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Archive 2010

Date / Time Speaker Title Location
1 February 2010
16:15-17:15 		A. Lang
University of Mannheim, Germany
Event Details
Speaker invited by C. Schwab
Abstract Lax's equivalence theorem is a well-known result in the approximation theory of partial differential equations. It states: If an approximation is consistent, stability and convergence are equivalent. In this talk I will generalize this result to Hilbert space valued stochastic differential equations. Furthermore, examples will illustrate consistency, stability, and convergence for some approximations of Hilbert space valued stochastic differential equations.
Lax's equivalence theorem for stochastic differential equations
HG E 1.2
17 February 2010
16:15-17:15 		Prof. Dr. Cornelis W. Oosterlee
TU Delft, The Netherlands
Event Details
Speaker invited by C. Schwab
Abstract In this presentation we discuss the Heston model with stochastic interest rates driven by Hull-White or Cox-Ingersoll-Ross processes. We present approximations in the Heston-Hull-White hybrid model, so that a characteristic function can be derived and derivative pricing can be efficiently done using the Fourier Cosine expansion technique. This pricing method, called the COS method, is explained in some detail. We furthermore discuss the effect of the approximations in the hybrid model on the instantaneous correlations, and check the influence of the correlation between stock and interest rate on the implied volatilities.
The Heston model with stochastic interest rates and pricing options with Fourier-cosine expansions
HG E 1.2
24 February 2010
16:15-17:15 		Prof. Dr. Rolf Stenberg
Aalto University, Espoo, Finland
Event Details
Speaker invited by C. Schwab
Analysis of finite element methods for the Brinkman problem
HG E 1.2
2 March 2010
16:15-17:15 		Dr. Philipp Grohs
TU Wien, Austria
Event Details
Speaker invited by C. Schwab
Abstract Im my talk I will consider the problem of representing a function in a non-adaptive fashion such that oscillatory behavior (singularities, shocks, edges, ...) can be handled effectively. In one dimension the wavelet transform does a good job at this task. But as the dimension becomes larger, the wavelet transform does not possess sufficient frequency resolution to describe the subtle geometric phenomena that occur at microscopic scales. At least on the theoretical side, and for bivariate functions, a satisfactory solution to this problem has been given by the introduction of the curvelet transform by Candes/Donoho and later the shearlet transform by Labate et. al (the latter possessing certain computational advantages). However, all of the known curvelet- and shearlet constructions up to date are rather specific and not localized in space. I will discuss some more general constructions of shearlets which can be localized in space while still retaining the desirable theoretical properties of previous constructions.
Capturing directional phenomena with the shearlet transform
HG G 19.2
3 March 2010
16:15-17:15 		Prof. Dr. Fabio Nobile
University of Milano, Italy
Event Details
Speaker invited by C. Schwab
Abstract We consider the problem of numerically approximating statistical moments of the solution of a linear elliptic or parabolic partial differential equation (PDE), whose coefficients and/or forcing terms are spatially correlated random fields. After approximating the stochastic coefficients by truncated expansions, the original stochastic PDE turns into a new deterministic parametric PDE of the same type, the dimension of the parameter set being equal to the number of random variables introduced. In this talk, we review Galerkin and Collocation type polynomial approximations of the stochastic PDE in the probability space based either on full or sparse tensor product polynomial spaces. In particular, we will consider approximations in total degree polynomial spaces, hyperbolic cross spaces as well as the polynomial spaces induced by the standard Smolyak construction for sparse grid approximation. For each of them we will also introduce an anisotropic version which takes into account the different influence that each random variable might have on the solution. We review some available convergence results for isotropic and anisotropic sparese collocation and show a numerical comparison between the Galerkin and Collocation approaches in terms of error versus computational cost. The numerical results for a linear elliptic SPDE indicate a slight computational work advantage of Collocation over Galerkin. Also, for anisotropic problems, an optimal tuning of anisotropy ratios in the polynomial space lead to highly improved convergence rates both for Galerkin and Collocation approaches.
Stochastic polynomial approximations for PDEs with random input data
HG E 1.2
10 March 2010
16:15-17:15 		Prof. Dr. Stig Larsson
Chalmers University, Göteborg, Sweden
Event Details
Speaker invited by C. Schwab
Abstract A unified approach is given for the analysis of the weak error of the spatially semi-discrete finite element method for linear stochastic evolution equations driven by additive noise. An error representation is found in an abstract form which is then applied to the stochastic heat, linearized Cahn-Hilliard, and wave equations.
Finite element approximation of stochastic partial differential equations driven by noise
HG E 1.2
17 March 2010
16:15-17:15 		Prof. Dr. Yousef Saad
University of Minnesota, USA
Event Details
Speaker invited by D. Kressner
Abstract This talk will present preconditioning techniques which emphasize robustness. One such technique is based on combining two-sided permutations with a multilevel approach. The nonsymmetric permutation technique exploits a greedy strategy to put large entries of the matrix in the diagonal of the upper leading submatrix. This leads to an effective incomplete factorization preconditioner for general nonsymmetric, irregularly structured, sparse linear systems. The algorithm is implemented in a multilevel fashion and borrows from the Algebraic Recursive Multilevel Solver (ARMS) framework. A parallel implementation using a Domain Decomposition framework will be briefly discussed. We are currently interested in several applications. One is the use of these techniques for indefinite systems arising from wave phenomena (e.g. the Helmholtz equations and the Maxwell equation). Another emerging application in Dynamic Mean Field Theory (DMFT) requires to compute the diagonal of the inverse of a (sparse) matrix. The method of `probing' used to solve this problem leads to the solution of many linear systems with different right-hand sides. We will show some results on both applications and will also present a few preliminary experiments using GPUs (with CUDA) for solving sparse linear systems of equations.
Multilevel preconditioning techniques with applications
HG E 1.2
24 March 2010
16:15-17:15 		Dr. Stephen Langdon
University of Reading, UK
Event Details
Speaker invited by R. Hiptmair
Abstract In this talk, we discuss a new approach for determining the pressure and velocity fields for a weakly compressible fluid flowing in a three-dimensional reservoir, in the presence of vertical wells injecting or extracting fluid. In the application to oil reservoir recovery the depth scale of the reservoir is often small compared to the horizontal length scale. This makes full numerical solution of the problem computationally expensive, but here we take advantage of the small aspect ratio to derive asymptotic expansions in both the inner (near the wells) and outer regions. This leaves us with simple two-dimensional problems to solve numerically, allowing the computation of all significant process quantities in a matter of minutes, compared to computing times measured in hours for more conventional solvers. This is joint work with Dave Needham (Birmingham, UK), and the research was supported by Schlumberger Oilfield UK Plc.
Unsteady flow of a weakly compressible fluid in a thin porous layer
HG E 1.2
31 March 2010
16:15-17:15 		Prof. Dr. Yves Achdou
Université Paris VII, France
Event Details
Speaker invited by C. Schwab
Abstract Mean field type models describing the limiting behavior of differential game problems as the number of players tends to $+\infty$ have recently been introduced by J-M. Lasry and P-L. Lions. The main assumptions are that all the $N$ players are identical and that each player chooses his optimal strategy in view of a global (partial) information on the game. At the limit a system of two coupled equations is obtained: a forward in time Hamilton-Jacobi-Bellman for a value function and a backward in time Fokker-Planck equation for a probability measure. Uniqueness is obtained under some reasonable assumptions. Infinite horizon games can also be considered. This talk is focused on numerical methods for these models. Both finite and infinite horizon problems will be discussed. Discrete optimal planning problems in the context of mean field games will be considered too.
Mean field games: numerical methods
HG E 1.2
12 April 2010
08:00-09:00 		Prof. Dr. Daniel Kressner
SAM, ETHZ
Event Details
Speaker invited by Seminar for Applied Mathematics (SAM)
Abstract The aim of this talk is to provide an overview of current research on the numerical solution of matrix equations, as they appear in the design and analysis of control systems, in particular in model reduction and optimal control. The first part is concerned with classical eigenvalue based methods. The application of such methods to large unstructured problems requires the use of high performance computers and a careful design of the eigenvalue solver. A novel variant of the QR algorithm is presented, which significantly outperforms the current state-of-the-art ScaLAPACK implementation on heterogeneous parallel architectures. The second part of the talk is concerned with iterative methods based on Krylov subspaces and variants thereof. These methods are extended to deal not only with matrix equations but also with a certain class of discretized, high-dimensional partial differential equations. The convergence analysis and experiments reveal that the computational complexity of these methods grows only linearly (instead of exponentially) as the dimension increases. Finally, it is shown how to deal with more general high-dimensional and parametrized PDEs by combining Krylov subspace methods with low tensor rank approximations. This is joint work with Peter Benner (MPI Magdeburg), Bo Kagstrom (U Umea), Enrique Quintana-Orti (U Castellon), and Christine Tobler (ETH Zurich).
Numerical solution of matrix equations in systems, control, and beyond
HG D 7.2
12 April 2010
10:15-11:15 		Prof. Dr. Thorsten Hohage
University of Göttingen, Germany
Event Details
Speaker invited by Seminar for Applied Mathematics (SAM)
Abstract The topic of the this talk are inverse problems with random data which are described by a Poisson process. Examples include deconvolution problems in fluorescence microscopy, astronomy, and mass spectroscopy, SPECT, PET as well as phase retrieval and inverse scattering problems in optics and quantum mechanics.. We study variational regularization methods in Banach spaces with the Kullback-Leibler divergence as natural data misfit functional. Convergence and convergence rate results are derived as the expected total number of counts (often proportional to an illumination time) tends to infinity. We conclude our talk with some real data examples from 4Pi fluorescence microscopy and x-ray optics.
Inverse problems with Poisson data
HG D 7.2
12 April 2010
13:30-14:30 		Prof. Dr. Ansgar Jüngel
TU Vienna, Austria
Event Details
Speaker invited by Seminar for Applied Mathematics (SAM)
Abstract Numerical simulations of highly integrated electronic circuits help to reduce costly experiments and to accelerate the technological progress in microelectronics. Typically, only simplified models are used in industrial applications for reasons of computing time. However, modern semiconductor devices have extremely small structures such that a precise modeling of, for instance, temperature or quantum effects becomes necessary for accurate simulation results. In this talk, some numerical techniques are presented which allow for an efficient simulation of such effects. In the first part, fluid dynamical diffusion models for semiconductors with temperature effects are approximated using mixed finite elements. Numerical simulations of two- and three-dimensional transistors and electric circuits illustrate the influence of the temperature. The second part is concerned with the numerical analysis of a finite-volume scheme of a quantum diffusion model. Furthermore, time-dependent simulations of the switching behavior of a quantum transistor, modeled by the Schroedinger equation and discretized by a pseudo-spectral method, are presented.
Numerical approximation of ultrasmall semiconductor devices
HG F 7
12 April 2010
15:45-16:45 		Dr. Massimo Fornasier
RICAM, Austrian Academy of Sciences, Linz, Austria
Event Details
Speaker invited by Seminar for Applied Mathematics (SAM)
Abstract Solutions of certain PDEs and variational problems may be characterized by a few relevant degrees of freedom and one may want to take advantage of this feature in order to design efficient numerical solutions based on nonlinear and adaptive iterations. In this context we will address three levels of increasing complexity. We start with rather classical elliptic problems associated with strictly convex and smooth energies, then we will address problems associated with convex nonsmooth energies, and eventually we will conclude with certain nonconvex problems. For each of these levels we will point out interesting challenges. More specifically we will be concerned with the following problems: 1) Adaptive numerical methods for elliptic PDEs by means of redundant frame discretizations (strictly convex and smooth energies). We will emphasize challenges related to the geometrical construction of wavelets on domains and the optimal complexity of redundant discretizations. This is perhaps the most traditional part of the talk which introduces us to the next level. 2) Numerical methods for 1-Laplacian equations associated to total variation minimization (convex and nonsmooth energies). Traditional methods of error estimate do not apply and techniques from variational calculus (e.g., compactness arguments, Gamma-convergence) have to be employed. Nevertheless we will be able to show preliminary results of a rate of convergence theory as well as the first successful analysis of subspace/domain decomposition methods in this context. Next we will abandon definitively convex problems to address even more complex situations. 3) Inverse free-discontinuity problems (nonconvex energies). Free-discontinuity problems describe situations where the solution of interest is defined by a smooth function a.e. and a lower dimensional set consisting of the points of discontinuity of the function. First we will present new preliminary analytic results on the existence of solutions for certain inverse free-discontinuity problems. Then we will show that, when such problems are discretized (via FE or FD), the numerical approximation of solutions is NP-hard. Eventually, with the aim of formulating anyway tractable computational approaches in such a complicated situation, we will address iterative thresholding algorithms which intertwine gradient-type iterations with thresholding steps, and we will discuss their convergence properties.
Compressed numerical methods for well-posed and degenerate elliptic PDEs
HG F 7
13 April 2010
08:00-09:00 		Prof. Dr. Siddhartha Mishra
SAM, ETHZ
Event Details
Speaker invited by Seminar for Applied Mathematics (SAM)
Abstract Most systems of conservation laws, particularly those arising in physics and engineering are equipped with an entropy framework. The entropy inequality serves to single out physically relevant weak solutions as well as provides global stability estimates. We design finite volume (difference) schemes that satisfy a discrete version of such an entropy inequality. The schemes are based on a judicious combination of very high order accurate entropy conservative fluxes along with suitable numerical diffusion operators. The numerical diffusion operators are also accurate to an arbitrary order, being based on an essentially non-oscillatory (ENO) reconstruction in scaled entropy variables. The resulting schemes are formally accurate to arbitrary order, parameter free and are essentially non-oscillatory around discontinuities. Furthermore, they are shown to be entropy stable for non-linear systems and convergent for linear symmetrizable systems thus providing some of the first global stability results for schemes that approximate systems of conservation laws with arbitrary order of accuracy. Several numerical examples demonstrating the computational efficiency of these very-high order schemes are provided. The talk presents results obtained jointly with U.S. Fjordholm (SAM, ETHZ) and E. Tadmor (CSCAMM, Univ. of Maryland, U.S.A).
Very high-order essentially non-oscillatory entropy stable schemes for systems of conservation laws
HG D 7.2
14 April 2010
16:15-17:15 		Prof. Dr. Ralf Kornhuber
Freie Universität Berlin, Germany
Event Details
Speaker invited by C. Schwab
Abstract Richards equations for saturated/unsaturated groundwater flow is based on state equations relating saturation to capillary pressure. The numerical solution of the resulting degenerate parabolic problems typically suffers from strong nonlinearities and ill-conditioning in the presence of strongly varying saturation. As a remedy, we suggest a solver-friendly discretization based on Kirchhoff transformation which can be reinterpreted in physical variables in terms of suitable quadrature rules. In this way ill-conditioning is separated from the numerical solution process. This approach is extended to heterogeneous state equations by domain decomposition methods based on nonlinear transmission conditions. We show convergence and illustrate the theoretical results by numerical computations. In order to account for uncertain parameters, we consider a polynomial chaos approach to stochastic variational inequalities arising as spatial problems in time-discretized stochastic versions of Richards equations.
On the numerical simulation of saturated/unsaturated ground water flow
HG E 1.2
16 April 2010
08:00-18:00
Event Details

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