In the iRTG lecture series members or PostDocs 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 Roland Griesmaier and Roland Schnaubelt.

Björn de Rijk: Stability of Periodic Waves

The lectures take place on November 10, November 17, and November 24.

Periodic waves are ubiquitous in dynamical processes across the sciences, including nonlinear optics, fluid dynamics, ecology, and biology. The classical Turing mechanism shows that stable periodic patterns typically emerge after a short-wave destabilization of a homogeneous background state. This lecture series provides an introduction to the stability theory of periodic waves in one and multiple spatial dimensions. The first part develops the spectral analysis of periodic waves, employing tools such as Floquet-Bloch theory, exponential dichotomies, and Lyapunov-Schmidt reduction. In the second part, I demonstrate how spectral information can be leveraged to establish nonlinear stability, both in dissipative and in conservative settings.

Soumen Senapati: Some Multidimensional Inverse Problems for Hyperbolic Equations

The lectures take place on December 01, December 08, and December 15.

This lecture series is devoted to the study of several multidimensional inverse problems associated with the standard wave operator. The primary objective here is to establish uniqueness results for the recovery of certain intrinsic properties of a medium from boundary measurements.Depending on the type of boundary operator under consideration, which may correspond to infinitely many, finitely many, or local measurements, we will employ different analytical techniques to address these problems. A unifying theme throughout the lectures will be the application of Carleman estimates, which constitute a fundamental tool in addressing such inverse problems. These estimates also play a crucial role in related areas, including unique continuation for partial differential equations and control theory.

Martin Halla: Numerical methods for transparent boundary conditions

The lectures take place on January 12, January 19, and January 26.

We consider wave propagation problems posed in unbounded domains. The application of many discretization schemes (e.g., finite elements, finite differences) require a bounded domain. To this end the domain is truncated to an ("interior") bounded domain of interest and a so-called transparent boundary condition (TBC) is imposed at the artificial boundary. The definition of the TBC is such that the solution to the original PDE remains unmodified in the interior domain. For simulations the TBC has to be approximated as well. We review existing techniques to construct such numerical schemes, and discuss the popular perfectly matched layer (PML) method in detail. We highlight settings for which the PML fails or requires modifications, and draw a connection to the more sophisticated Hardy space infinite element method.

Sebastian Krumscheid: An Introduction to Bayesian inverse problems in function spaces

The lectures take place on February 02, February 09, and February 16.

Uncertainties are prevalent across scientific and engineering disciplines, where mathematical models are used to describe complex systems and interpret noisy observational data. Inverse problems, that is, reconstructing unknown parameters, functions, or dynamically changing states so that model predictions best explain observed data, provide a rigorous framework for quantifying and reducing these uncertainties. This short course offers an introduction to the Bayesian approach to inverse problems, with a particular focus on infinite-dimensional settings, where the unknowns are modeled as functions or fields rather than finite-dimensional parameter vectors. Emphasizing the function-space viewpoint yields a mathematically coherent framework that remains well-posed under discretization and leads to efficient computational algorithms for large-scale applications. Participants will gain an understanding of the theoretical foundations of Bayesian inverse problems in function spaces, as well as practical insight into their numerical implementation.

In the iRTG lecture series members or PostDocs 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 Roland Griesmaier and Roland Schnaubelt.

Marvin Knöller: Convolution Quadrature in Acoustic Obstacle Scattering

The lectures take place on November 04, November 11, and November 18.

In this course we study theoretical and practical aspects of acoustic wave propagation in presence of an impenetrable scattering object. The content of the lectures is twofold. In the first part we study the scattering problem for the wave equation and briefly recall existence and uniqueness of solutions by proceeding through the Laplace domain. Afterwards, we derive the all-at-once convolution quadrature approach for a semi-discretization in time. The second part is about the implementation of this method in Python using the boundary element library Bempp (see Bempp). This is planned as a live coding lecture, i.e., the code is developed in real time. Participants of this course are encouraged to bring their own laptops and actively engage in hands-on programming during the event.

Louis Garenaux: Normal Form and Space-Time Resonances for Wave-Type Equations

The lectures take place on November 25, December 02, and December 09.

In this part of the lecture, we will discuss two useful methods when studying the long-time behavior of PDE solutions. The normal form transform and the space-time integrations rely on a description of linear mode interactions through nonlinear terms. Thanks to a relatively simple algorithm, they allow to identify the irrelevant terms and to remove them from the equation. We will first set the stage, and present the impact of dimension and power exponents on the asymptotic behavior of solutions. Then, we will successively present both methods in various contexts and illustrate them on examples.

Carsten Rockstuhl: Materials and Wave Phenomena in Optics: From Natural Media to Engineered Structures

The lectures take place on Dezember 16, January 16, and January 20.

In this part of the lecture series, we will explore the crucial role of materials in observing and controlling wave phenomena in optics and their mathematical description. Initially, we focus on natural materials, examining how their intrinsic optical properties can be integrated into Maxwell's equations. We discuss material-specific optical phenomena and their mathematical foundations. Then, we address artificial materials with subwavelength structures, demonstrating how these engineered materials can be treated as homogenous while exhibiting properties not found in nature. Finally, we investigate artificially periodically structured materials on length scales comparable to the wavelength of light. Considering these artificial materials enables new opportunities to manipulate light at the nanoscale, offering insights into novel applications in photonics.

Benjamin Dörich: Space and Time Discretization of Nonlinear Wave Equations

The lectures take place on January 27, February 03, and February 10.

In this part of the lecture, we study nonlinear wave equations and their discretization in space and time. For completeness, we will introduce the main concepts of the spatial discretization by finite elements methods as well as basic tools from the numerical analysis of ordinary differential equations. First, we briefly recall the wellposedness theory of specific nonlinear wave equations using both semigroup and energy techniques. In a second step, we transfer these (two) approaches to the spatial discretization and show how rigorous error bounds can be obtained. We finally consider the fully discrete case where we combine both the spatial and temporal discretization to establish error bounds also for this case.

In the iRTG lecture series members or PostDocs 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 Roland Griesmaier and Roland Schnaubelt.

Roland Griesmaier: Acoustic Scattering in the Time Domain

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

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.

Roland Maier: Efficient simulation of wave phenomena in highly heterogeneous media

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

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.

Roland Schnaubelt: Strichartz estimates for wave equations

The lectures take place on January 08, January 15, and January 22.

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.

Thomas Bohlen: Wave phenomena in geophysics

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

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.

In the iRTG lecture series members or PostDocs 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 Roland Griesmaier and Roland Schnaubelt.

Lukas Pieronek: The interior transmission eigenvalue problem

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

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.

Lucrezia Cossetti: Spectral stability for perturbed operators

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

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.

Fatima Goffi: Homogenization of optical metamaterials from two perspectives

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

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.

Stefan Schrammer: Dynamical low-rank integrators for matrix differential equations

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

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.

In the iRTG lecture series members or PostDocs 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 Roland Griesmaier and Roland Schnaubelt.

Barbara Verfürth: Localized Orthogonal Decomposition for Helmholtz problems

The lectures take place on November 8, November 15, and November 22.

In this series of lectures, the Localized Orthogonal Decomposition (LOD), a computational multiscale method, will be introduced and analyzed for the Helmholtz equation with heterogeneous coefficients and large wavenumber. The numerical simulation of this problem using standard discretization schemes (e.g., the finite element method) would require a very fine mesh resolution because of (i) the heterogeneity of the coefficients and (ii) the high-frequency regime. The LOD is based on splitting the solution space into a finescale component, characterized as the kernel of an interpolation operator, and an „orthogonal“ multiscale space. Using this (low-dimensional) multiscale space in a Galerkin method yields much better approximations than the finite element method on an a comparable coarse mesh.
We will first introduce the Helmholtz problem under consideration and collect some important analytical and numerical results. Then, we will introduce the LOD in detail and rigorously analyze its discretization error. In particular, we will explain how and why the results differ from those for the finite element method. If time permits, we will finally give an outlook on how the methodology can be extended to high-contrast coefficients and/or nonlinear problems.

Tilo Arens: Boundary Integral Equation Methods for Time-Harmonic Scattering Problems

The lectures take place on November 29, December 12, and December 13.

We consider the formulation of boundary integral equations for time-harmonic scattering problems in acoustics and electromagnetics, i.e. for the Helmholtz equation and the Maxwell system. Starting from the definitions and mapping properties of potentials and boundary operators in appropriate Sobolev spaces, we discuss existence and uniqueness of solution results for selected boundary integral equations. For the numerical solution of boundary integral equations of the first kind, operator preconditioning techniques derived from the Calderon projector are particularly useful and will be presented.

Xian Liao: Conserved energies for the one dimensional Gross-Pitaevskii equation

The lectures take place on December 20, January 10, and January 17.

In this minicourse I will present a family of conserved energies for the one dimensional Gross-Pitaevskii equation, which is the defocusing cubic nonlinear Schrödinger equation but with nonzero boundary condition at infinity. I will speak more precisely about

- The (generalized) energy spaces, in the first lecture,
- The Lax pair structure and the transmission coefficient, in the second lecture,
- The conserved energies and the conservation of the energy norms, in the last lecture.

Dorothee Frey: Wave packet analysis

The lectures take place on January 24, January 31, and February 7.

We give an introduction into aspects of wave packet analysis with applications to the well-posedness of wave equations with low regularity coefficients. In dispersive PDEs, the wave packet transform of Córdoba-Fefferman serves as a suitable replacement and refinement of Littlewood-Paley theory, realising a dyadic-parabolic decomposition in phase space. We will investigate how this can be used in the study of the wave equation.

Topics include wave packets and their interaction with Fourier integral operators, Hardy spaces as function spaces built from wave packets, Sobolev embeddings, and well-posedness results of wave equations on Hardy and $L^p$ spaces.

In the iRTG lecture series members or PostDocs 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 Roland Griesmaier and Roland Schnaubelt.

Tobias Jahnke: High-frequency nonlinear optics

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

We consider a class of semilinear hyperbolic systems which includes the Maxwell-Lorentz equations in a dispersive medium as one special case. Both the evolution equation and the initial data involve a factor with a small parameter . As a consequence, typical solutions oscillate with frequency in time and space and have to be computed on time intervals of length to observe physically interesting effects. Approximating such solutions numerically is a very challenging task.

In the three lectures, we will focus on the analytical instead of the numerical approximation. Following the classical approach, we derive a nonlinear Schrödinger equation with a non-oscillatory solution which allows to approximate the solution of the original problem up to an error of . This will be discussed in detail. Finally, an approach for the construction of numerical methods for high-frequency nonlinear optics will be sketched.

Roland Griesmaier: Inverse source problems

The lectures take place on November 30, December 07, and December 14.

In this lecture series we discuss theoretical aspects and numerical reconstruction algorithms for the inverse source problem for acoustic and electromagnetic waves. This is a classical inverse problem which is well-known to be underdetermined and notoriously instable.

After briefly introducing the basic concepts of time-harmonic wave propagation and recalling some traditional regularization schemes for the inverse source problem, we will focus on more recent developments. We consider the convex source support, which is a novel solution concept for the inverse source problem. Then we discuss the inverse problems of wave splitting and data completion and show how these can be utilized in the inverse source problem. Finally we present uncertainty principles for acoustic and electromagnetic waves that can be used to establish stability estimates for the aforementioned inverse problems.

Peer Kunstmann: Wave front sets and propagation of singularities

The lectures take place on December 21, January 11, and January 18.

A description is missing.

Ivan Fernandez-Corbaton: An alternative starting point for electromagnetism

The lectures take place on January 25, February 01, and February 08.

The aim of these lectures is to provide a simple introduction to an unconventional approach to electromagnetism. From the start, the prominent role of the electric and magnetic fields is taken over by two other fields. These fields represent the two handedness that Maxwell solutions can have. The use of this alternative set of fields has notable advantages over the electric and magnetic fields: Decoupled evolution equations, relativistic invariance, and the remarkable ability to split the two possible handedness in electromagnetism.

In the iRTG lecture series members or PostDocs 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 Roland Griesmaier and Roland Schnaubelt.

Ivan Fernandez-Corbaton: An alternative starting point for electromagnetism

The lectures take place on April 20, April 27, and May 4.

The aim of these lectures is to provide a simple introduction to an unconventional approach to electromagnetism. From the start, the prominent role of the electric and magnetic fields is taken over by two other fields. These fields represent the two handedness that Maxwell solutions can have. The use of this alternative set of fields has notable advantages over the electric and magnetic fields: Decoupled evolution equations, relativistic invariance, and the remarkable ability to split the two possible handedness in electromagnetism.

Tobias Jahnke: High-frequency nonlinear optics

The lectures take place on Mai 11, Mai 18, and Mai 25.

We consider a class of semilinear hyperbolic systems which includes the Maxwell-Lorentz equations in a dispersive medium as one special case. Both the evolution equation and the initial data involve a factor $1/\epsilon$ with a small parameter $\epsilon$. As a consequence, typical solutions oscillate with frequency $O(1/\epsilon)$ in time and space and have to be computed on time intervals of length $O(1/\epsilon)$ to observe physically interesting effects. Approximating such solutions numerically is a very challenging task.

In the three lectures, we will focus on the analytical instead of the numerical approximation. Following the classical approach, we derive a nonlinear Schrödinger equation with a non-oscillatory solution which allows to approximate the solution of the original problem up to an error of $O(\epsilon)$. This will be discussed in detail. Finally, an approach for the construction of numerical methods for high-frequency nonlinear optics will be sketched.

Peer Kunstmann: Wave front sets and propagation of singularities

The lectures take place on June 8, June 15, and June 22.

A description is missing.

Roland Griesmaier: Inverse source problems

The lectures take place on June 29, July 6, and July 13.

In this lecture series we discuss theoretical aspects and numerical reconstruction algorithms for the inverse source problem for acoustic and electromagnetic waves. This is a classical inverse problem which is well-known to be underdetermined and notoriously instable.

After briefly introducing the basic concepts of time-harmonic wave propagation and recalling some traditional regularization schemes for the inverse source problem, we will focus on more recent developments. We consider the convex source support, which is a novel solution concept for the inverse source problem. Then we discuss the inverse problems of wave splitting and data completion and show how these can be utilized in the inverse source problem. Finally we present uncertainty principles for acoustic and electromagnetic waves that can be used to establish stability estimates for the aforementioned inverse problems.

In the iRTG lecture series members or PostDocs 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.

Axel Loewe: Cardiac Depolarization Waves in the Bidomain Setting

The lectures take place on April 29, May 6, and May 13.

In this series of lectures, we will explore and discuss cardiac depolarization waves as described by bidomain theory. The series is structured according to the multiscale nature of the problem: starting from the ion channel level, we will work our way up to the cellular and eventually tissue scale.

1) Ion Channel Electrophysiology
We will first introduce the basic biophysical principles governing cardiac electrophysiology on the ion channel level. Starting from the Nernst equilibrium and the gating behavior of Hodgkin-Huxley type ion channels, we will develop suitable models to describe current flow through ion channels in the cell membrane. Aspects of these models will be implemented in Matlab.

2) Cellular Electrophysiology
Based on the concept of ion channel gating, we will move to more comprehensive descriptions of the membrane and intracellular calcium cycling. The interplay of different ionic currents and their coupling via the transmembrane voltage and the intracellular ion concentrations will be considered. The focus will be on aspects which are crucial for the propagation of excitation waves and arrhythmogenesis such as the initiation of action potentials and refractoriness. The resulting system of coupled ODEs will be solved numerically using Matlab.

3) Excitation Propagation - Cardiac Depolarization Waves
We will employ bidomain theory to couple a set of cardiac myocytes to a syncitial tissue. Starting with cable theory in the 1D setting, we will model the spatial distribution of the potential in the intracellular and extracellular domains and the transmembrane currents between them. This concept will be extended to a multidimensional setting yielding the bidomain reaction-diffusion PDEs, which will allow us to study depolarization waves in physiological and arrhythmic scenarios.

Martin Spitz: Quasilinear wave equations with small data

The lectures take place on May 20, May 27, and June 3.

In this series of lectures we will investigate the long-time behavior of quasilinear wave equations with small initial data.

1) Energy estimates and local wellposedness
In the first lecture we will recall the basic wellposedness results for the linear wave equation. Based on the corresponding energy estimates, we then prove the local wellposedness of the nonlinear problem.

2) Invariant vector fields
We introduce the invariant vector fields and show the Klainerman-Sobolev inequality. We also discuss the significance of the latter for the long-time behavior of quasilinear wave equations.

3) Global and almost global existence
We prove global existence for solutions of quasilinear wave equations with small intitial data in higher spatial dimensions ($d \geq 4$). In the other dimensions, we provide lower bounds for the lifespan of the solutions. In particular, we will see that in $d = 3$ we obtain almost global ones. If time permits, we will also discuss the null condition and global existence in this dimension.

Ruming Zhang: Scattering Problems in Periodic Domains

The lectures take place on June 17, June 24, and July 1.

In this lecture series we will consider the scattering problems in periodic domains.

1) Quasi-periodic scattering problems.
The first lecture introduces the mathematical formulation and regularity results for the quasi-periodic scattering problems. Both the integral equation and the variation formulation will be introduced, and applied to the investigation of the scattering problems.

2) The Floquet-Bloch transform and its application to scattering problems
In the second lecture, first we introduce an important tool, the Floquet-Bloch transform, and some of its important properties. Then we presents its application to the scattering problems with (locally perturbed) periodic structures. We also study the regularity of the transformed scattering problems.

3) Numerical analysis for Floquet-Bloch transform based method
In the third lecture, we consider the numerical solutions of the scattering problems with (locally perturbed) periodic structures, based on the Floquet-Bloch transform. We present the Galerkin discretization of the transformed problem, and then consider the convergence results numerical method.

References:
A. Kirsch, Diffraction by periodic structures, Lecture Notes in Physics 422, Springer, 1993.
A. Lechleiter & R. Zhang, A Floquet-Bloch transform based numerical method for scattering from locally perturbed periodic surfaces. SIAM J. Sci. Comput., 39(5), B819 – B839, 2017.

Jonas Köhler: Error analysis of full discretizations of wave-type problems

The lectures take place on July 8, July 11, and July 15. On Thursday, July 11, the lecture takes place in SR 1.067.

In this series of lectures we investigate the error of full discretizations of linear wave-type problems. In particular, we apply a discontinuous Galerkin method (dG) in space and either the Crank-Nicolson or the leapfrog (or Verlet) method in time.

1) Framework: Friedrichs' operators.
We consider linear wave-type problems of the form $\partial_t u = \mathcal{L} u + f$, where $\mathcal{L}$ is a first-order spatial differential operator belonging to the class of Friedrichs' operators. Well-posedness of such problems (supplied with suitable initial and boundary conditions) is shown by proving that $\mathcal{L}$ is maximal dissipative and therefore generates a strongly continuous semigroup by the Lumer-Phillips theorem.

2) Spatial and temporal discretization.
Next, we discretize the Friedrichs' operator $\mathcal{L}$ using a central fluxes dG method and provide some basic properties of the resulting discrete operator. To discretize in time we consider the Crank-Nicolson and the leapfrog scheme applied to the spatially discrete wave-type equation.

3) Error analysis.
Finally, we analyze the error of the fully discretized schemes obtained in 2). With $k$ denoting the polynomial degree used in the dG method, we show that the discrete solution converges to the exact solution with order $k$ in space and order two in time for both schemes if the exact solution is sufficiently smooth.

In the iRTG lecture series members or PostDocs 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.

In the iRTG lecture series members or PostDocs 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.

Carsten Rockstuhl

The lectures take place on 24 April, 8 May, and 15 May.

As a physicist in the SFB, the lectures I will concentrate on applied aspects of electromagnetic waves; but of course explored on mathematical grounds. Specifically, I will discuss the interaction of light with matter that has critical features at the nanoscale. I will emphasise the importance of exploiting resonances to notably enhance the optical response. These resonances lay the ground for various applications that are equally sketched. Three lectures are given on the following subjects.

In a first lecture, the notion of plasmonics is introduced and particularly localized surface plasmon polaritons are discussed. The latter being hybrid states of excitation where the electromagnetic field is coupled to the charge density oscillation in metals that are spatially confined to form nanoparticles. In the first lecture we concentrate on spherical particles for which an analytical solution to the scattering problem exists. Once a localized surface plasmon polariton is excited, a huge field enhancement close to the nanoparticle and a large scattering and absorption cross section in a narrow spectral region are observed.

In a second lecture, the further developments are discussed when not just spherical particles are considered but more complexly shaped objects. This is the wide field of optical nanoantennas. It eventually exploits the opportunity to induce a response in the nanoparticle that has different contributions in terms of electric and magnetic multipole moments. These multipole moments can be individually considered in the context of, e.g. meta-atoms as the basic building blocks of metamaterials. However, the interference among multiple multipole moments is equally important as it provides excellent control on the radiation characteristics of these nanoantennas.

In a third lecture, we discuss the possibility to observe many of the effects studied in plasmonics, that rely on metallic nanostructures, with dielectric structures only. This approach has the clear advantage that absorption is heavily suppressed at the expenses of a lower confinement of electromagnetic fields. However, using high-permittivity materials such as semiconductors, many interesting effects can be equally observed.

References.

• Stefan Maier: Plasmonics. Fundamentals and Applications.
• Lukas Novotny and Bert Hecht: Principles of Nano-Optics.
• Craig F. Bohren and Donald R. Huffman: Absorption and Scattering of Light by Small Particles.

Willy Dörfler

The lectures take place on 22 May, 29 May, and 12 June.

The lecture addresses the following two relevant topics for finite element computations: Is there a theoretical or practical reliable measure to estimate the error of the approximated solution? Can error estimation be used to significantly diminish the computational effort in an optimal way? The main techniques to control the approximation are local mesh-refinement, the (local) polynomial degree of the finite elements and the timestep size for time-dependent problems. We present an overview of main results for elliptic and parabolic equations.

References.

• M. Ainsworth and J. T. Oden: A Posteriori Error Estimation in Finite Element Analysis. John Wiley, New York, 2000.
• C. Schwab: p- and hp-finite element methods. Theory and applications in solid and fluid mechanics. Clarendon Press, Oxford, 1998.

Andreas Kirsch

The lectures take place on 19 June, 26 June, and 3 July.

A) Introduction and Basic Linear Theory
After an introduction into the notions of an inverse problem and the ill-posedness of a problem we will consider some examples, in particular formulated as integral equations of the first kind, and introduce the general regularization technique with filter functions.

(B) Particular Regularization Strategies
In this part we will study the classical Tikhonov regularization technique with a priori and a posteriori choices of the regularization parameter. Then we will introduce the Landweber method as an example of an iterative regularization strategy.

(C) The Problem of Impedance Tomography
This is an example of a nonlinear inverse problem. After a short repetition of the direct problem - an elliptic boundary value problem - We will discuss the Factorization Method to determine the support of the contrast and comment on other (iterative) techniques.

References. We will essentially follow the monograph:

• A. Kirsch: An Introduction to the Mathematical Theory of Inverse Problems (2nd Edition). Springer, 2011.

Jens Rottmann-Matthes

The lectures take place on 7 July, 14 July, and 21 July.

In this series of lectures we will consider patterns in equivariant evolution equations and their stability properties.

I) Equivariant evolution equations
We begin by introducing the abstract notion of equivariant evolution equations. Equivariance plays an important role for the appearance of patterns like traveling and rotating waves. We look at concrete examples which exhibit traveling or rotating waves and also generalize to other symmetry solutions.

II) Stability of traveling waves
We present a method of how to capture traveling waves and other patterns in equivariant evolution equations numerically. Then we show that spectral stability implies nonlinear stability of traveling waves in first order hyperbolic systems of partial differential equations.

III) The case of second order evolution equations
In the last lecture we will consider systems of semilinear wave equations. We show how the approach of capturing traveling and rotating waves generalizes to this case. Furthermore we present a stability traveling waves result and look at the spectrum of the linearized operators.

References.

• W.-J. Beyn, D. Otten, J. Rottmann-Matthes: Stability and Computation of Dynamic Patterns in PDEs.
• W.-J. Beyn, D. Otten, J. Rottmann-Matthes: Freezing traveling and rotating waves in second order evolution equations
• J. Rottmann-Matthes: Stability and freezing of nonlinear waves in first order hyperbolic PDEs.