CRC seminar

Abstract:
Pattern-forming systems admitting a conservation law structure occur naturally in hydrodynamic stability problems with a free surface. Examples are the Bénard-Marangoni problem, which models a free-surface fluid on a heated bottom surface, and the flow down an inclined plane. As a parameter increases, the homogeneous ground state of the system destabilizes and spatially periodic traveling waves bifurcate. These patterns often arise in the wake of an invading heteroclinic front which connects the unstable ground state to the periodic traveling wave.
In this talk, I outline how this invasion process can be mathematically described by constructing so-called modulating traveling front solutions and present recent results for two model problems.

Abstract:
For the first time, a method has recently become available for fast computation of near-best rational approximations on arbitrary sets in the real line or complex plane: the AAA algorithm (Nakatsukasa-Sete-T. 2018). We will present the algorithm and demonstrate a number of applications, including a subset of
* detection of singularities
* model order reduction
* analytic continuation
* functions of matrices
* nonlinear eigenvalue problems
* interpolation of equispaced data
* smooth extension of multivariate real functions
* extrapolation of ODE and PDE solutions into the complex plane
* solution of Laplace problems
* conformal mapping
* Wiener-Hopf factorization
(Joint work with Stefano Costa and others)

Abstract:
In water limited regions, it has been observed that vegetation can self-organize into patterns, such as spots and stripes. While much is known analytically about large amplitude vegetation stripe patterns, less is known about far-from-onset radially symmetric solutions such as spots and gaps. We study a 2-component reaction-diffusion dryland ecosystem model and describe the construction of radially symmetric spot and gap solutions using singular perturbation methods. The solutions are constructed as slow/fast heteroclinic orbits in a radial spatial coordinate. We draw connections between the construction of spots/gaps and that of planar bistable fronts between the bare soil and vegetated states, which also provide insight into 2D (in)stability mechanisms for large spots.

Abstract:
Front propagation into unstable states plays an important role in organizing structure formation in many spatially extended systems. When a trivial background state is pointwise unstable, localized perturbations typically grow and spread with a selected speed, leaving behind a selected state in their wake. A fundamental question of great interest is to predict the propagation speed and the state selected in the wake. The marginal stability conjecture postulates that speeds can be universally predicted via a marginal spectral stability criterion. In this talk, we will present background on the marginal stability conjecture and present some ideas of our recent conceptual proof of the conjecture in a model-independent framework focusing on systems of parabolic equations.

Abstract:
One of the challenges for graphene is to modify the sample in order to turn it into a semiconductor, that is, open a gap at zero, the Fermi energy. In this talk, I will present some quantum Hamiltonians based on Dirac operators to model sample of graphenes with periodic holes, and we shall give the mathematical results obtained in the analysis of the gap opening in its spectrum near zero.

Abstract:
Order reduction, i.e., the convergence of the solution at a lower rate than the formal order of the chosen time-stepping scheme, is a fundamental challenge in stiff problems. Runge-Kutta schemes with high stage order provide a remedy, but unfortunately high stage order is incompatible with DIRK schemes. We first highlight the spatial manifestations of order reduction in PDE IBVPs. Then we introduce the concept of weak stage order, and (a) demonstrate how it overcomes order reduction in important linear PDE problems; and (b) how high-order DIRK schemes can be constructed that are devoid of order reduction.

Abstract:
The interaction between electromagnetic and elastodynamic vibrations has a long and distinguished history, dating from the work of Brillouin in the early 20th century. More recently researchers have begun to rediscover these interactions in the context of nanophotonics, in which light is trapped or guided within structures that possess features that are typically as large as the wavelength of light (and sound) in the material. These interactions can lead to several interesting and unusual effects, including "slow-light", by which the speed of light is reduced to a fraction of its value in vacuum. However at these small scales the mathematics of the different types of waves, and of the forces that cause them to interact, can become complicated. This task is complicated by the inclusion of noisy processes, which arise from thermal effects and transform the three-wave interaction into a set of coupled stochastic PDEs. Here we discuss the journey towards a comprehensive and accurate numerical description of light-sound interactions, and review recent progress in using these models for comparison with experiments.

Abstract:
We consider obstacles being metamaterials, which are artificial materials constituted of a periodic arrangement of subwavelength inclusions. The response of a metamaterial to an exiting electromagnetic wave is given through a nonlocal constitutive relation. We use this nonlocal constitutive relation to model the effective medium. Afterwards, we use the model for analyzing a particular case which is considered to be a slab waveguide.

Abstract:
Maxwell’s equations are a set of four coupled partial differential equations. They can explain all classical phenomena regarding the propagation of light and its interaction with matter. However, since microscopic equations, where the space-time dynamics of all charges and currents thereof are tracked, are impossible to solve, macroscopic equations are usually considered. In these macroscopic equations, the response of materials to an electric and magnetic field is introduced on phenomenological grounds that is, nevertheless, always motivated by physics-based considerations.
The purpose of this talk is to elaborate on and derive a larger set of possible constitutive relations that can be applied to specific situations. Depending on time, we will consider linear and nonlinear, local and nonlocal, isotropic and anisotropic, dispersive and frequency independent material laws. Finally, a glimpse is given on more advanced constitutive relations as they occur in classes of materials of contemporary research interests, e.g., metamaterials or materials with time-varying properties.
The purpose of the talk is to trigger an exchange among mathematicians and physicists to work towards a joint base on how to formulate material laws.

Abstract:
We prove resolvent estimates for time-harmonic Maxwell equations in $L^p$-spaces with pointwise, spatially homogeneous, and possibly anisotropic material laws. The resolvent estimates allow for the proofs of Limiting Absorption Principles and construction of solutions. In the fully anisotropic case, which is joint work with Rainer Mandel, the construction relies on new Bochner-Riesz estimates with negative index for non-elliptic surfaces.
The talk is based on the preprints arXiv:2103.16951 and arXiv:2103.17176.

Abstract:
The full dispersion Kadomtsev-Petviashvili (FDKP) equation $$ u_t + m(D)u_x + 2uu_x =0, $$ where $$ m(D) = (1+\beta|D|^2)^{\frac{1}{2}}\left(\frac{\tanh(|D|)}{|D|}\right)^{\frac{1}{2}}\left(1+\frac{2D_2^2}{D_1^2}\right)^{\frac{1}{2}}, $$ and $D = −i(\partial x, \partial y)$, is a model equation describing three-dimensional long waves of small amplitude. The FDKP equation is a fully dispersive version of the classical KP equation, similar to how the Whitham equation is a fully dispersive version of the KdV equation. Recently Ehrnström and Groves (2018) proved existence of solitary wave solutions for the FDKP equation for strong surface tension $(\beta > 1/3)$ using a variational approach. The solitary waves they found can be approximated by rescalings of KP-solitary waves. In my talk I will consider the weak surface tension regime $(\beta < 1/3)$ and describe how to prove existence of FDKP-solitary waves that can be approximated by rescalings of Davey–Stewartson-solitary waves.
This talk is based on a joint work with Mats Ehrnström (Norwegian University of Science and Technology) and Mark Groves (Saarland University).

Abstract:
We consider the semiclassical magnetic Schrödinger equation, which describes the dynamics of particles under the influence of a magnetic field. The solution to the Schrödinger equation is approximated by Gaussian wave packets. Apart from error bounds in $L^2$ and observable error bounds with respect to the semiclassical parameter, we will see equations of motion for the parameters of the approximating Gaussian wave packet.

Abstract:
In this talk, I will discuss the numerical solution of the Maxwell–Schrodinger system equations in the Coulomb gauge. A Crank–Nicolson finite element scheme is proposed for solving the equations and the conservation of energy of the discrete equations is proved. The (almost) unconditional optimal error estimates are derived for the numerical scheme. Numerical experiments are provided to support the theoretical results.

Abstract:
Consider a family of continuity equations where the velocity field is given by the convolution of the solution with a regular kernel. In the singular limit where the convolution kernel is replaced by a Dirac delta, one formally recovers a conservation law: can we rigorously justify this formal limit? This question was posed by P. Amorim, R. Colombo and A. Teixeira and a positive answer was suggested by numerical simulations. In the talk we will exhibit counterexamples showing that in general convergence of the solutions does not hold. We will also show that the answer to the above question is positive if we consider viscous perturbations of the nonlocal equations. In this case, in the singular local limit the solutions converge to the solution of the viscous conservation law. We will also discuss the possible role of numerical viscosity in numerical simulations, as well as some more recent results dealing with the case of an anisotropic convolution kernel, a more realistic case for applications to traffic modelling.
The talk will be based on some joint works with Maria Colombo, Marie Graff, Elio Marconi, and Laura Spinolo.

Abstract:
In this talk, we discuss the construction and analysis of higher-order time integration schemes for the full discretization of linear Maxwell equations on locally refined spatial grids. While there is a rigorous analysis of locally implicit methods of order two for linear problems, it is not clear how to prove the stability for higher-order methods.
We thus follow a different approach. The idea is to use the numerical linear algebra part of the integrator to exploit the fact that the stiffness of the problem only stems from a few tiny mesh elements. We suggest do use a Krylov subspace method with preconditioning, which – roughly speaking – consists of solving the problem on the fine part of the mesh with a direct solver. The advantage of this approach is that it is applicable to any implicit scheme and also works for exponential integrators. It is even interesting for nonlinear problem, where such linear systems arise within the Newton iterations.
We will prove that the convergence of the preconditioned quasi-minimal-residual method is independent of size of the small mesh elements. For the sake of presentation, we focus on the fourth order Gauss–Legendre Runge–Kutta method and verify our theoretical results with numerical experiments.

Abstract:
To capture accurately the properties of metamaterials at the effective level, nonlocal constitutive relations have shown recently to be more suitable when compared to the constitutive relations that are local. Under the assumption that the metamaterial is made from nonmagnetic constituents, the nonlocal constitutive relation is written in its most comprehensive form as a convolution between a nonlocal response function and the exciting electric field. Even though the nonlocal response function provides the exact description, dealing with it for applications remains limited. Therefore, the use of an approximated response function, followed with the interface conditions describing the behaviour on an interface, simplifies much better the task. As a real application, we analyse the propagation of light in cylindrical waveguides made of metamaterials. We formulate the system of equations in the cylindrical coordinates and look for special solutions for defining the guided modes.

Abstract:
In this talk we consider second-order time integration schemes applied to linear wave-type problems in first order formulation and provide error and stability bounds in strong energy norms. In particular, this means that we identify suitable discrete objects obtained from the approximations given by the scheme that provide approximations to the first derivatives of the exact solution. We then show that they converge to the continuous objects with the same rates as the original approximation if we assume a bit more regularity.
For the sake of presentation, in this talk we restrict ourselves to the Crank-Nicolson scheme applied to the wave-equation in div-grad-formulation. However, we point out that the analysis can be easily generalized to more general wave-type problems and to other second-order time-integration schemes like the Leapfrog/Verlet scheme or the Peaceman-Rachford splitting scheme (where the generalization to the latter is not as easy anymore).

Abstract:
In this talk, we will present some numerical methods to simulate wave propagation in periodic layers. This problem is modelled by the Helmholtz equation with the upward propagation radiation condition. As is well known, this problem is not always unique solvable. Thus we adopt the limiting absorption principle to get the solution which coincides with the wave in the physical world. In this talk, we first approximate the problem by a new problem in a closed periodic waveguide, and then get an explicate solution for the closed waveguide problem with the help of the Floquet-Bloch transform. The numerical methods are developed based on the formulations for the closed waveguide problems.

Abstract:
Full-waveform inversion (FWI) is becoming a popular geophysical technique to characterize the physical properties of subsurface models on a broad range of scales. There are several problems that FWI faces, such as high computational cost, high ill-posedness, and lack of uncertainty information. We developed a random-objective waveform inversion (ROWI) method to address these problems. In ROWI, we build multi-objective functions and use a stochastic gradient descent algorithm for optimization. Three different measure functions: the least-squares waveform misfit, the f-k spectra misfit, and the envelope misfit, are used in the multi-objective functions to account for the properties of the shallow-seismic wavefield. The combination of every single shot and every single measure function formulates one of the multi-objective functions independently. This multi-objective framework reduces the ill-posedness of the inverse problem. Additionally, we can estimate the uncertainty information of the model parameters by analyzing the distribution of Pareto optimal solutions in the model space. In the optimization, we randomly choose and treat only one of the multi-objective functions per iteration. Therefore, we avoid using redundant data during the iteration to improve computational efficiency.

Abstract:
2D full-waveform inversion (FWI) has become a powerful method for reconstructing shallow subsurface with high resolution. Despite the recovery of the S-wave velocity model, multiparameter models nowadays are of interest in geophysical and geotechnical site investigation. However, the multiparameter reconstruction in FWI is challenging due to the potential presence of crosstalk between different parameters and the unbalanced sensitivity of Rayleigh-wave data with respect to different parameter classes. For multiparameter reconstruction, accounting for the inverse Hessian during the inversion has been proven as an effective tool to mitigate the crosstalks caused by the coupling between different parameters. Truncated Newton method provides a nice way to account for the Hessian operator. It computes the Hessian-vector product based on second-order adjoint state method. We apply a preconditioned truncated Newton method (PTN) to shallow-seismic FWI to simultaneously invert for multiparameters for near-surface models (P- and S-wave velocities, attenuation of P and S wave, and density). Spatially correlated and uncorrelated models demonstrate that PTN improves the accuracy and resolution of the reconstructed results.

Abstract:
Optical frequency comb is an optical signal consisting of hundreds or even thousands of highly stable, narrow spectral, and equidistantly spaced optical tones, which enables the coherent phase link between the RF and optical domains. It revolutionized the optical frequency metrology, spectroscopy, optical frequency synthesizer, and optical atomic clocks. Recently, with the rapid development of chip-scale Kerr soliton comb, it offers the prospect of realizing optical frequency comb devices with the advantages of highly compact, portable, robust, mass produced and low power consumption, which has great potential for the large-scale use both in scientific and commercial applications. In this talk, I will show the synchronization of a RF signal with the repetition-rate of the Kerr soliton comb via injection locking. The theoretical models, numerical and analytical analysis will be presented./div>

Abstract:
We study the space discretization of nonautonomous wave equations in materials, which depend smoothly on time but are rough in space. In particular, this implies that a space discretization using a standard finite element approach leads to an unreasonably high computational effort. Thus, we apply a numerical homogenization scheme in the spirit of the localized orthogonal decomposition (LOD).
In this talk, we give a brief introduction to the LOD and discuss the application to nonautonomous wave equations. Finally, we prove convergence of the spatially discrete scheme based only on regularity assumptions on the data.

Abstract:
FWI entails the full nonlinear inverse problem of seismic imaging where one wants to reconstruct the interior structure of earth (parameters) from recordings of reflected waves. Due to ill-posedness, regularization needs to be applied, whose general theory is so far only established for Hilbert spaces and for smooth and reflexive Banach spaces. Since parameters are naturally assigned to $L^{\infty}$, however, customized solution strategies are barely known. In this talk we try to exhibit the underlying mathematical difficulties and present possible approaches for a convergence theory in $L^{\infty}$.

Abstract:
In this talk I will present a joint work with P. Idzik and W. Reichel on the existence of two-dimensional traveling waves for a quasilinear equation appearing in the context of electromagnetic waves. The traveling wave solution is localized in one and periodic in the complementary horizontal direction. Using a Fourier decomposition, the problem can essentially be reduced to studying spectral properties of a family of Schrödinger type operators. Eventually, we apply bifurcation theory to obtain the existence of nontrivial solutions.

Abstract:
We discuss the question whether the Galerkin discretization of the Helmholtz problem with impedance boundary conditions can lead to a singular system matrix. We show that in 1d the discretization is always stable independent of the mesh or polynomial degree. On the other hand, we present an example of a mesh in 2d where the resulting matrix is singular for some wave number. We introduce an algorithm that checks if a triangular mesh is stable and propose a strategy to fix stability if the mesh is unstable. The algorithm depends on the graph of the mesh, in particular it is independent of the mesh width, and therefore is an alternative to the Schatz argument to prove well-posedness of the discrete problem.

Abstract:
The electromagnetic helicity of the free electromagnetic field and the static magnetic helicity are shown to be two different embodiments of the same physical quantity, the total helicity. The total helicity is the sum of two terms that measure the difference between the number of left-handed and right-handed photons of the free field, and the screwiness of the static magnetization density in matter, respectively. This unification provides the theoretical basis for studying the conversion between the two embodiments of total helicity upon light-matter interaction.

Abstract:
Quantifying uncertainties in hyperbolic equations comes with several challenges. First, the solution forms shocks which lead to oscillatory solution approximations. Second, the number of unknowns to represent the solution grows exponentially with the dimension of the uncertainty, yielding high numerical costs and memory requirements. These challenges can be tackled with an efficient solution representation such as adequate basis functions. The generalized polynomial chaos polynomials allow such an efficient representation when the distribution of the uncertainty is known. Since this distribution is usually only available for input uncertainties such as initial conditions, the efficiency of this ansatz is lost during runtime.
In this talk, we employ a dynamical low-rank approach to obtain an efficient solution representation on a low-rank manifold. By using the projector splitting integrator as well as the unconventional integrator, we derive time evolution equations for the spatial and uncertain basis functions. This guarantees an efficient solution approximation even if the underlying probability distribution changes over time. Moreover, the low-rank approach allows for an memory-efficient solution representation in the case of multi-dimensional uncertainties.

Abstract:
New Strichartz estimates for Maxwell's equations in two dimensions with rough permittivity are proved. We use the FBI transform to carry out the analysis in phase space. For this purpose, Maxwell's equations are conjugated to a system of half-wave equations with rough coefficients, for which Strichartz estimates are derived similarly as previously for wave equations with rough coefficients by Tataru. We use the estimates to improve the local well-posedness theory for quasilinear Maxwell equations in two dimensions. The talk is based on joint work with Roland Schnaubelt.

Abstract:
Isomerizations are chemical reactions where the reactant and the product have the same chemical formula but different chemical structure, i.e., spatial configuration. In this talk we are going to consider isomerization reaction between two heavy molecules described by pseudorelativistic Schrödinger operators. We will show how the problem of finding the activation energy of the reaction is a mountain pass problem between two local stable configurations, where one minimizes over all possible paths the highest value of the energy along the path. Finally, we will prove boundedness of the sequence of minimizing paths.
From a joint work with I. Anapolitanos and S. Zalczer.

Abstract:
Interchanging the role of space and time is widely used in nonlinear optics for modeling the evolution of light pulses in glass fibers. A phenomenological model for the mathematical description of light pulses in glass fibers with a periodic structure in this set-up is the so called dispersion management equation. It is the purpose of this lecture to answer the question whether the dispersion management equation or other modulation equations are more than phenomenological models in this situation.
Using Floquet theory we prove that in case of comparable wave lengths of the light and of the fiber periodicity the NLS equation and NLS-like modulation equations with constant coefficients can be derived and justified through error estimates under the assumption that rather strong stability and non-resonance conditions hold.
We explain that the failure of these conditions allows us to prove that these modulation equations in general make wrong predictions. The reasons for this failure and the behavior of the system for a fiber periodicity much larger than the wave length of light shows that interchanging the role of space and time for glass fibers with a periodic structure is at least a questionable modelling.

Abstract:
In this seminar we investigate the $L^p-L^q$ mapping properties of the resolvent associated with the time-harmonic isotropic Maxwell operator. As spectral parameters close to the spectrum are also covered by our analysis, we establish an $L^p-L^q$ type limiting absorption principle for this operator. Our analysis relies on new results for Helmholtz systems with zero order non-Hermitian perturbations.
The talk is based on a joint work with R. Mandel.

Abstract:
Time integration of high-dimensional problems, arising from the discretization of PDEs such as the Vlasov-Poisson equation of plasma physics and Schroedinger equation in many-body quantum mechanics, is a challenging numerical task: the total amount of information required to be stored and computed exceeds standard computational capacity. Time dependent model order reduction techniques are desirable. In the present talk, dynamical low-rank approximation for matrices together with the matrix and the Tucker projector splitting integrator is introduced. Two remarkable properties of these integrators are presented, namely: the exactness property and the robustness with respect to small singular values.
Then, the memory storage requirements for tensors in Tucker format are discussed and Tree Tensor Networks are introduced. Based upon a new compact formulation of the Tucker projector splitting integrator, the Recursive Tree Tensor Network integrator is presented.
This is joint work with Ch. Lubich and H. Walach.

Abstract:
From an experimental point of view it is quite attractive to study the generation of Kerr frequency combs by pumping multiple modes. As an example, such techniques allow to excite bright square pulses in normal dispersion resonators with small waveguide cross-sections, which could feature significantly improved power conversion efficiency. We discuss a new variant of the Lugiato-Lefever equation which corresponds to the situation where two modes are pumped. We will show how a-priori bounds and degree theory can be used to obtain existence results. We will also demonstrate some numerical results which fit with our analytic results.

Abstract:
Appropriate models of the human heart involve multiple physiological systems, ranging from intracellular ion exchange over electric potential diffusion to hyperelastic deformation. This deformation follows the rules of nonlinear elastodynamics and requires passive material laws suited for finite elasticity.
In this talk, we introduce the challenges of cardiac simulations coupling the ordinary and partial differential equations involved in physiological models. We focus on solution methods for the mechanical contraction which is described by a nonlinear Cauchy-Navier-equation with an additional internal active strain and discuss the discretization with finite elements and their application in cardiac modeling.

Abstract:
We use a biharmonic (elastic plate-type) operator in order to introduce a rigid version of the wave maps equation, a geometric wave equation that has been studied intensively for its subtle interplay of gauge invariance, nonlinear dispersion and singularity formation. In the talk, we will briefly explain a local wellposedness result for biharmonic wave maps with bounded initial velocities. This is obtained by a limit of viscosity solutions with improved a priori energy estimates in joint work with S. Herr, T. Lamm and R. Schnaubelt. Similar arguments in dimension d = 1,2 combined with energy conservation lead to global regularity of the Cauchy problem (with smooth initial data). In the main part of the talk, we explain a recent result of global regularity for biharmonic wave maps into spheres with small initial data in the scaling critical Besov space (in high dimensions). This result hinges on the non-generic quadratic nonlinearity, which is exploited similarly as in Tataru's solution of the decomposition problem for wave maps. Compared to the first part of the talk, the proof relies on Fourier restriction methods and lateral Strichartz estimates. If time permits, we will briefly discuss ongoing joint work with B. Schörkhuber on the dynamics of type II blow up solutions of an energy critical focusing biharmonic wave equation, following an idea of Krieger, Schlag and Tataru.

Abstract:
We consider a nonlinear Klein-Gordon system on a discrete necklace graph. On the one hand it is known that solutions with symmetric initial conditions with respect to the periodic branching of the discrete graph decay like the linear problem. On the other hand we know that strongly localized breather solutions exist which are not symmetric with respect to the periodic branching of the discrete graph if a certain nonresonance condition is fulfilled. Our goal is to establish asymptotic stability of the vacuum state in a weighted $\ell^2$ norm for even nonlinearities if this nonresonance condition is not met. This is a work in progress.

Abstract:
We consider an elliptic partial differential equation with a random diffusion parameter discretized by a stochastic collocation method in the parameter domain and a finite element method in the spatial domain. We prove convergence of an adaptive algorithm which adaptively enriches the parameter space as well as refines the finite element meshes.

Abstract:
With the development of integrated photonics, optical waveguides are of particular interest, since they are connecting the circuit components. Waveguides of various shapes are required for integrated circuits design. However, with bending the transmitted information might have losses, so it is necessary to solve the following optimization problem: find a form of a waveguide bending such that energy losses are minimal.
In this talk we compare a 2D simulation of a thin waveguide with the derived asymptotic form. We discuss the observed features of the behavior of the reflected coefficient and usage of the 1D model to a 2D waveguide shape optimization.

Abstract:
In this talk we present our progress on the space-time discontinous Galerkin method for wave equations. Space-time methods are based on treating space and time simultaneously in a variational manner resulting in a fully discretized system of linear equations. Since our future aim is to solve three dimensional problems (in space), hence four dimensions in space-time, the performance and memory consumption have to be improved. To increase the performance to of solving the linear system, we consider different techniques of multilevel-preconditioning. This will be shown in visco acoustic-examples with application in seismic imaging.

Abstract:
Multilevel Monte Carlo methods have gained much popularity in recent years. Their ability to estimate expectation values on high dimensional probability spaces have made them the method of choice for computing expected solutions of PDEs with random coefficients. Furthermore, their non-intrusive implementation schemes have allowed them to build on existing theory and software.
In this talk, we present a short overview on the method, the used heuristics and present our applications on elliptic, parabolic and hyperbolic PDEs with different sources of randomness. We want to finish the talk by presenting numerical results and giving an outlook on further applications for wave propagation and nonlinear parabolic PDEs.

Abstract:
A starting point for every heartbeat is an electrical signal propagating through the heart and triggering the contraction for pumping the blood through the body. The monodomain equation is a model to simulate this signal mathematically. It is a reaction-diffusion equation consisting of a PDE coupled to a complex ODE system representing the mechanisms on the cell level. In this talk, we show some analytical results from the literature and present our numerical experiments. On the cell level we explain the physiology and introduce different models used for the ODE system. An operator splitting method for solving the monodomain equations is presented. We show convergence results numerically and illustrate the problems coming with the stiffness of the ODE part.

Abstract:
In this talk we discuss fixed-time $L^p$ estimates and Strichartz estimates for wave equations with low regularity coefficients. It was shown by Smith and Tataru that wave equations with $C^{1,1}$ coefficients satisfy the same Strichartz estimates as the unperturbed wave equation on $\mathbb{R}^n$, and that for less regular coefficients a loss of derivatives in the data occurs. We considerably improve these results for a specific class of $C^{0,1}$ coefficients, for which we show that no loss of derivatives occurs at the level of fixed-time $L^p$ estimates. The permitted class in particular excludes singular focussing effects. We discuss possible generalisations of the results based on our new operator-theoretic approach. This is joint work with P. Portal (ANU).

Abstract:
Nonlinear effects can be observed easily in sound waves with sufficiently large amplitudes. The nonlinearity will be apparent even sooner in high-frequency waves because these effects accumulate over the distance measured in wavelengths. This makes ultrasonic waves inherently nonlinear. Their many applications range from non-invasive surgery to industrial welding and motivate the mathematical investigation into nonlinear acoustics.
In this talk, we will give a brief overview of challenges arising in the mathematics of nonlinear sound waves and then further focus on the analysis of finite-element based discretizations in this context. In particular, we will discuss a high-order discontinuous Galerkin discretization in space of the Westervelt wave equation in second-order form. Westervelt’s equation is a strongly damped wave equation that models sound propagation through dissipative fluids and accounts for nonlinearities of quadratic type. The general approach in the a priori error analysis combines the stability and convergence analysis of its linearization with the Banach fixed-point theorem. Numerical experiments will illustrate the theoretical results.
The talk is based on joint research with Paola F. Antonietti, Ilario Mazzieri (Politecnico di Milano), Markus Muhr, and Barbara Wohlmuth (TU Munich).

Abstract:
We consider an inverse obstacle scattering problem for the Helmholtz equation with obstacles that carry mixed Dirichlet and Neumann boundary conditions. We discuss far field operators that map superpositions of plane wave incident fields to far field patterns of scattered waves, and we derive monotonicity relations for the eigenvalues of suitable modifications of these operators. These monotonicity relations are then used to establish a novel characterization of the support of mixed obstacles in terms of the corresponding far field operators. We apply this characterization in reconstruction schemes for shape detection and object classification, and we present numerical results to illustrate our theoretical findings.

Abstract:
Electromagnetic chirality describes the scattering behavior of an object in three-dimensional free space corresponding to illuminations with fields of pure positive or negative helicity. If the scattering behavior of the object with respect to incident waves of one helicity cannot be reproduced with the other one, we define the scatterer to be electromagnetically chiral (em-chiral), else em-achiral. As maximal em-chiral we therefore refer to objects, which are invisible to incident fields of one helicity. In previous works of Fernandez-Corbaton et al. a scalar measure was introduced, which assigns a value between 0 and 1 to three-dimensional shapes.
Here, 0 stands for an achiral object, 1 on the other hand for a maximal em-chiral object. However, the construction of a maximal em-chiral object, i.e. an optimization of the scalar measure with respect to the scattering object, is a current research topic.
In this talk, we establish an optimization scheme based on an asymptotic perturbation formula for approximating scattered fields corresponding to Maxwell's equations. Due to this approximation, no partial differential equation has to be solved during the whole optimization process, what leads to an efficient and reliable shape optimization.
We first discuss the optimization problem and establish the asymptotic perturbation formula. Afterwards, we investigate the Fréchet derivative of the shape-to-chirality measure map and set up the optimization algorithm. In the final section we show numerical results of the algorithm and discuss the final scattering objects that the algorithm finds.

Abstract:
In this talk, we consider a preconditioner or solver for large linear systems arising from finite element discretizations of the time-harmonic Maxwell's equations. By replacing the boundary condition used to truncate auxiliary problems, we arrive at a method we can use recursively, reducing the size of the involved matrices until a direct solver can be applied. This scheme's properties follow from those of basic sweeping preconditioners and the numerical properties of Hardy space (or complex scaled) infinite elements.

Abstract:
In this talk we investigate time dependent linear Maxwell systems in a medium with highly oscillatory parameters. We use analytical homogenization results to derive the effective Maxwell system with corresponding cell problems. The solution of this effective system is the macroscopic part of the oscillating solution in which we are interested. Due to homogenization the system includes a convolution with a new time-dependent parameter whose corresponding cell corrector is also time-dependent. This time-dependence is the challenge, both in the implementation and in the analysis of the numerical scheme. We apply the Finite Element Heterogeneous Multiscale Method (FE-HMM) combined with recursive convolution to solve the effective Maxwell system.

Abstract:
The radiative transfer equation is a linear integro-differential equation describing the movement of particles traveling through a background medium, through which particles can be absorbed or scattered. It plays a key role in various physics applications such as radiation therapy, nuclear engineering, high-energy astrophysics, supernovae and fusion. In addition to time and space, the solution to the radiative transfer equation depends on an angular domain, which represents the direction in which particles travel. Performing a nodal discretization of the angular domain yields the discrete ordinates $(S_N )$ method. Unfortunately, the $S_N$ method leads to a non-physical imprint of the chosen angular directions on the solution. These spurious artifacts are often called ray effects and stem from the fact that particles are only allowed to travel in finitely many directions. Furthermore, several problems in radiative transport require implicit time discretizations which must be chosen carefully to allow for an efficient time update.
To mitigate ray effects, we propose a modification of the $S_N$ method, which we call artificial scattering $S_N (as-S_N)$. The method adds an artificial forward-peaked scattering operator which generates angular diffusion to the solution and thereby mitigates ray effects. This method can be understood as a filter acting on the angular domain. Our method allows an efficient implementation of explicit and implicit time integration according to standard SN solver technology, which is outlined in this talk. For two test cases, we demonstrate a significant reduction of the error for the as-SN method when compared to the standard SN method and we show that a prescribed numerical precision can be reached with less memory due to the reduction in the number of ordinates.
This is joint work with Martin Frank, Thomas Camminady, and Cory D. Hauck.

Abstract:
In this talk, we consider semilinear hyperbolic systems with highly oscillatory initial data. Typical solutions oscillate with high frequency in time and space, and relevant effects appear only on very long time intervals. As a consequence, applying standard schemes to compute numerical approximations is bound to fail because the total cost of such a scheme would be very high to be practicable. Therefore, we prove that the solution can be approximated analytically by a modulated Fourier expansion. The advantage of this approach is that the coefficient functions in the modulated Fourier expansion do not oscillate in space.

Abstract:
In this talk we consider the time integration with leapfrog--Chebyshev (LFC) schemes for linear second order differential equations consisting of a stiff, 'cheap' and a non-stiff, 'expensive' part. After the introduction of LFC schemes we state some conditions showing the importance of the stabilization parameter of the schemes. We further show that these conditions are necessary for ensuring stability and correct long-time behavior as well as the error analysis. The results are illustrated by numerical examples.Moreover, we point out connections between these multirate methods and local time-stepping methods (based on the leapfrog method) for the linear acousticwave equation.

Abstract:
In this talk we consider acoustic wave equations with dynamic boundary conditions. In contrast to standard boundary conditions, as of Dirichlet or Neumann type, dynamic boundary conditions do not neglect the momentum of the wave on the boundary. The mathematical modeling of such effects leads to an evolution equation in the interior domain coupled to an evolution equation on the boundary.
We discuss the numerical analysis of such equations. After shortly summarize recent results for the semilinear case, we will focus on ongoing work for equations with nonlinear damping terms.

Abstract:
In this talk, I give an overview on some new breather examples in semilinear wave equations. The breathers are constructed variationally as ground states of an energy functional defined on a suitable Hilbert space. Since the energy functional is unbounded from above and below, saddle point techniques are used. A key role is played by the linear wave operator whose spectrum is separated into a positive and negative part with 0 lying in a spectral gap.

Abstract:
In this talk we introduce a low-rank integrator for second-order matrix ODEs $ A''(t) = F(A(t)), t \in [0,T]$, typically stemming from space discretization of a PDE. The integrator is constructed by combining ideas from dynamical low-rank integration (DLR) for first-order matrix ODES with a Strang-splitting.
We briefly sketch the theory of DLR and present the construction of our proposed algorithm. Additionally, we show how to achieve rank-adaptivity and discuss how to ensure efficient computation. The potential of the new algorithm is illustrated on a nonlinear wave equations modeling laser-plasma interaction.

Abstract:
In this talk we study the time integration of linear isotropic Maxwell equations. The domain under consideration is a heterogeneous cuboid Q that contains several smaller separated subcuboids. The subcuboids and the remainder of Q consist of (different) homogeneous media being characterized by material parameters.
Under certain assumptions on the material parameters, we first introduce an appropriate analytical framework and provide a regularity result for the solution of the Maxwell system. The regularity statements at hand, we afterwards state a rigorous error result for the Peaceman-Rachford ADI scheme and give insights into the corresponding error analysis.

Abstract:
In this talk we introduce a class of second-order exponential schemes for the time-integration of semilinear wave equations. They are constructed such that the established second-order error bounds only depend on quantities obtained from a well-posedness result of a classical solution. We sketch how appropriate filter functions as well as the integration-by-parts and summation-by-parts formulas are used in order to compensate missing regularity of the solution. We include numerical examples to illustrate the advantage of the proposed methods.

Abstract:
We consider the Landau-Lifshitz-Gilbert-equation (LLG) on a bounded domain $\Omega$ with Lipschitz-boundary $\varGamma$ coupled with the linear Maxwell equations on the whole space. As the material parameters outside of $\Omega$ are assumed to be constant, we are able to reformulate the problem to a MLLG system in $\Omega$ coupled to a boundary equation on $\varGamma$.
We define a suitable weak solution (which still has a reasonable trace for the boundary equation) and propose a time-stepping algorithm which decouples the Maxwell part and the LLG part of the system and which only needs linear solvers even for the nonlinear LLG part. The approximation of the boundary integrals is done with convolution quadrature.
Under weak assumptions on the initial data and the input parameters we show convergence of the algorithm towards weak solutions, which especially guarantees the existence of solutions to the MLLG system.

Abstract:
The first part of the presentation deals with the numerical integration in time of the nonlinear Schrödinger equation with power law nonlinearity and random dispersion. We will then present preliminary results on the time discretisation of a coupled system of stochastic Schrödinger equations.

Abstract:
I report on progress in project B5 concerning the investigation of threshold phenomena for energy supercritical wave equations.
For geometric wave equations such as wave maps and the Yang-Mills equation it is well-known from numerical experiments that self-similar blowup solutions may appear as intermediate attractors close to the threshold for singularity formation. From an analytic point of view this is poorly understood. In this talk, I discuss as toy models the supercritical radial wave equation with a cubic and a quadratic nonlinearity, respectively.

Abstract:
We establish global-in-time Strichartz estimates for the linear Maxwell system on $\mathbb{R}^3$ with scalar time-independent coefficients of class $C^2$. For the coefficients we require decay conditions at infinity and a one-sided bound on their radial derivative, ensuring non-trapping. There is no global smallness assumption. Using its divergence equations, the Maxwell system is rewritten as a wave system with coefficients in front of the Laplacian and a coupling in the first-order terms. Our approach relies on known local-in-time Strichartz estimates for the wave equation due to Metcalfe-Tataru and new global-in-time weighted energy estimates for our wave system. The latter rely on a detailed analysis of the corresponding Helmholtz problem.
This is joint work with Piero D'Ancona (Rome).

Abstract:
When the period of unit-cells constituting metamaterials is no longer much smaller than the wavelength but only smaller, local material laws fail to describe the propagation of light in such composite media when considered at the effective level. Instead, nonlocal material laws are required. They have to be derived by approximating a general response function of the electric field in the metamaterial at the effective level. At first, we overcome this limitation by considering a Padé-type approximation of the response function of the electric field. It is shown that nine different constitutive relations can be derived. We present a checklist each of these constitutive relation has to pass in order to be physically and mathematically liable. It turned out that only one of these nine Padé approximations passes the checklist. Alternatively, we propose a general higher order approximation for the same purpose. For such nonlocal constitutive relations, the classical interface conditions for the Maxwell equations are not sufficient to fix the amplitudes of all the propagating modes. Additional interface conditions are then required. We give a general formula of these interface conditions at any order of the constitutive relation.

Abstract:
Solutions to boundary value problems of differential equations are not necessarily unique. As parameters are varied, solutions to a problem can bifurcate. The simplest example is a fold bifurcation, where two solutions merge and annihilate each other. More complicated, high-codimensional phenomena occur if several parameters are present. To understand the structure of the solutions space it is of interest to locate bifurcation points of high codimension because they act like organising centres. I will present results on how variational structures of ODEs or PDEs influence which bifurcations occur persistently. Moreover, I will show how the variational structure can be exploited to locate bifurcation points in PDEs numerically.

Abstract:
Maxwell's equations describe the wave-like evolution of electromagnetic fields. In addition to the dynamical equations, there are two constraints corresponding to fundamental physical laws: the nonexistence of magnetic monopoles and the conservation of charge. The dynamical equations automatically preserve these constraints, but when numerical methods are used to simulate Maxwell's equations, one or both of these constraints may be violated. I will give a basic introduction to this problem and describe some recent work (joint with Yakov Berchenko-Kogan) on numerical methods that automatically preserve these constraints, within the framework of 'hybrid' finite element methods.

Abstract:
We consider a molecule in the Born-Oppenheimer approximation interacting with an infinite metallic plate. We prove that the interaction energy $W$ of the system is given by $W(r,v)= -C(v)r^{-3} + O(r^{-4})$, where $r$ is the distance between the molecule and the plate and $v$ indicates their relative orientation. Here $C(v)$ is a positive and continuous function of v. This proves a long range attraction independently of the relative orientation of the molecule and the plate. This asymptotic bahavior is in the physics literature known, however we are not aware of previous rigorous results. Moreover, as far as we know this is the first result providing the leading term of a van der Waals force with no need of assumptions about the multiplicity of the ground state energy of the molecule.
For pedagogical reasons the talk will start with the simple case of a hydrogen atom interacting with a metallic plate and then we will explain how to generalize the arguments for a general molecule. This is a joint work with Dirk Hundertmark and Mariam Badalyan.

Abstract:
While numerical methods typically come with a refinement strategy that allows to control accuracy, mathematical models for physical processes mostly come as monolithic theories and the actual model error are difficult to be assessed. Constructing hierarchical models the physical accuracy can be increased successively by traversing a model cascade. Hence, model error estimates can easily be obtained allowing true predictivity including model adaptivity.
We will construct and investigate hierarchical models for two different applications: Slow non-equilibrium gases where the model hierarchy bridges from classical fluid dynamics to Boltzmann equation, and shallow flows where the hierarchy bridges from depth-averaged St. Venant equations to free-surface Navier-Stokes. In both cases we construct a cascade of models based on function expansions and projections of the respective complex reference model mimicking the approach of numerical methods.

Abstract:
In this talk, we will discuss the local (or global) energy decay for the wave equation with damping at infinity. Being particularly interested in the case of a periodic (or asymptotically periodic) setting, we will mainly describe the contribution of low frequencies and observe that it behaves like the solution of some heat equation. We will see how this emerges from the spectral analysis of the damped wave equation.

Abstract:
The Stokes equation is a basic model in fluid mechanics and can be obtained from the Navier-Stokes equation by taking solutions that are constant in time and neglecting the convective, non-linear term. The discretization of the Stokes equation by finite element methods has attracted sustained attention at least since the seventies. From the point of view of mixed finite elements, the challenge is to define two finite element spaces $X_h \subseteq H^1(S)^n$ and $Y_h \subseteq L^2(S)$ (where $S$ is the domain) such that the divergence operator gives a surjection from $X_h$ to $Y_h$, with a right inverse uniformly bounded in $h$. Since this has proved difficult, the goal has been relaxed in several ways, for instance by non-conforming methods and methods that do not guarantee the incompressibility of the flow.
We approach this problem through the lens of constructing subcomplexes of the de Rham complex with sufficient regularity. We achieve the goal as initially formulated by defining composite elements that are piecewise polynomial with respect to different simplicial refinements at each index in the complex. For instance the complexes typically start with a variant of Clough-Tocher elements for scalar functions of class $C^1(S)$. The gluing conditions between cells, that insure the required regularities of the fields, is taken care of by a framework of finite element systems, which is a natural discrete analogue of sheaf theory. It guarantees the existence of degrees of freedom that provide commuting interpolators. This is joint work with Kaibo Hu.
S. H. Christiansen, K. Hu., Generalized Finite Element Systems for smooth differential forms and Stokes' problem, Numer. Math. 140(2):327--371, 2018.​

Abstract:
We study the NLS in one dimension with nonlinearity $|u|^{\alpha-1} u$ with initial data $u_0$ in $L^2(\mathbb{R})+H^1_{per}(\mathbb{R})$. We show local wellposedness for $\alpha\in[2,5)$, discuss several approaches to conserved quantities, and prove global existence for $\alpha=2$.

Abstract:
We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic nonlinear Schrödinger equation in one dimension with initial data $u_0$ in $H^s(R)+H^r(T), 0\leq s\leq r$. In addition, we show that if $u_0$ is in $H^s(R)+H^{\frac12+\epsilon}(T)$, where $\epsilon>0$ and $\frac16\leq s\leq\frac12$, the solution is unique in $H^s(R)+H^{\frac12+\epsilon}(T)$.

Abstract:
In this talk we investigate time dependent Maxwell's equations coupled with the Debye model for orientation polarization in a medium with highly oscillatory parameters. The goal is to characterize the macroscopic behavior of the solution to the resulting integro-differential system. We use analytical homogenization results to derive the effective Maxwell system with the corresponding cell problems. The Finite Element Heterogeneous Multiscale Method (FE-HMM) is applied to solve the homogenized Maxwell system and we give first insights into the semidiscrete error analysis.

Abstract:
Finite time break-down of solutions to nonlinear physical systems (also called singularity formation or blowup) is one of the central problems in partial differential equations theory today. For equations of the so called supercritical type it is widely believed that large initial data lead to formation of singularities in finite time. Furthermore, numerical simulations in supercritical hyperbolic systems like wave maps and Yang-Mills equations indicate that the generic blowup profile is self-similar.
Lead by the intuition from above, we consider the problem of self-similar blowup for wave maps from $(1+d)$-dimensional Minkowski space into negatively curved Riemannian manifolds. Due to their resemblance to defocussing nonlinear wave equations, this kind of wave maps were conjectured to not exhibit generic (stable) blowup. We proceed by constructing for each dimension $d\geq8$ a negatively curved, $d$−dimensional target manifold that allows for the existence of a self-similar wave map. What is more, we prove that our solutions provide a stable blowup mechanism for the corresponding Cauchy problem. This, being the first example of stable blowup for wave maps with negatively curved targets, shows that the conjecture from above was in fact false. The talk is based on a joint work with Roland Donninger, University of Vienna, Austria.

Abstract:
In this talk we discuss the stability of leap-frog type methods. The standard test problem to study the stability is the unforced harmonic oscillator with a fixed frequency. It is well known that the leap-frog method is stable (in the sense that the approximation remains bounded uniformly w.r.t. the simulation time) if the product of the frequency with the time step size is strictly smaller than two. Modifications of the leap-frog method which weaken this strong step size restriction have been recently proposed in the literature. However, these schemes lose the stability property of the leap-frog method.
In this talk we present a general stability result for such time integration methods and show how to construct stable variants of the leap-frog method allowing for larger time step sizes. Numerical results show the superior stability and convergence properties of these new methods compared to recent schemes.
This is joint work with Andreas Sturm

Abstract:
In this talk I will report about joint work with A. Hirsch (KIT). We give a variational existence proof for time-periodic standing waves of a 1+1 dimensional semilinear wave equation with periodic potentials. Theses waves are localized in the unbounded spatial direction. Using Fourier decomposition in time we can solve the resulting variational problem via constrained minimization. It appears that the admissible growth of the nonlinearity is limited by a Sobolev-exponent introduced via the regularity of the potential.

Abstract:
Modelling a dispersion-managed optical fiber cable leads to a nonlinear Schrödinger equation where the linear part is multiplied by a large, piecewise constant but rapidly changing coefficient function. As a consequence, typical solutions are highly oscillatory, which imposes a severe step-size restriction for traditional numerical methods. In this talk, we present and analyze a tailor-made time-integrator which does not suffer from such a restriction, because it converges at least with order one and the error constant does not depend on the speed of oscillations. The main goal of the talk, however, is to discuss a somewhat surprising property of the integrator: the accuracy improves significantly if special step-sizes are chosen.
Joint work with Marcel Mikl.

Abstract:
In this talk, we are going to address the numerical solution of the nonlinear Klein-Gordon equation in the highly oscillatory regime, i.e. the so-called the nonrelativistic limit. The developments of numerical methods with the aim of using large step size have reached a state-of-art. We are going to review a series of numerical integrators from classical methods to some recent developed multiscale methods. A high order multiscale expansion of the solution will be presented in the end and based on which a new optimally uniformly accurate method is proposed.

Abstract:
This talk is concerned with the singular mean field problem of Liouville type on compact surfaces. This equation appears in the prescribed Gaussian curvature problem in Geometry and also arises in the Electroweak Theory and in the abelian Chern-Simons-Higgs model in Physics.
We will focus on the existence and compactness of solutions for this type of problems, which have been extensively studied for positive potentials, see [1, 3]. However, the case of sign changing potentials has not been much considered in the literature. For the latter case, we present some new results on the solvability using variational techniques. Concerning the compactness, we deal with the possible blow–up phenomena in order to establish a general criterion.
References
[1] D. Bartolucci, G. Tarantello, Liouville type equations with singular data and their applications to periodic multivortices for the electroweak theory, Comm. Math. Phys. 229 (2002), no. 1, 3–47.
[2] F. De Marchis, R. López-Soriano, D. Ruiz. Compactness, existence and multiplicity for the singular mean field problem with sign-changing potentials, accepted for publication on J. Math. Pures Appl.
[3] Z. Djadli, A. Malchiodi. Existence of conformal metrics with constant Q-curvature. Ann. of Math. 168 (2008), no. 3, 813–858.

Abstract:
In the current talk we revisit one of the classical problems in homogenization theory - homogenization of the Dirichlet Laplacian in a domain with a lot of tiny holes. It is also known as crushed ice problem. Namely, let $\Omega_\varepsilon := \Omega \ (U_i D_{i\varepsilon})$. Here $\Omega \subset\mathbb{R}^n$ and $\{D_{i\varepsilon}\}_i$ is a family of tiny identical holes (ice pieces) distributed periodically in $\mathbb{R}^n$ with period $\varepsilon$. We denote by $cap(D_{i\varepsilon})$ the capacity of a single hole. It was known for a long time that $-\Delta_{\Omega_\varepsilon}$ (the Dirichlet Laplacian in $\Omega_\varepsilon$) converges to the operator $-\Delta_\Omega + q$ in strong resolvent sense provided the limit $q := \lim_{\varepsilon\rightarrow 0} cap(D_{i\varepsilon})\varepsilon^{-n}$ exists and is finite.
In this talk an improvement of this result will be discussed. We present estimates for the rate of convergence in terms of operator norms. As an application, we establish (for bounded $\Omega$) an estimate for the difference of the eigenvalues of $-\Delta_{\Omega_\varepsilon}$ and $-\Delta_{\Omega_\varepsilon} + q$.
Our analysis relies on an abstract scheme for studying the convergence of operators in varying Hilbert spaces developed in [2]. This is a joint work with Olaf Post (University of Trier) [1].
References:
[1] A. Khrabustovskyi, O. Post, Operator estimates for the crushed ice problem, arXiv:1710.03080 (2017).
[2] O. Post, Spectral convergence of quasi-one-dimensional spaces, Ann. Henri Poincare 7 (2006), 933-973.

Abstract:
Computational models based on the phase-field method have become an indispensable tool for modeling the microstructural evolution in material science and physics. The combination of phase-field modeling with multiphysics applications such as heat and mass transfer, continuum mechanics, fluid flow, micromagnetism and electrochemistry has been achieved. The models typically operate on a mesoscopic length scale and provide valuable information about structural changes in materials through describing the interface motion. Several examples of large scale simulations are presented showing solidification patterns, droplet formations and fluid propagation in cellular structures in the first part of this talk.
The specific parametrization method of the model complicates to satisfy the jump conditions and the formulation of the driving forces at the interfaces, since the sharp interface is stretched over a volumetric region. In the second part of this talk, local homogenization methods of mechanical material parameters are analyzed and a method is presented that guarantees the fulfillment of the mechanical jump conditions and reflects the mechanical configuration forces at diffusely parameterized interfaces [2]. This model is based on multi component and multiphase-field model [1] and is extended for applications in multiphase systems and applied to the martensitic phase transformation process [5,6]. The mechanically driven interface motion also includes the propagation and development of cracks. To analyze these processes, a multiphase-field model is presented to study the crack propagation in polycrystalline materials coupled with a phase transformation process. The model is applied to polycrystalline materials with heterogeneous crack resistance. Additionally, the influence of the grain boundary energy between the solid phases on the resulting crack path is demonstrated [3,4].
References
1. B. Nestler, H. Garcke, B. Stinner Multicomponent alloy solidification: Phase-field modeling and simulations. Physical Review E (2005) 4:041609.
2. D. Schneider, O. Tschukin, M. Selzer, T. Böhlke, B. Nestler Phase-field elasticity model based on mechanical jump conditions. Comp. Mech. (2015) 5:887-901.
3. D. Schneider, E. Schoof, Y. Huang, M. Selzer, B. Nestler Phase-field modeling of crack propagation in multiphase systems. Computer Methods in Applied Mechanics and Engineering (2016) in print.
4. B. Nestler, D. Schneider, E. Schoof, Y. Huang, M. Selzer Modeling of crack propagation on a mesoscopic length scale. GAMM-Mitt. (2016) 1:78-91.
5. D. Schneider, E. Schoof, O. Tschukin, A. Reiter, C. Herrmann, F. Schwab, M. Selzer, B. Nestler Small strain multiphase-field model accounting for configurational forces and mechanical jump conditions. Com. Mech. (2017)
6. D. Schneider, E. Schoof, O. Tschukin, A. Reiter, C. Herrmann, F. Schwab, M. Selzer, B. Nestler, Small strain multiphase-field model accounting for configurational forces and mechanical jump conditions, Com. Mech. (2017) accepted

Abstract:
The current miniaturization trends in nanophotonics augment the need for comprehensive models of light-matter couplings in the near field. In my talk, I will explain a theoretical approach which provides both an intuitive understanding of evanescent light-matter interactions, and the means for making rigorous quantitative experimental predictions. I will use the approach to explain recent experimental results.

Abstract:
In this talk I am going to present some results on the nonlinear Helmholtz equation on $\mathbb{R}^n$ for radial potentials as well as first ideas concerning variational methods for treatment of periodic potentials. As we will see, a major difficulty in the latter is to etablish a so-called limiting absorption principle, i.e. resolvent estimates, for the involved linear Helmholtz operator. The techniques originate mainly from harmonic analysis.

Abstract:
We consider electromagnetic wave propagation through highly oscillatory media. More precisely, we present two Finite Element Heterogeneous Multiscale Methods (FE-HMM) for time dependent Maxwell’s equations. One for the second order in time formulation and one for the first order in time formulation. In this talk, we compare these two schemes and show how both of them can be embedded in a unified framework for non-conforming wave-type problems to derive a-priori error estimates.

Abstract:
Wave type phenomena are common in many fields of science, such as seismology, acoustics and electromagnetics. The propagation of waves is often modeled by partial differential equations (PDEs), for which it is important to have accurate and efficient numerical solvers. In the presence of small geometric features or re-entrant corners in the spatial domain, locally refined meshes around the obstacles permit accurate simulations without introducing too many spatial unknowns and are thus computationally efficient.
Local mesh refinement, however, significantly decreases the performance of explicit time-stepping methods, as the smallest mesh elements dictate the size of the time-step in the entire domain. To circumvent this problem we propose multi-level local time-stepping (MLTS) schemes. In the presence of meshes with several levels of refinement, MLTS methods allow for an appropriate time-step on each of the levels. The methods are based on explicit Runge—Kutta (RK) schemes and an extension of the LTS-RK methods, their 2-level counterpart. They retain the explicitness and the one-step nature of the underlying RK method and thus require no starting procedure and facilitate adaptivity in time. We show that the novel schemes keep the accuracy of the underlying RK scheme and present numerical results that illustrate the versatility of the approach.

Abstract:
As an attempt to develop structure preserving numerical schemes for some non linear wave equations arising in theoretical physics, we focus on the Maxwell Klein Gordon equation in dimension 2. We propose to develop a numerical discretization framework that takes advantage of the Hamiltonian structure of the equation. The gauge invariance is recovered at the discrete level with the help of the Lattice Gauge theory. We then propose and analyse a fully discrete scheme. The strategy of the proof of convergence, based on discrete energy principle, is developed in a more general context and next applied in the particular case of the Maxwell Klein Gordon equation.
This is a joint work with S. Christiansen and T. Halvorsen from the University of Oslo.

Abstract:
In this talk we consider linear hyperbolic operators with low regularity coefficients. In particular, we focus on the case of scalar wave equations and of first order hyperbolic systems.
It is well-known that regularity assumptions weaker than Lipschitz on the coefficients of the operator entail in general a loss of regularity of the solution during the time evolution. Then, well-posedness of the Cauchy problem can be recovered just in $H^1$, with a loss of a finite number of derivatives. In addition, a sharp classification is given about the loss of derivatives which occurs, depending on the modulus of continuity of the coefficients.
Here we will see how to improve these results for weaker hypotheses, the so-called Zygmund conditions. These assumptions are somehow second order conditions, since they concern the second variation of a function, rather than its modulus of continuity. They reveal to be still important for the analysis of hyperbolic Cauchy problems, thanks to a suitable modification of the energy functional.
This talk is based on joint works with Ferruccio Colombini (Pisa), Daniele Del Santo (Trieste) and Guy Métivier (Bordeaux).

Abstract:
We prove global existence for the one-dimensional cubic non-linear Schrodinger equation in modulation spaces $M_{p, p'}$ for $p$ sufficiently close to $2$. In contrast to known results, our result requires no smallness condition on initial data. The proof adapts a splitting method inspired by work of Vargas-Vega, Hyakuna-Tsutsumi and Grunrock to the modulation space setting and exploits polynomial growth of the free Schodinger group on modulation spaces.

Abstract:
This talk is about diffusion equations with time-fractional derivatives of space-varying order. I will examine the well-posedness issue and prove that the space-dependent variable order coefficient is uniquely determined, among other coefficients of these equations, by the knowledge of a suitable time-sequence of partial Dirichlet-to-Neumann maps.
This is joint work with Y. Kian (Marseille) and M. Yamamoto (Tokyo).

Abstract:
The search for time-harmonic solutions of nonlinear Maxwell equations in the absence of charges and currents leads to the following equation $$\begin{cases} \nabla\times(\mu(x)^{-1}\nabla\times E)-\omega^2\varepsilon(x)E=f(x,E) & \text{in }\Omega\\ \nu\times E=0 & \text{on }\partial\Omega \end{cases}$$ for the field $E :\Omega\to\mathbb{R}^3$, where $\Omega\subset\mathbb{R}^3$ is a bounded Lipschitz domain with exterior normal $\nu:d\Omega\to\mathbb{R}^3$, $\varepsilon(x)\in\mathbb{R}^{3\times 3}$ is the (linear) permittivity tensor of the material, and $\mu(x)\in\mathbb{R}^{3\times 3}$ denotes the magnetic permeability tensor. The nonlinearity $f:\Omega\times\mathbb{R}^3\to\mathbb{R}^3$ comes from the nonlinear polarization. If $f=\nabla_EF$ is a gradient then this equation has a variational structure. Our goal is to present ground state and bound state solutions for superlinear and subcritical nonlinearities $f$, e.g. of the form $\Gamma(x)|E|^{p−2}E$ with $2 < p < 2* = 6$, obtained jointly with Thomas Bartsch. Moreover we discuss the critical case when $p = 6$.

Abstract:
In the recent years there has been considerable interest in the transmission eigenvalue problem associated with the scattering by an inhomogneous media. Transmission eigenvalues are related to non-scattering frequencies, they can be determined from the scattering operator and carry information about the refractive index of the scattering medium [1]. However the use of transmission eigenvalues in nondestructive testing has two major drawbacks. The first drawback is that in general only first few transmission eigenvalues can be accurately determined from the measured data and the determination of these eigenvalue means that the frequency of the interrogating wave must be varied in a frequency range around these eigenvalues. In particular, multifrequency data must be used in an a priori determined frequency range. The second drawback is that only real transmission eigenvalues can be determined from the measured scattering data which means that transmission eigenvalues cannot be used for the nondestructive testing of inhomogeneous absorbing media. In our presentation we show how to overcome these difficulties by modifying the far field operator (or the scattering operator). Properties of the modified far field operator are linked to a new eigenvalue problem, such as the Stekloff eigenvalue problem [2] or a different version of the transmission eigenvalue problem. The key idea is that, as oppose to transmission eigenvalues, the eigenvalue parameter in these problems is not related to interrogating frequencies. Nevertheless, these new eigenvalues (possibly complex) can still be determined form scattering data and hence can be used to determine changes in the refractive index of more general type of inhomogeneous media.
References:
[1] F. Cakoni, D. Colton and H. Haddar, CBMS Series, SIAM Publications, 88 (2016).
[2] F.Cakoni, D. Colton, S. Meng and P. Monk, SIAM J. Appl. Math, 76, 1737-1763 (2016)

Abstract:
For the Klein-Gordon-Zakharov (KGZ) system a number of singular limits occur. We group these limits in different classes and explain for each class how to prove that the regular limit system makes correct predictions about the dynamics of the KGZ system in the singular limit.
We explain that in the NLS limit the KGZ system is a normal form of a very general class of dispersive wave systems such as the water wave problem over a periodic bottom, the poly-atomic FPU problem, or the description of modulations of general periodic wave trains.
This is still work in progress and joint work with Simon Baumstark, Patrick Cummings, Markus Daub, Katharina Schratz, and Dominik Zimmermann.

Abstract:
We introduce a novel substructuring discretization scheme for first order systems in space-time. It is based on a skeleton reduction procedure related to the recently introduced discontinuous Petrov-Galerkin (DPG) methods. While being applicable to a variety of problems, the substructuring approach is flexible with respect to the selection of trial and test spaces allowing for problem-specific nonconforming choices with desired approximation and conservation properties. The scheme yields a discrete equation that can be solved in two steps, where first a representation of the solution restricted to the skeleton is obtained by solving a symmetric positive definite linear system. In a second step the approximate solution on each cell is reconstructed from the skeleton values. As a model problem, we consider the linear acoustic wave equation on a bounded interval in one dimension formulated as a first order system. We compare the performance of various discretization schemes, e.g., leap-frog finite differences, space-time LSFEM of different orders, to the new substructuring approach.
Further, an outlook for upcoming work for two space dimensions ist given.

Abstract:
We study an initial-boundary-value problem for a quasilinear thermoelastic plate of Kirchhoff & Love-type, which is simply supported and held at the reference temperature on the boundary of a bounded domain with the heat conduction resulting from the application of Fourier's law. For this problem, we show the short-time existence and uniqueness of classical solutions under appropriate regularity and compatibility assumptions on the data. Further, we prove an exponential decay rate for solutions to the nonlinear equations and exploit barrier techniques to show the global existence and stability of solutions under a smallness condition on the initial data.
This is joint work with Irena Lasiecka (University of Memphis, TN) and Xiang Wan (University of Virginia at Charlottesville, VA).

Abstract:
We develop a semi-Lagrangian scheme adapted to the case where the solution present low oscillations in some fixed direction. The method is analyzed for the 2D constant advection and applied to the numerical resolution of the gyrokinetic equations.​

Abstract:
The common geometrical definition of chirality lacks meaningful upper bounds. In this talk, I will introduce a definition for the electromagnetic chirality of an object which solves this problem. I will then show that the objects that achieve the upper bound have promising applications, and discuss two of them: A two-fold resonantly enhanced and background free circular dichroism measurement setup and angle independent helicity filtering glasses. Finally, I will talk about the practical realization of such objects, including its challenges.

Abstract:
We shall discuss the quasi-neutral limit of the Vlasov Poisson system which is a basic system describing the motion of electrons and ions in plasma physics. We will prove in particular the local well-posedness of the limit system, which is a Vlasov type equation with Dirac interaction potential, for initial data with Sobolev regularity for which the profile in the velocity variable satisfies a stability condition.
Joint work with D Han-Kwan.

Abstract:
In this talk we present and analyze a parallel space-time multigrid solver by using the Fourier mode analysis. We will consider parabolic and hyperbolic problems and we will discuss the difficulties concerning the hyperbolic case. We then explain how to implement the new multigrid algorithm in parallel, and show with numerical experiments its excellent strong and weak scalability properties.

Abstract:
It has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properties as a fractional Laplacian. This has led to the hope that the homogenous Boltzmann equation enjoys similar regularity properties as the heat equation with a fractional Laplacian.
In particular, the weak solution of the fully nonlinear non-cutoff homogenous Boltzmann equation with initial datum in $L^1_2(\mathbb{R}^d)\cap L\log L(\mathbb{R}^d)$, i.e., finite mass, energy and entropy, should immediately become Gevrey regular for strictly positive times. We prove this conjecture for Maxwellian molecules.

Abstract:
An error analysis of trigonometric integrators (or exponential integrators) applied to spatial semidiscretizations of semilinear wave equations with periodic boundary conditions in one space dimension is given. In particular, optimal second-order convergence is shown requiring only that the exact solution is of finite energy. The analysis is uniform in the spatial discretization parameter. It covers the impulse method which coincides with the method of Deuflhard and the mollified impulse method of García-Archilla, Sanz-Serna, and Skeel as well as the trigonometric methods proposed by Hairer and Lubich and by Grimm and Hochbruck. The analysis can also be used to explain the convergence behavior of the Störmer-Verlet/leapfrog time discretization. Read more on https://doi.org/10.1137/140977217.

Abstract:
In this talk, I will discuss some questions related to the nonlinear theory of electromagnetism formulated by Born and Infeld in 1934. I will discuss the link between this theory and the curvature operators in the Euclidean and in the Lorentz-Minkowski space. I will address the solvability of the electrostatic Born-Infeld equation with sources (and a model driven by the nonlinear Klein-Gordon equation) emphasizing the open questions and the partial recent progress we made. Finally, I will discuss more academic results related to the curvature operator in the Lorentz-Minkowski space and the olvability of some scalar field type equations.