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Ringvorlesung Wavephenomena (Winter Semester 2023/24)

Acoustic Scattering in the Time Domain

Prof. Dr. Roland Griesmaier

Scattering of transient acoustic waves by compactly supported scattering obstacles can be modeled by exterior boundary value problems for the wave equation in unbounded free space. After giving a brief introduction to this class of scattering problems, we will discuss existence and uniqueness of solutions using retarded potentials and boundary integral equations. The analysis of the time-domain integral operators will be performed in the Laplace domain.

The lectures take place on November 06, November 13, and November 20.

Efficient simulation of wave phenomena in highly heterogeneous media

Prof. Dr. Roland Maier

In this part of the lecture, we deal with wave phenomena in heterogeneous media, where effects can occur on multiple different scales (e.g., composite materials). In the underlying partial differential equations, this multiscale nature is present in the form of highly oscillatory coefficients. Classical numerical discretization methods need to resolve these oscillations in order to obtain reasonable results in the first place. To avoid costly computations on very fine scales, specifically designed discretization spaces present an alternative. They are particularly useful when discretizing time-dependent and/or nonlinear problems. Here, we focus on the discretization of the classical wave equation and a nonlinear Helmholtz equation, both involving highly oscillatory and rough coefficients. Theoretical aspects of the multiscale construction (stability and error estimates) and also numerical illustrations are treated.

The lectures take place on November 27, December 04, and December 11.

Strichartz estimates for wave equations

Prof. Dr. Roland Schnaubelt

Wave-type equations typically exhibit dispersive behavior which means that wave packets smear out if time evolves. These properties are quantified by Strichartz estimates for solutions to linear problems. Compared to the preservation of L^2-based norms, they provide increased spatial integrability of solutions at the price of decreased time integrability and (for the wave equation) of a loss of regularity. Since the 90's these and related estimates have been crucial for the tremendous progress in the wellposedness and qualitative theory of semilinear dispersive problems.

The lectures focus on Strichartz estimates for the wave equation on \mathbb{R}^d and explain their proofs. Some applications to the wellposedness theory for semilinear wave equations will be sketched. We also give an overview on results for problems with coefficients or on domains. Here we can only discuss main difficulties and indicate some ideas how to solve them.

The lectures take place on January 08, January 15, and January 22. You can download lecture notes here.

Wave phenomena in geophysics

Prof. Dr. Thomas Bohlen

In the application part of the lecture series, we deal with seismic wave phenomena that occur regularly in geophysics and are evaluated to produce detailed images of the Earth's interior. For the general basic understanding, we first consider the wave types and their properties and study their propagation behavior using various application examples. Unusual wave fields allow special "insights" in special cases. Since wave propagation simulation is an important tool, we learn how to discretize the wave equation with finite differences in an efficient way and how corresponding algorithms work. Last, we deal with perhaps the most advanced inversion technique in seismics/seismology, the so-called full waveform inversion (FWI), which allows to exploit the complete signals and wave fields for the exploration of the Earth's interior.

Course material

The lectures take place on January 29, February 05, and February 19.

Lecture: Monday 14:00-15:30 20.30 1. OG R. 1.66/ 1.67
Lecturer Prof. Dr. Roland Schnaubelt
Office hours: Tuesday at 12:00 - 13:00, and by appointment.
Room 2-047 (Englerstr. 2) Kollegiengebäude Mathematik (20.30)
Email: schnaubelt@kit.edu
Lecturer Prof. Dr. Roland Griesmaier
Office hours: Tuesday, 2:00-3:00 PM
Room 1.040 Kollegiengebäude Mathematik (20.30)
Email: roland.griesmaier@kit.edu
Lecturer JProf. Dr. Roland Maier
Office hours: by appointment
Room 3.009 Kollegiengebäude Mathematik (20.30)
Email: roland.maier@kit.edu