Abstracts of talks given at an GRK-Event
Winter term 2014/15
Nanophotonics is a field of physics that allows to observe new very interesting phenomena, in particular through the illumination of nanometric devices at optical frequencies. In this context, media, their geometry and scales are complex and pose problems in terms of numerical modelling. The mathematical modelling rely on a coupling between Maxwell's equations, that describe the propagation of waves, with a dispersive media model. We present a numerical study of these types of models. We propose a numerical framework based on a Time Domain Discontinuous Galerkin approach adapted to the challenges encountered in nanophotonics. Numerical simulations will illustrate this work.
Linear electromagnetic waves or linear elastic waves are solutions to hyperbolic systems of partial differential equations (PDE). The propagation of waves in bounded regions leads to boundary value problems where the system of PDE gets augmented by a boundary condition. We review the classical theory of hyperbolic boundary value problems which is largely due to Friedrichs (1954), Hersh (1963), Sakamoto(1970), and Kreiss (1970). The main result is the well-posedness in the function space of square integrable functions.
Not all boundary conditions which are of practical interest fit the classical theory. The most interesting class of boundary conditions may be characterized as conservative. In this case, the well-posedness of the problem still holds; however, there is a loss of regularity along the boundary. This is a relevant scenario for Maxwell's equations with a perfect conductor as a boundary or for the elastic wave equations with traction forces along the boundary.
We close by classifying hyperbolic boundary problems as either strongly stable, strongly unstable or weakly regular of real type. This characterization is due to Benzoni-Gavage, Rousset, Serre, and Zumbrun (2002).
In this talk we study the eigenvalue sums of magnetic Dirichlet Laplacians on bounded two dimensional domains. We establish Li-Yau and Berezin bounds in the presence of a constant magnetic field and as an application we get the estimate for eigenvalue moments of three dimensional magnetic Dirichlet Laplacians.
This is a joint work with Pavel Exner and Timo Weidl.
Summer term 2014
In this lecture, I will mainly consider the numerical integration of the nonlinear Schrödinger equation over long times and discuss in particular the numerical stability of solitons in the one dimensional case. I will explain the phenomenon of time and space resonances, and show that under some restriction between the time and space discretization parameter, we can prove some orbital stability result for numerical schemes.
We present a generalization of Lubich's Convolution Quadrature based on Runge-Kutta methods which allows for variable time steps. The main application of our method is the time integration of retarded potentials arising in wave scattering problems. The algorithmic realization of the new method relies on contour integral techniques in the complex plane. Numerical experiments are provided to show the potential of our approach.
The interaction of light with metallic nanostructure can be driven into resonance by exploiting surface plasmon polaritons (SPP). There, the electromagnetic field is coupled to the oscillation of the charge density in the metals. While distinguishing between propagating and localised SPPs, bound to an infinitely extended interface between a metal and a dielectric or bound to a small metallic nanoparticle, respectively, SPPs allow to localise light into tiny volumes, provide a spectral response in narrow frequency ranges, and provide unorthodox scattering characteristics for nanostructures with complicated shapes. These features are extremely appealing and can be beneficially used for various purposes. Examples thereof are optical metamaterials, for optical information processing at the nanoscale, for the photon management in solar cells, or to enhance chemical, nonlinear and quantum effects. The resonant character and the critical geometrical size of the nanostructures that is comparable to and in most cases even smaller than the wavelength of light, usually requires to solve Maxwell’s equations without further approximation to grasp all the properties. This is a major challenge which asks for contributions from multiple perspectives. In this talk, I will give an overview on my research in this field with emphasis on numerical and analytical aspects of light propagation in and its interaction with plasmonic nanostructures, where linear and nonlinear optical aspects are important.
Accelerated charged particles emit electromagnetic radiation. At modern particle accelerators, this radiation is in the x-ray regime, but recent advances in accelerator physics made it possible to create intense THz-radiation as well. Their size ranges from 10m to 27km and applications range from material science to arts to cancer therapy and collision experiments that give insight into the structure of elementary particles. Usually up to a billion particles are accelerated simultaneously and the design and understanding of an accelerator requires solving the equations of motion for many particles in a given setup of electric and magnetic fields. A full understanding of the emitted radiation requires novel numerical tools for solutions to Maxwell equations. Besides the "wanted" radiation, the electrons create electromagnetic wake fields, propagating in a wave guide. The solution to the inverse problem would then yield information about important accelerator parameters.
This talk gives an introduction to accelerator physics and the different numerical methods used in modeling modern particle accelerators.
Winter term 2013/2014
We investigate parabolic equations in a general non-smooth setting with emphasis on mixed interface and boundary conditions. In particular, we allow for dynamics and diffusion on a Lipschitz interface and on the boundary, where diffusion coefficients are only assumed to be bounded, measurable and positive semidefinite. In the bulk, we additionally take into account diffusion coefficients which may degenerate towards a Lipschitz surface. For this problem class, we introduce a unified functional analytic framework based on sesquilinear forms and show maximal regularity for the corresponding abstract Cauchy problem. This is joint work with K. Disser, A.F.M. ter Elst and J. Rehberg.
Summer term 2012
Exponential time integrators are a powerful tool for numerical solution of large time dependent problems. The actions of matrix functions on vectors, necessary for exponential integrators, an be efficiently computed by different elegant numerical techniques, such as Krylov subspaces. Unfortunately, in some situations the additional work required by exponential integrators per time step is not paid off because the time step can not be increased too much due to the accuracy restrictions. We consider one class of problems when we can get around this difficulty. This problem class is formed by linear systems of ordinary differential equations (ODEs) where the non-autonomous term lives in a small-dimensional subspace. We show that in this case the problem can be solved with a single block Krylov subspace and there is no time stepping process involved. We refer to this method as EBK, exponential block Krylov method. An extension of EBK to nonlinear ODE systems is an open problem and we discuss possible ways for such an extension.
We will deal with the following operators in :
Here are bounded above and bounded away from zero -periodic functions. We denote by the set of such operators.
In the talk we will discuss the following result: for an arbitrary and for arbitrary pairwise disjoint intervals we construct the family of operators such that the spectrum of has exactly gaps in when is small enough, and these gaps tend to the intervals as . The corresponding functions can be chosen in such a way that their ranges have at most values.
The idea how to construct the family is based on methods of the homogenization theory.
Also we will discuss a similar result obtained for periodic Laplace-Beltrami operators.
In this talk, we will consider the nonlinear Schrödinger equation in where the coefficients and are composed of periodic functions and on the left and right of an interface plane.
We will discuss the existence of -ground state solutions for this equation assuming that . Using constrained minimization on a generalized
Nehari manifold, we derive an abstract existence criterion based on the ground state energies of the purely periodic problems with and and a more practical criterion based on the ground states themselves. We conclude with examples where these criteria are satisfied.
This is joint work with Wolfgang Reichel and based on a paper by Tomáš Dohnal, Michael Plum and Wolfgang Reichel.
Winter term 2011/2012
We consider a Reissner-Mindlin plate with Green & Naghdi type II hyperbolic heat conduction modeled by a conservative system of second order PDEs. Under certain conditions on the geometry of the domain as well as physical parameters, we prove the exact solv- ability for the Neumann boundary controllability problem: given null Dirichlet boundary conditions on one part of the boundary, there ex- ist Neumann L2-boundary controls on the other part of the boundary steering the system from an “arbitrary” initial into an “arbitrary” final state. The proof is based on a general control theory in reflexive Banach spaces and consists in showing the admissibility of the control operator as well as an observability inequality for the dual operator. Time permitting, a generalization for the case of a Reissner-Mindlin plate with Cattaneo heat conduction will be presented.
This talk is about the design of mirrors and lenses that transmit radiation with a prescribed magniﬁcation. It is assumed that radiation emanates from a source point or in a beam of parallel rays. As an example, I will explain and solve the problem of designing the passenger mirror in a car without a blind spot. It turns out that this problem leads to a ﬁrst order diﬀerential equation that can be solved by elementary methods. The very same idea can be used to design lenses magnifying the image with a prescribed factor. References:
C. E. Gutiérrez. Reﬂection, refraction and the Legendre transform. Journal Optical Society of America A, 28(2):284–289, 2011.
C. E. Gutiérrez and F. Tournier. Surfaces refracting and reﬂecting collimated beams. Journal Optical Society of America A, 28(9):1860–1863, 2011.
We are interested in positive exponentially decreasing solutions with of the following NLS system
where and . The main ideas of the Nehari manifold approach for NLS systems are sketched. In the case a short proof of the existence of bound and ground states, i.e. of a finite energy and a least energy solution of the above equation is presented.
Optofluidics is an exciting new field of research that leverages the rapid progress in microfluidic technology and the more mature field of optics to create low-cost versatile devices for applications in e.g. bio-analysis, communication and imaging. One of the developments that has attracted much attention is the creation of dynamically reconfigurable optical components that are based on liquid-liquid interfaces. In the interest of being able to study such components through numerical simulation, a model for immiscible two-phase flow is presented. The main challenge is to accurately compute the evolution of the interface. For this purpose, I use a Level Set method that is discretized using a Least Squares Finite Element Method. The resulting advection equation is coupled to the incompressible Navier-Stokes equations that describe the evolution of the flow field. In the presentation, I will discuss the model and aspects of its discretization, and present some preliminary numerical results.
A very efficient solution method to the problem of linear water-wave diffraction by a large number of bodies is presented. Several bodies are assembled in modules, which are grouped in periodic infinite line arrays. Then, using an iterative method, a finite number of these infinite arrays are stacked together. While the method is general and can be used in a variety of situations, it is particularly suitable to predict the attenuation of ocean waves by vast fields of ice floes as they occur in the Marginal Ice Zones. Comparisons with two sets of experimental data give convincing agreement. Moreover, the method can also be used to calculate the band structure of periodic lattices of arbitrary bodies so as to construct arrangements of bodies effectively having a negative refractive index (for some frequency ranges).
Summer term 2011
For a class of linear parabolic equations we propose a nonadaptive sparse space-time Galerkin least squares discretization. We formulate criteria on the trial and test spaces for the well-posedness of the corresponding Galerkin least squares solution. In order to obtain discrete stability uniformly in the discretization parameters, we allow test spaces which are suitably larger than the trial space. The problem is then reduced to a finite, overdetermined linear system of equations by a choice of bases. We present several strategies that render the resulting normal equations well-conditioned uniformly in the discretization parameters. The numerical solution is then shown to converge quasi-optimally to the exact solution in the natural space for the original equation. Numerical examples for the heat equation confirm the theory.
We consider nonlinear time dependent PDE’s on unbounded domains, the solutions of which show speciﬁc spatio-temporal patterns. Examples are provided by semilinear reaction diffusion systems on , such as
If the nonlinearity f is of excitable type such systems exhibit travelling or rotating waves for d = 1, rigidly rotating or meandering spiral waves for d = 2, and scroll waves for d = 3. The idea of the freezing method is to determine during the numerical process a moving coordinate frame in which the aforementioned patterns become stationary. For this purpose the Cauchy problem for the PDE is transformed into a partial differential algebraic equation (PDAE). Additional algebraic variables are introduced that describe the position of the pattern, and extra constraints are imposed that try to minimize the temporal changes of the spatial proﬁle. The method generalizes to evolution equations that are equivariant with respect to the action of a Lie group The numerical solution of the PDAE involves several approximation processes, such as restriction to a bounded domain with asymptotic boundary conditions and discretization in space and time. We show a series of applications to systems of Ginzburg-Landau and FitzHugh-Nagumo type. Finally, we report on some analytical results related to the freezing approach. These are concerned with the preservation of asymptotic stability for speciﬁc patterns and the inﬂuence of numerical approximations on dicrete and continuous spectra of linearizations.
Winter term 2010/2011
Nano-Photonics deals with light propagation and light-matter interaction within (mostly artificially) structured optical materials. For the theoretical description this means that complex geometries with several relevant length- and time-scales have to be dealt with. In addtion, suitable material models have to be developed. Here, the word "suitable" encompasses two meanings. First, the material models have to capture the relevant physics in an (almost) quantitative manner. Second, the material models have to be compatible with the (typically numerical) methods for solving the resulting equations of motions. For instance, linear material models can (mostly) be utilized without much ado but clearly fail to describe nonlinear optical properties. Conversely, even for rather small nano-particles it is quite impossible to solve the full quantum-mechanical problem (even if it were, it would often be useless). In this talk, certain examples that illustrate these issues will be presented. These examples include coupled dielectric waveguide resonator systems and plasmonic nanostructures.
Since its first proposed use in the early 90's, the ring cavity fiber laser mode-locked by utilizing the nonlinear polarization technique has become one of the most reliable and compact sources for robust ultra-short optical pulses. However, due to the limitations in the energy output, these fiber lasers have lagged well behind the solid-state lasers in the key performance parameters. In this talk I will present recent developments in achieving high-energy pulses in a ring cavity laser that is passively mode-locked by a series of waveplates and a polarizer. Specifically, I will show how the multi-pulsing instability can be circumvented in favor of bifurcating to higher-energy single pulses by appropriately adjusting the group-velocity-dispersion in the fiber and the waveplate/polarizer settings in the saturable absorber. The findings may be used as a practical guideline for designing high-power lasers since the theoretical model relate directly to the experimental settings. I will also extend the model to describe the mode-locking dynamics in multi-mode fibers as well as nonlinear crystals.
Splitting methods intend to break down a complicated problem into a series of simpler sub-problems. In the context of time integration a common idea is to split up the right-hand side and to decompose the given evolution equation into a sequence of locally one-dimensional problems. In many situations, the latter can be solved more efficiently than the original problem. In this talk we study the convergence properties of splitting methods for inhomogeneous evolution equations. We work in an abstract Banach space setting of maximal dissipative operators. This framework allows us to study certain parabolic equations and their spatial discretizations. An example of a parabolic equation illustrating the theoretical assumptions is given. Numerical results are included.
In laser-plasma physics, many phenomena can be described by a nonlinear wave equation coupled to an equation for the plasma response. Since the interesting physical problems are huge, fast and efficient numerical solvers are required. Often, storage limitations also affect the simulation. In this talk, we present a dynamical low-rank approximation to the solution of the wave equation. If the equation is spatially discretized on a tensorized grid, we obtain a matrix differential equation. For localized solutions such as traveling laser pulses, these matrices can be approximated by matrices of small rank for long times. This property can be exploited to describe the solution by a matrix decomposition, which for small ranks needs considerably less storage. To derive a new system of differential equations directly advancing the small matrix factors in time, we project the original matrix differential equation onto the manifold of rank-r matrices. This projection is motivated by an energy conservation property, which such problems posses. We also adress the implementation especially with respect to an adaptive rank control. The latter is important to avoid additional storage costs as well as numerical instabilities during the computation.
Photonic band-gap materials can be useful in constructing waveguides. The idea is that by introducing a linear defect of infinite extent in a photonic crystal, guided modes corresponding to frequencies in a band-gap of the bulk crystal may arise; these modes decay into the bulk and may be interpreted as waves propagating along the defect. What happens when we have a straight, semi-infinite waveguide inside a PBG material and a radiation source is present? Physical intuition tells us that the e.m. waves should propagate along the waveguide towards infinity. Mathematically, it is not at all easy to prove that exactly this happens. In the talk, I will introduce a radiation condition for this problem and discuss a corresponding new existence and uniqueness result.
Summer term 2010No Abstracts available.
Winter term 2009/2010
Dielectric and metallic photonic crystals are promising materials for controlling and manipulating electromagnetic waves 1. For frequency independent material models considerable mathematical progress has been made 2. In the frequency dependent case, however, the nonlinearity of the spectral problem complicates the analysis. We study the spectrum of a scalar operator-valued function with periodic coefficients, which after application of the Floquet transform become a family of spectral problems on the torus. The frequency dependence of the material parameters lead to spectral analysis of a family of holomorphic operator-valued functions. We show that the spectrum for a passive material model consists of isolated eigenvalues of finite geometrical multiplicity. These eigenvalues depend continuously on the quasi momentum and all non-zero eigenvalues have a non-zero imaginary part whenever losses (absorption) occur 3. Lorentz permittivity model, which is a common model for solid materials, lead to a rational eigenvalue problem. We study both the self-adjoint case and the non-self-adjoint case. Moreover, a high-order discontinuous Galerkin method is used to discretize the operator-valued function, and the resulting matrix problem is transformed into a linear eigenvalue problem. Finally, we use an implicitly restarted Arnoldi method to compute approximate eigenpairs of the sparse matrix problem. 1 K.Sakoda, Optical properties of photonic crystals, Springer-Verlag, Heidelberg, 2001. 2 P.Kuchment, Floquet theory for partial differential equations, Birkhäuser, Basel, 1993. 3 C.Engström, On the spectrum of a holomorphic operator-valued function with applications to absorptive photonic crystals, To appear.
For the solution of Laplacian eigenvalue problems we propose a boundary element method which is used to solve equivalent nonlinear eigenvalue problems for related boundary integral operators. We use the concept of eigenvalue problems for holomorphic Fredholm operator functions to establish a convergence and error analysis for a Galerkin discretization of boundary integral operator eigenvalue problems. The Galerkin discretization of such problems leads to algebraic nonlinear eigenvalue problems which can be solved by iterative schemes. We analyze different methods as the inverse iteration, the Rayleigh functional iteration and Kummer's method. For the latter method we give convergence rates with respect to the multiplicity of the eigenvalues. Finally, numerical examples are presented which confirm the theoretical results.
For a selfadjoint periodic differential operator we investigate the limiting absorption limit in local L^2 topology. The proofs are based on some facts about the Hilbert transform and surprisingly elementary.