Webrelaunch 2020

Ringvorlesung Wavephenomena (Winter Semester 2022/23)

The interior transmission eigenvalue problem

Dr. Lukas Pieronek

The interior transmission eigenvalue problem arises in the study of inverse scattering problems where one tries to reconstruct the support of an unknown scatterer on the basis of its scattering response with time-harmonic waves. In this context, interior transmission eigenvalues correspond to those wave numbers for which the scatterer can give a non-scattering response, leading to spurious reconstructions in practice. They are mathematically modelled as the spectrum of a non-selfadjoint operator which is in many ways non-standard. In this lecture series we develop a theoretical framework for analyzing interior transmission eigenvalues and discuss open questions in this field.

The lectures take place on November 14, November 21, and November 28.

Spectral stability for perturbed operators

Dr. Lucrezia Cossetti

In these lectures we will investigate spectral properties of operators of the following form

H = H_0 + V.

These operators can be seen as a perturbation through the operator V of a reference operator H_0.
If one is interested in the spectrum of the perturbed Hamiltonian H, then the following natural questions arise:

1. Under which perturbations V the spectral properties of the free operator H_0 are preserved?

Or, from a different point of view:

2. How and to what extend spectral properties of H deviate from the ones of H_0?

The object of these lectures is to collect and elaborate on different tools, both well-established and more recent ones, which have been developed in the last decades to give a satisfactory answer to the questions posed above, both in a self-adjoint and non-self-adjoint context. More specifically, we will show how Hardy-type and Sobolev inequalities, together with Virial theorems and Birman-Schwinger principles enter into play in the analysis of the spectrum of these Hamiltonians.

The lectures take place on December 5, December 12, and December 19.

Homogenization of optical metamaterials from two perspectives

Dr. Fatima Goffi

We give an introduction into aspects of homogenization of periodic composite materials with respect to two different approaches, with applications to optical metamaterials. For the case of Maxwell equations the type of the considered constitutive relations is decisive for the choice of the followed approach. One way is the asymptotic homogenization which considers local constitutive relations, and it is based on the two scale-scale convergence. The other way considers nonlocal effects described through nonlocal constitutive relations. We will see how the effective equations as well as the parameters describing the effective medium can be obtained for the Maxwell equations with time harmonic dependency.

The lectures take place on January 09, January 16, and January 23.

Dynamical low-rank integrators for matrix differential equations

Dr. Stefan Schrammer

Many natural phenomena can be modeled by (1+2)-dimensional partial differential equations (PDEs). In the space discretization often a fine resolution has to be used, which yields large matrix differential equations (MEDs). While standard time integration schemes usually suffer from the large size of the problem, dynamical low-rank integrators (DLRIs) have been observed to give very good approximations in a reasonable amount of time if the exact solution of the respective MDE can be well approximated by a low-rank matrix.

We motivate and introduce the concept of dynamical low-rank integration. Afterwards, we construct DLRIs for first and second-order MDEs and discuss their properties. Moreover, we derive variants of the original schemes tailored to stiff problems and comment on implementation details.

The lectures take place on January 30, February 06, and February 13.

Lecture: Monday 14:00-15:30 20.30 1. OG R. 1.66/ 1.67 Begin: 14.11.2022, End: 13.2.2023