solutions depending on a parameter. This motivates using methods from model order reduction such as the proper orthogonal decomposition (POD) or the Greedy algorithm to approximate this set. However, both the POD and the Greedy algorithm rely on a training set of finite cardinality that is chosen such that every point in the admissible parameter set is close to a point in the training set. Therefore, both algorithms suffer from the curse of dimensionality. We thus employ randomization and consider the parameter (here: boundary data) as a random variable with values in a Hilbert space. By choosing a suitable distribution we can then exploit the concentration of measure phenomenon, which is also sometimes called the “blessing of dimensionality” to break the curse.

In detail, we will present a randomized greedy algorithm that provides with high probability a certification for the whole parameter set rather than only for the parameters in the training set. Moreover, we will present a randomized POD and a corresponding error analysis that shows that for exponentially decaying eigenvalues of the randomized POD which uses the exact correlation operator (integral in the expectation) the approximation error between any solution corresponding to a parameter in the admissible parameter set and the approximation with the POD that uses a Monte-Carlo approximation converges exponentially as well.

We study the scattering of waves from a planar inclusion with constant refractive index, governed by the Helmholtz equation. It is well understood that singular inclusions—those whose boundary has a singular point—generically scatter every incident wave. Far less is known about the scattering behavior of regular inclusions. We consider a large class of regular inclusions and show that they generically scatter any (complex-analytic) incident wave. Focusing on incident plane waves, we prove that they always scatter from any regular inclusion whose refractive index is less than one. For inclusions with refractive index greater than one, under an additional convexity assumption, we obtain explicit wavenumber intervals in which scattering occurs. These intervals drift to infinity and expand, and are given by explicit formulas in terms of the geometry (directional widths) of the inclusion. If the inclusion is elongated (its maximal width is at least twice its minimal width), the union of these intervals covers the entire spectrum, and hence such inclusions scatter every incident plane wave. Our approach makes use of a connection between this scattering problem and the Schiffer/Pompeiu problem.

This is based on a joint work with Michael Vogelius.