In the iRTG lecture series four members of CRC 1173 will talk about topics whithin the analysis and numerics of wave phenomena. The lectures are directed to Ph.D. students (in particular of the integrated research training group of CRC 1173) and to advanced master students with a solid background in partial differential equations. The series is organized by Willy Dörfler and Roland Schnaubelt.

Thomas Bohlen: Geophysical Applications of Full Waveform Inversion

The lectures take place on 16 April, 23 April, and 30 April.

Overview
Seismic waves bring to the surface information gathered on the physical properties of the earth. Full waveform inversion (FWI) is a challenging data-fitting procedure based on full-wavefield modeling to extract quantitative information from seismograms with maximum resolution. In this part of the Ringvorlesung we will explain the fundamentals of the widely used adjoint-state method (lecture 1) and will discuss applications of adjoint-state FWI to body waves (lecture 2) and shallow seismic surface waves (lecture 3).

1) Adjoint-state FWI in Geophysics
The most widely used implementations of FWI in Geophysics apply the adjoint-state technique to calculate the gradients of the misfit function. We will derive the corresponding equations using the Born-approximation and the concept of adjoint operators. This way we can show the mathematical analogy to other seismic imaging methods. We will discuss the conventional workflow of FWI including seismic data pre-processing, regularization and source time function inversion.

2) Applications of FWI to body waves
In geophysical exploration FWI is mainly applied to body compressional waves which can be computed by solving the acoustic wave equation. Using several synthetic examples we illustrate the workflow and imaging potentials of FWI. We will discuss several field data examples to demonstrate the challenges of FWI to real word applications. This examples will include the application to top-salt imaging and the characterization of sub-marine gas accumulations. Finally we will discuss first applications to ultrasonic imaging, which is widely used in medical cancer screening and in non- destructive testing of materials.

3) Applications of FWI to shallow seismic surface waves
In the third lecture we will discuss applications of elastic FWI to shallow seismic surface waves. These wave can be computed by solving the (visco-) elastic wave equations. Shallow seismic surface waves penetrate only up to a few tens of meter into the earth and can carry information about the elastic shear properties of the very shallow subsurface. We will discuss the characteristics of surface waves (depth penetration, dispersion) and will show different applications of (visco-) elastic FWI to field data.

References

• Virieux, J., Operto, S., 2009, An overview of full-waveform inversion in exploration geophysics, Geophysics, Vol. 74, 6, WCC127–WCC152.
• Kurzmann, A., Przebindowska, A., Köhn, D. and Bohlen, T. 2013. Acoustic full waveform tomography in the presence of attenuation: a sensitivity analysis. Geophysical Journal International 195(2), 985-1000.
• Groos, L., Schäfer, M., Forbriger, T. & Bohlen, T. (2017), Application of a complete workflow for 2D elastic full-waveform inversion to recorded shallow-seismic Rayleigh waves, Geophysics 82(2), 1–9.

Andreas Rieder: An abstract framework for inverse wave problems with applications

The lectures take place on 7 May, 14 May, and 28 May.

In this short series of lectures we present a general theory for nonlinear inverse problems related to abstract evolution equations. We study the corresponding parameter-to-solution map and provide its Fréchet derivative and the adjoint operator thereof. Access to both operators is needed, e.g., for setting up Newton-like solvers. Further, we show that the inverse problem is ill-posed in the strict mathematical sense. Finally, the abstract results are applied to parameter identification problems related to the following first order hyperbolic systems: elastic wave equation (seismic imaging) and Maxwell's equation (electromagnetic scattering in conducting media).

References

• A. Kirsch, A. Rieder, Inverse problems for abstract evolution equations with applications in electrodynamics and elasticity, Inverse Problems 32 (2016) 085001.

Michael Feischl: Computational Micromagnetism

The lectures take place on 4 June, 11 June, and 18 June.

Magnetic processes play an important role in a variety of technological applications, e.g., magnetic sensors, recording heads, and magnetoresistive storage devices. On a microscale, the quantity which describes the magnetic condition of a ferromagnetic body is the magnetization, a three-dimensional vector field. In the literature, it is well accepted that the dynamics of the magnetization is governed by the Landau-Lifshitz-Gilbert equation, which describes the behavior of the magnetization under the influence of the so-called effective field, which is characterized by a multitude of physical effects. Mathematical challenges of this evolution equation are due to the strong nonlinearity, possibly complicated and nonlocal field contributions, as well as an inherent nonconvex side constraint which enforces length preservation.

In the lecture, first we introduce the theory of micromagnetism. Then, we discuss the mathematical formulation of both the stationary and the dynamical problems in micromagnetism, giving an overview of the available analytical results. Finally, we focus on the convergent numerical integrators for the LLG equation which are available in the mathematical market.

Birgit Schörkhuber

The lectures take place on 25 June, 2 July, and 9 July.

In this series of lectures we will discuss some aspects in the analysis of nonlinear wave equations.

1) The linear wave equation
First, we will review the basic theory for the linear wave equation on R^d including energy and Strichartz estimates. Then we will focus on the behavior of solutions in backward lightcones. By introducing adapted coordinates, we will obtain a natural framework to study lightcone solutions by using semigroup methods.

2) Semilinear problems
In this lecture we will focus on wave equations with a power nonlinearity. Although this is the simplest class of semilinear wave equations, it allows for very complex dynamics and has been the subject of intensive research in recent past. After a short introduction we will discuss the existence of local/global solutions and the possibility of finite-time blowup. Finally, we will consider the existence of special solutions including solitons and self-similar blowup solutions.

3) Blowup dynamics
In this final part we will deepen the discussion on blowup dynamics for nonlinear wave equations. In particular, we will see how the framework introduced in the first part of this lecture series can be used to study the stability of self-similar blowup solutions.