Zurich colloquium in applied and computational mathematics

1 October 2012
16:15-17:15

Prof. Dr. Karen Willcox
MIT, USA

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Zurich Colloquium in Applied and Computational Mathematics

Title	Model reduction for systems with high-dimensional parameter spaces
Speaker, Affiliation	Prof. Dr. Karen Willcox, MIT, USA
Date, Time	1 October 2012, 16:15-17:15
Location	HG D 3.2
Abstract	Recent advances in projection-based model reduction methods for nonlinear and parametrically varying systems have opened up a broad new class of potential applications. Problems with large parameter dimension present a signiﬁcant opportunity for model reduction to accelerate solution of large-scale systems with applications in optimization, inverse problems and uncertainty quantiﬁcation. However, large parameter dimension also poses a signiﬁcant challenge, since most model reduction methods rely on sampling the parameter space to build the reduced-space basis. This talk highlights recent progress on model reduction for large-scale problems with many parameters. Our approaches use a goal-oriented philosophy combined with optimization methods to guide the selection of samples over the parameter space in an adaptive manner. We also show how reduced basis approximations of the state space can be extended to reduce the dimension of the parameter space. We demonstrate our methods in the context of applications in optimization, inverse problems and uncertainty quantiﬁcation with a variety of engineering examples.

Model reduction for systems with high-dimensional parameter spacesread_more

HG D 3.2

3 October 2012
16:15-17:15

Dr. Zhen Peng
ElectroScientLab, Ohio State University, USA

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Zurich Colloquium in Applied and Computational Mathematics

Title	Domain decomposition methods for time-harmonic Maxwell equations
Speaker, Affiliation	Dr. Zhen Peng, ElectroScientLab, Ohio State University, USA
Date, Time	3 October 2012, 16:15-17:15
Location	HG E 1.2
Abstract	The domain decomposition methods (DDMs) have been demonstrated effective in solving the time- harmonic Maxwell equations in R3. One category of the DDMs is the non-overlapping DDMs. The basic idea is to decompose the original entire problem domain into several non-overlapping sub- domains. The continuities of electromagnetic fields at the interfaces between adjacent sub-domains are enforced through some boundary conditions. Moreover, the non-conformal DDMs which relieve the restriction of mesh conformity have been proposed and shown to be accurate and efficient. The first topic we will be discussing is a geometrically non-conformal finite element domain decom- position method (FE-DDM). A true second order transmission condition (SOTC), which involves two second-order transverse derivatives, has been proposed and allows convergence of both propa- gating and evanescent electromagnetic waves across domain interfaces. Here, we introduce a new optimal second order transmission condition to further improve the convergence in the DDM iter- ations. A global plane wave deflation technique is utilized to derive an effective global-coarse-grid preconditioner to promote fast convergence of the cutoff or near cutoff modes in the vicinity of domain interfaces. With these ingredients, the proposed non-conformal FE-DDM scales very well with respect to mesh size, the electric size of the sub-domain, as well as the electric size of the entire problem domain. Recently, a surface integral equation domain decomposition method (SIE-DDM) has been proposed and surfaced as a very appealing alternative for the computations of electromagnetic scatterings from composite penetrable objects. Each local sub-domain is enclosed by a closed surface and solved individually through the generalized combined field integral equation (G-CFIE) with excitations that include radiations coming from all the other sub-domains. In the method to be present, the continuities of tangential fields, both electric and magnetic fields, on the interfaces between adjacent sub-domains are enforced via a complex CFIE type transmission condition. We proceed to examine the eigenvalue distributions of the proposed SIE-DDM using a modular problem using both one-way and general geometrical partitions. Systematic numerical experiments for some model problems are included which lead to some insights on the behavior of the method.

Domain decomposition methods for time-harmonic Maxwell equationsread_more

9 October 2012
10:00-12:00

Stig Larsson
Chalmers University, Sweden

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Zurich Colloquium in Applied and Computational Mathematics

Title	Stochastic Evolution PDEs: Existence and Regularity
Speaker, Affiliation	Stig Larsson, Chalmers University, Sweden
Date, Time	9 October 2012, 10:00-12:00
Location	HG G 43
Abstract	SNF Prodoc Minicourses "Numerik" on Numerical Solution of Inverse Problems and of Stochastic PDEs More information at http://www.fim.math.ethz.ch/lectures/Minicourses

Stochastic Evolution PDEs: Existence and Regularityread_more

11 October 2012
10:10-12:00

Stig Larsson
Chalmers University, Sweden

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Zurich Colloquium in Applied and Computational Mathematics

Title	Stochastic Evolution PDEs: Existence and Regularity
Speaker, Affiliation	Stig Larsson, Chalmers University, Sweden
Date, Time	11 October 2012, 10:10-12:00
Location	HG G 43
Abstract	SNF Prodoc Minicourses "Numerik" on Numerical Solution of Inverse Problems and of Stochastic PDEs More information at http://www.fim.math.ethz.ch/lectures/Minicourses

Stochastic Evolution PDEs: Existence and Regularityread_more

11 October 2012
10:15-11:15

Prof. Dr. Holger Rauhut
University of Bonn, Germany

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Zurich Colloquium in Applied and Computational Mathematics

Title	Compressive sensing with structured random matrices
Speaker, Affiliation	Prof. Dr. Holger Rauhut, University of Bonn, Germany
Date, Time	11 October 2012, 10:15-11:15
Location	HG E 1.1
Abstract	Compressive sensing predicts that sparse and compressible signals can be recovered from what was previously believed to be incomplete linear measurements via efficient algorithms such as l1-minimization. Remarkably, all optimal measurement matrices modelling the linear information acquisition process known so far are based on randomness. While Gaussian and Bernoulli matrices provide simple models of such random matrices, they are of limited practical use. In fact, applications demand for structure, both for physical/modelling reasons and in order to have fast matrix computations. This motivates to study structured random matrices and we will discuss several types of them including partial random Fourier matrices and their generalizations, partial random circulant matrices as well as random scattering matrices arising in remote sensing. For instance, partial random circulant matrices model subsampled convolutions with random vectors and are motivated by tasks in wireless communications and radar. We will present the currently best available sparse recovery guarantees for the various structured random matrices. Moreover, we will discuss computational aspects of recovery algorithms.

Compressive sensing with structured random matricesread_more

HG E 1.1

11 October 2012
13:15-14:15

Prof. Dr. Annalisa Buffa
University of Pavia, Italy

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Zurich Colloquium in Applied and Computational Mathematics

Title	New tools in the numerical analysis of PDEs: isogeometric methods
Speaker, Affiliation	Prof. Dr. Annalisa Buffa, University of Pavia, Italy
Date, Time	11 October 2012, 13:15-14:15
Location	HG E 1.1
Abstract	I am interested in the discretization of partial differential equations having a relevant geometric structure which needs to be preserved at discrete level in order to avoid spurious behaviors, numerical instability or simply non-physical solutions. In this context, I will propose completely new schemes, enjoying features which are difficult to obtain with standard techniques. These methods are called isogeometric because they are inspired to the paradigm of isogeometric analysis (IGA). IGA has been introduced in 2005 in the mechanical engineering community and it is growing very fast as a new powerful approach to the discretization of PDEs.

New tools in the numerical analysis of PDEs: isogeometric methodsread_more

HG E 1.1

17 October 2012
16:15-17:15

Prof. Dr. Dominique Chapelle
INRIA, France

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Zurich Colloquium in Applied and Computational Mathematics

Title	Improving convergence in numerical analysis using observers - Applications in cardiac modeling
Speaker, Affiliation	Prof. Dr. Dominique Chapelle, INRIA, France
Date, Time	17 October 2012, 16:15-17:15
Location	HG E 1.2
Abstract	We propose an observer-based approach to circumvent the issue of unbounded approximation errors - with respect to the length of the time window considered - in the discretization of wave-like equations in bounded domains, which covers the cases of the wave equation per se and of linear elasticity as well as beam, plate and shell formulations, and so on. Namely, taking advantage of some measurements available on the system over time, we adopt a strategy inspired from sequential data assimilation and by which the discrete system is dynamically corrected using the discrepancy between the solution and the measurements. In addition to the classical cornerstones of numerical analysis made up by stability and consistency, we are thus led to incorporating a third crucial requirement pertaining to observability - to be preserved through discretization. Numerical assessments confirm the a priori error estimates found in the theory, and we will also present a real-life application of the approach with the model of a beating heart and measurements provided by medical imaging.

Improving convergence in numerical analysis using observers - Applications in cardiac modelingread_more

25 October 2012
12:00-13:00

Prof. Dr. Laurent Demanet
MIT, USA

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Zurich Colloquium in Applied and Computational Mathematics

Title	The butterfly algorithm for high-frequency wave propagation and related oscillatory integrals
Speaker, Affiliation	Prof. Dr. Laurent Demanet, MIT, USA
Date, Time	25 October 2012, 12:00-13:00
Location	HG G 19.2
Abstract	The butterfly algorithm is an O(N log N) alternative to the FFT for computing oscillatory integrals -- yet it is based on completely different ideas. As in the fast multipole method, it is the low rank of interactions between faraway cubes that makes the butterfly work. The availability of this relatively new algorithmic construction in numerical analysis has had implications much beyond the FFT: 1) for the design of fast solvers for the acoustic wave equation in smooth media, and 2) for the fast computation of generalized Radon transforms that arise in synthetic aperture radar and seismic imaging. In this talk I will review the nuts and bolts of the butterfly algorithm, and explain why its success in the context of wave equations owes to the geometry of nondispersive propagation of singularities.

The butterfly algorithm for high-frequency wave propagation and related oscillatory integralsread_more

HG G 19.2

26 October 2012
10:15-11:15

Prof. Dr. Martin Burger
WWU Münster, Germany

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Zurich Colloquium in Applied and Computational Mathematics

Title	Inverse scale space methods in biomedical imaging
Speaker, Affiliation	Prof. Dr. Martin Burger, WWU Münster, Germany
Date, Time	26 October 2012, 10:15-11:15
Location	HG E 23
Abstract	Sparse recovery of signals and images and related methods such as total variation regularization have become a popular topic recently with respect to theory, computation and application. In this talk we discuss advances of such variational techniques for image reconstruction, with main focus on inverse scale space methods, which yield a technique without bias and thus results of superior quality, which can be decisive in applications, as we will demonstrate for biomedical imaging applications. Moreover we discuss adaptive inverse scale space methods for efficient computational solutions. Surprisingly the latter connect various different computational approaches for sparse recovery such as variational, iterative, and greedy methods, and they can be generalized from l1-minimization to a wider range of variational problems. This is also relevant for some applications we discuss, ranging from image reconstruction to the unmixing of spectral data.

Inverse scale space methods in biomedical imagingread_more

HG E 23

30 October 2012
10:00-12:00

Andrew Stuart
University of Warwick, UK

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Zurich Colloquium in Applied and Computational Mathematics

Title	Beysian Inverse Problems in Differential Equations
Speaker, Affiliation	Andrew Stuart, University of Warwick, UK
Date, Time	30 October 2012, 10:00-12:00
Location	HG D 16.2
Abstract	SNF Prodoc Minicourses "Numerik" on Numerical Solution of Inverse Problems and of Stochastic PDEs More information at http://www.fim.math.ethz.ch/lectures/Minicourses

Beysian Inverse Problems in Differential Equationsread_more

31 October 2012
16:15-17:15

Prof. Dr. Magnus Svard
University of Bergen, Norway

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Zurich Colloquium in Applied and Computational Mathematics

Title	Convergence rates and computational efficiency of high-order finite difference schemes
Speaker, Affiliation	Prof. Dr. Magnus Svard, University of Bergen, Norway
Date, Time	31 October 2012, 16:15-17:15
Location	HG E 1.2
Abstract	In this talk, I will discuss various approaches to deduce the optimal global convergence rate of finite difference schemes approximating initial-boundary value problems. The main conclusion is that optimal rates are obtained for schemes satisfying an energy estimate. Furthermore, I will discuss computational efficiency of high-order finite difference schemes.

Convergence rates and computational efficiency of high-order finite difference schemesread_more

1 November 2012
10:00-12:00

Andrew Stuart
University of Warwick, UK

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Zurich Colloquium in Applied and Computational Mathematics

Title	Beysian Inverse Problems in Differential Equations
Speaker, Affiliation	Andrew Stuart, University of Warwick, UK
Date, Time	1 November 2012, 10:00-12:00
Location	HG D 16.2
Abstract	SNF Prodoc Minicourses "Numerik" on Numerical Solution of Inverse Problems and of Stochastic PDEs More information at http://www.fim.math.ethz.ch/lectures/Minicourses

Beysian Inverse Problems in Differential Equationsread_more

5 November 2012
16:15-17:15

Prof. Dr. Adi Adimurthi
TIFR Bangalore, India

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Zurich Colloquium in Applied and Computational Mathematics

Title	Exact controllability of entropy solutions of scalar conservation laws
Speaker, Affiliation	Prof. Dr. Adi Adimurthi, TIFR Bangalore, India
Date, Time	5 November 2012, 16:15-17:15
Location	CLA E 4
Abstract	In general the cotrollability problems in non linear equations is hard because the linearization technique fails to implement. One such equation is the conservation law. In this talk I will consider the scalar conservation laws in one space dimension with strict convex flux. Assuming the target function satisfying the Lax - Olenik compatibility condition which is a necessary condition I will show that the controllability problem has a solution.

Exact controllability of entropy solutions of scalar conservation lawsread_more

CLA E 4

6 November 2012
10:00-12:00

Andrew Stuart
University of Warwick, UK

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Zurich Colloquium in Applied and Computational Mathematics

Title	Date Assimilation in PDEs
Speaker, Affiliation	Andrew Stuart, University of Warwick, UK
Date, Time	6 November 2012, 10:00-12:00
Location	HG D 16.2
Abstract	SNF Prodoc Minicourses "Numerik" on Numerical Solution of Inverse Problems and of Stochastic PDEs More information at http://www.fim.math.ethz.ch/lectures/Minicourses

Date Assimilation in PDEsread_more

8 November 2012
10:00-12:00

Andrew Stuart
University of Warwick, UK

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Zurich Colloquium in Applied and Computational Mathematics

Title	Bayesian Inverse Problems in Differential Equations
Speaker, Affiliation	Andrew Stuart, University of Warwick, UK
Date, Time	8 November 2012, 10:00-12:00
Location	HG D 16.2

Bayesian Inverse Problems in Differential Equations

20 November 2012
10:00-12:00

Arnulf Jentzen
ETH Zürich

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Zurich Colloquium in Applied and Computational Mathematics

Title	Numerical Analysis of Nonlinear Stochastic Differential Equations
Speaker, Affiliation	Arnulf Jentzen, ETH Zürich
Date, Time	20 November 2012, 10:00-12:00
Location	HG G 43
Abstract	SNF Prodoc Minicourses "Numerik" on Numerical Solution of Inverse Problems and of Stochastic PDEs More information at http://www.fim.math.ethz.ch/lectures/Minicourses

Numerical Analysis of Nonlinear Stochastic Differential Equationsread_more

21 November 2012
16:15-17:15

Prof. Dr. Ilya Krishtal
Northern Illinois University, USA

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Zurich Colloquium in Applied and Computational Mathematics

Title	Dynamical sampling
Speaker, Affiliation	Prof. Dr. Ilya Krishtal, Northern Illinois University, USA
Date, Time	21 November 2012, 16:15-17:15
Location	HG E 1.2
Documents	Abstractfile_download

Dynamical samplingread_more

22 November 2012
10:10-12:00

Arnulf Jentzen
ETH Zürich

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Zurich Colloquium in Applied and Computational Mathematics

Title	Numerical Analysis of Nonlinear Stochastic Differential Equations
Speaker, Affiliation	Arnulf Jentzen, ETH Zürich
Date, Time	22 November 2012, 10:10-12:00
Location	HG G 43
Abstract	SNF Prodoc Minicourses "Numerik" on Numerical Solution of Inverse Problems and of Stochastic PDEs More information at http://www.fim.math.ethz.ch/lectures/Minicourses

Numerical Analysis of Nonlinear Stochastic Differential Equationsread_more

28 November 2012
16:15-17:15

Prof. Dr. Manuel Castro Diaz
University of Malaga, Spain

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Zurich Colloquium in Applied and Computational Mathematics

Title	A new class of incomplete Riemann solvers based on uniform rational approximations to the absolute value function
Speaker, Affiliation	Prof. Dr. Manuel Castro Diaz, University of Malaga, Spain
Date, Time	28 November 2012, 16:15-17:15
Location	HG E 1.2
Abstract	In this talk we propose a new class of incomplete Riemann solvers, based on approximations in the L∞- norm to the absolute value function in [−1, 1] by means of rational functions, for the numerical approximation of the solution of hyperbolic systems of conservation laws. The main idea relies on the construction of a numerical approximation to the viscosity matrix by using an appropriate rational real function R(x), that approximates the function \|x\| uniformly in [−1, 1], evaluated at the Jacobian of the fluxes of the hyperbolic system computed at some average value (for example, Roe averages). In addition to the Jacobians of the fluxes we shall use either the maximum in absolute value of the characteristic speeds in each cell or an upper bound of them. Thus, the resulting approximate Riemann solver is incomplete in the sense that we do not use the complete spectral decomposition of the Jacobian. Moreover, the new class of Riemann solvers consists of a hierarchy of schemes running from the more dissipative to the less dissipative ones, and having as limiting case a Roe-like scheme. According to the order of the approximation of the generating rational function used, the degree of dissipation can be dosed for particular applications. We study different rational approximations: Newman-type functions, iterative generated Halley functions, and also Chebyshev polynomial approximants. We test our basic algorithms for different initial value Riemann problems for ideal gas dynamics (HD) and magnetohydrodynamics (MHD) to observe their behavior with respect to challenging scenarios in numerical simulations, including some standard numerical pathologies (e. g., heat conduction, postshock oscillations and overheating) and the formation of compound waves in ideal MHD. We also examine our proposed schemes, by computing the numerical approximation of different initial value problems for nonconservative multilayer shallow water equations, where it has been observed that intermediate waves can be properly captured for an appropriate degree of approximation of the generating rational function used. Our numerical tests indicate that the proposed schemes are robust, running stable and accurate with a satisfactory time step restriction CFL constant), and the computational cost is more advangeous with respect to schemes that use a complete spectral decomposition of the Jacobians.

A new class of incomplete Riemann solvers based on uniform rational approximations to the absolute value functionread_more

4 December 2012
10:00-12:00

Stig Larsson
Chalmers University, Sweden

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Zurich Colloquium in Applied and Computational Mathematics

Title	Stochastic Evolution PDEs: Finite Element Methods
Speaker, Affiliation	Stig Larsson, Chalmers University, Sweden
Date, Time	4 December 2012, 10:00-12:00
Location	HG G 43
Abstract	SNF Prodoc Minicourses "Numerik" on Numerical Solution of Inverse Problems and of Stochastic PDEs More information at http://www.fim.math.ethz.ch/lectures/Minicourses

Stochastic Evolution PDEs: Finite Element Methodsread_more

5 December 2012
16:15-17:15

Adam Andersson
Chalmers University, Sweden

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Zurich Colloquium in Applied and Computational Mathematics

Title	Malliavin calculus for SPDEs and weak convergence analysis for numerical schemes
Speaker, Affiliation	Adam Andersson, Chalmers University, Sweden
Date, Time	5 December 2012, 16:15-17:15
Location	HG E 1.2
Abstract	We give an introduction to the Malliavin calculus and define by means of it the Hilbert space valued Itô integral. About half of the talk will concern this topic. Then we show how the Malliavin integration by parts formula can be used in order to prove weak convergence of finite element approximations of the non-linear stochastic heat equation.

Malliavin calculus for SPDEs and weak convergence analysis for numerical schemesread_more

6 December 2012
10:00-12:00

Stig Larsson
Chalmers University, Sweden

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Zurich Colloquium in Applied and Computational Mathematics

Title	Stochastic Evolution PDEs: Finite Element Methods
Speaker, Affiliation	Stig Larsson, Chalmers University, Sweden
Date, Time	6 December 2012, 10:00-12:00
Location	HG G 43
Abstract	SNF Prodoc Minicourses "Numerik" on Numerical Solution of Inverse Problems and of Stochastic PDEs More information at http://www.fim.math.ethz.ch/lectures/Minicourses

Stochastic Evolution PDEs: Finite Element Methodsread_more

12 December 2012
16:15-17:15

Prof. Dr. Thorsten Hohage
Universität Göttingen

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Zurich Colloquium in Applied and Computational Mathematics

Title	Regularization of inverse problems with general data misfit functionals
Speaker, Affiliation	Prof. Dr. Thorsten Hohage, Universität Göttingen
Date, Time	12 December 2012, 16:15-17:15
Location	HG E 1.2
Abstract	We consider inverse problems formulated as operator equations $F(a)= y$ with a forward operator $F$ between Banach spaces $X$ and $Y$. Often the observed data $y^{\rm obs}$ are modelled as an element of the space $Y$ and the error is measured by $\\|y^{\rm obs}-y\\|_{Y}$. We will discuss a number of examples for which such models are inappropriate or lead to suboptimal results. We show recent general convergence results for Tikhonov- and Newton-type regularization methods with general data misfit terms. First we consider an impulsive noise model involving a parameter and the $L^1$-norm as data misfit functional. We show that classical noise models are suboptimal by several orders in the impulsiveness parameter. As a second example we consider inverse problems in which the data are drawn from a Poisson process with density $F(a)$, and the data misfit functional is the Kullback-Leibler divergence. Such problems occur in photonic imaging, e.g.~fluorescence microscopy, PET or x-ray diffraction imaging. We show a convergence in expectation result using a concentration inequality. As a final example we discuss parameter identification problems for stochastic differential equations.

Regularization of inverse problems with general data misfit functionalsread_more

19 December 2012
16:15-17:15

Dr. Alexey Chernov
University of Bonn, Germany

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Zurich Colloquium in Applied and Computational Mathematics

Title	Numerical methods for stochastic obstacle problems
Speaker, Affiliation	Dr. Alexey Chernov, University of Bonn, Germany
Date, Time	19 December 2012, 16:15-17:15
Location	HG E 1.2
Abstract	Obstacle problems arise in many fields in science and engineering. The main difficulty in these problems is their specific nonlinear character and the limited global regularity of the solution caused by the presence of inequality constraints. Nonetheless, combined with the state-of-the-art hardware, advanced numerical schemes are capable to produce a highly accurate and efficient deterministic numerical simulation, provided the problem data are known exactly. However, in real applications, the complete knowledge of the problem parameters is not realistic. One way to treat the lack of knowledge is to model uncertain parameters as random fields. In this talk we review some recent advances in numerical analysis and simulation for a class of stochastic obstacle problems. In particular, we discuss Stochastic Galerkin/Collocation and the Multilevel Monte Carlo approaches, present the asymptotic error estimates and illustrate the numerical performance of the above mentioned methods on several model problems. Joint work with Claudio Bierig, University of Bonn.

Numerical methods for stochastic obstacle problemsread_more