Departement Mathematik

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Zurich colloquium in applied and computational mathematics

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Frühjahrssemester 2014

Datum / Zeit Referent:in Titel Ort
26. Februar 2014
16:15-17:15 		Prof. Dr. Peng Chen
EPFL Lausanne
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Zurich Colloquium in Applied and Computational Mathematics

Titel Reduced basis method for uncertainty quantification problems
Referent:in, Affiliation Prof. Dr. Peng Chen, EPFL Lausanne
Datum, Zeit 26. Februar 2014, 16:15-17:15
Ort HG D 1.2
Abstract Several computational challenges, including high-dimensionality, low-regularity, arbitrary probability distribution, expensive deterministic solver, are commonly faced for the solution of some typical uncertainty quantification problems. In this talk, I will present an accurate and efficient reduced basis method to tackle these challenges, with particular applications to statistical moments evaluation, risk analysis and stochastic optimal control problems.
Reduced basis method for uncertainty quantification problems read_more
HG D 1.2
10. März 2014
16:15-17:15 		Dr. Oliver Sander
RWTH Aachen
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Zurich Colloquium in Applied and Computational Mathematics

Titel Geodesic Finite Elements
Referent:in, Affiliation Dr. Oliver Sander, RWTH Aachen
Datum, Zeit 10. März 2014, 16:15-17:15
Ort HG E 1.2
Abstract Geodesic finite elements are a novel way to discretize problems involving functions with values on a Riemannian manifold. Examples for such problems include Cosserat materials and liquid crystals. The basic idea is to rewrite Lagrangian interpolation as a minimization problem, which can be generalized to nonlinear spaces. Geodesic finite elements of any order can be constructed, and are conforming in the sense that they are first-order Sobolev functions. The construction is equivariant under isometries of the value manifold, which implies that frame-indifference in mechanics is preserved. Optimal discretization error bounds have been shown analytically, and can be observed in numerical experiments. We present the theory of geodesic finite elements and give a few example applications.
Geodesic Finite Elements read_more
HG E 1.2
12. März 2014
16:15-17:15 		Prof. Dr. Remi Abgrall
Universität Zürich
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Zurich Colloquium in Applied and Computational Mathematics

Titel High-order preserving residual distribution schemes for advection-diffusion scalar problems on arbitrary grids
Referent:in, Affiliation Prof. Dr. Remi Abgrall, Universität Zürich
Datum, Zeit 12. März 2014, 16:15-17:15
Ort HG E 1.2
Abstract In this talk, we first review the so-called Residual distribution scheme, a variant of non linear finite element methods for solving hyperbolic problems, and show how to extend the construction to advection-diffusion problems using conformal meshes. Second and third order methods will be detailed, the extension to higher order is straightforward. The problems considered range from pure diffusion to pure advection. The approximation of the solution is obtained using standard Lagrangian finite elements and the total residual of the problem is constructed taking into account both the advective and the diffusive terms in order to discretize with the same scheme both parts of the governing equation. To cope with the fact that the normal component of the gradient of the numerical solution is discontinuous across the faces of the elements, the gradient of the numerical solution is reconstructed at each degree of freedom of the grid and then interpolated with the same shape functions used for the solution. Linear and non-linear schemes are constructed and their accuracy is tested with the discretization of advection-diffusion and anisotropic diffusion problems. We end the talk by an extension to the Navier Stokes equations with 3D examples. This is a joint work with Dante de Santis (INRIA).
High-order preserving residual distribution schemes for advection-diffusion scalar problems on arbitrary gridsread_more
HG E 1.2
19. März 2014
16:15-17:15 		Prof. Dr. Helge Holden
NTNU Trondheim
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Zurich Colloquium in Applied and Computational Mathematics

Titel What is between conservative and dissipative solutions for the Camassa--Holm equation?
Referent:in, Affiliation Prof. Dr. Helge Holden, NTNU Trondheim
Datum, Zeit 19. März 2014, 16:15-17:15
Ort HG E 1.2
Abstract The Camassa--Holm (CH) equation reads $u_t-u_{txx}+\kappa u_x+3uu_x-2u_xu_{xx}-uu_{xxx}=0$ where $kappa$ is a real parameter. We are interested in the Cauchy problem on the line with initial data in $H^1$. There is a well-known and well-studied dichotomy between two distinct classes of solutions of the CH equation. The two classes appear exactly at wave breaking where the spatial derivative of the solution becomes unbounded while its $H^1$ norm remains finite. We here introduce a novel solution concept gauged by a continuous parameter $\alpha$ in such a way that $\alpha=0$ corresponds to conservative solutions and $\alpha=1$ gives the dissipative solutions. This allows for a detailed study of the difference between the two classes of solutions and their behavior at wave breaking. We also extend the analysis to a two-component Camassa--Holm system. This is joint work with Katrin Grunert (NTNU) and Xavier Raynaud (SINTEF).
What is between conservative and dissipative solutions for the Camassa--Holm equation? read_more
HG E 1.2
26. März 2014
16:15-17:15 		Prof. Dr. Max Wardetzky
Universität Göttingen
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Zurich Colloquium in Applied and Computational Mathematics

Titel Discrete Elastic Curves
Referent:in, Affiliation Prof. Dr. Max Wardetzky, Universität Göttingen
Datum, Zeit 26. März 2014, 16:15-17:15
Ort HG E 1.2
Abstract Elastic curves -- thin strips of elastic material -- have intrigued mathematicians and physicists for centuries, including Galileo, the Bernoullis, Euler, Born, and others. One way to describe elastic curves is to define them as minimizers of the so-called bending energy (the total squared curvature) of a curve. This view leads to the notion of discrete elastic curves -- polygonal curves that minimize a certain discrete bending energy. In my talk, besides presenting some of the fascinating history of elastic curves, I will discuss, using tools from variational analysis, how discrete elastic curves approximate their classical smooth counterparts.
Discrete Elastic Curvesread_more
HG E 1.2
2. April 2014
16:15-17:15 		Prof. Dr. Mario Bebendorf
Universität Bonn
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Zurich Colloquium in Applied and Computational Mathematics

Titel Improving hierarchical matrix preconditioners via preservation of vectors
Referent:in, Affiliation Prof. Dr. Mario Bebendorf, Universität Bonn
Datum, Zeit 2. April 2014, 16:15-17:15
Ort HG E 1.2
Abstract Hierarchical (H-)matrices provide a means to construct ecient and robust approximate preconditioners for the solution of elliptic boundary value problems. In order to cope with increasing condition numbers, the approximation accuracy, however, has to be adapted to the condition number, which deteriorates the eciency. We present a new approach, in which special vectors are preserved during the truncated H-matrix operations. This will be shown to lead to spectral equivalence even without adaptation of the approximation accuracy.
Improving hierarchical matrix preconditioners via preservation of vectorsread_more
HG E 1.2
16. April 2014
16:15-17:15 		Dr. Steffen Weisser
Universität Saarbrücken
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Zurich Colloquium in Applied and Computational Mathematics

Titel Conforming Trefftz-like basis functions on polygonal and polyhedral meshes with realization in BEM-based FEM
Referent:in, Affiliation Dr. Steffen Weisser, Universität Saarbrücken
Datum, Zeit 16. April 2014, 16:15-17:15
Ort HG E 1.2
Abstract In the development of numerical methods to solve boundary value problems the requirement of flexible mesh handling gains more and more importance. The BEM-based finite element method is one of the new promising strategies which yields conforming approximations on polygonal and polyhedral meshes, respectively. This flexibility is obtained by special trial functions which are defined implicitly as solutions of local boundary value problems related to the underlying differential equation. Due to this construction, the approximation space already inherit some properties of the unknown solution. These implicitly defined trial functions are treated by means of boundary element methods (BEM) in the realization. The presentation gives a short introduction into the BEM-based FEM and deals with recent challenges and developments. The basic idea in the construction of trial functions is generalized, and thus, trial functions of arbitrary order are obtained. With the help of an appropriate interpolation operator, it is possible to prove optimal rates of convergence in the $H^1$- as well as in the $L_2$-norm for the BEM-based FEM on uniform refined polygonal meshes with star-shaped elements. Furthermore, by using a posteriori error estimates, it is possible to achieve optimal rates of convergence even for problems with non-smooth solutions on adaptive refined polygonal meshes. Several numerical experiments confirm the theoretical results.
Conforming Trefftz-like basis functions on polygonal and polyhedral meshes with realization in BEM-based FEM read_more
HG E 1.2
30. April 2014
16:15-17:15 		Prof. Dr. Ernst Hairer
Universität Genf
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Zurich Colloquium in Applied and Computational Mathematics

Titel Control of parasitic oscillations in linear multistep methods
Referent:in, Affiliation Prof. Dr. Ernst Hairer, Universität Genf
Datum, Zeit 30. April 2014, 16:15-17:15
Ort HG E 1.2
Abstract Due to the presence of parasitic roots in linear multistep methods, the numerical solution of differential (and differential-algebraic) equations gives rise to non-physical oscillations. For strictly stable methods these oscillations are rapidly damped, so that the numerical solution behaves like that of a one-step method (Kirchgraber 1986, Stoffer 1993). For symmetric methods these oscillations, although with small amplitude in the beginning, can grow exponentially with time and soon dominate the error in the numerical approximation. Certain symmetric multistep methods for second order differential equations, when applied to (constrained) Hamiltonian systems, have the feature that these oscillations remain bounded and small (below the discretization error of the smooth solution) over very long time intervals. Numerical experiments are presented and a proof of the long-time behaviour is outlined. The technique of proof is backward error analysis combined with modulated Fourier expansions. The presented results have been obtained in collaboration with Christian Lubich and Paola Console.
Control of parasitic oscillations in linear multistep methods read_more
HG E 1.2
12. Mai 2014
16:15-17:15 		Prof. Dr. Jan van Neerven
Delft University of Technology
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Zurich Colloquium in Applied and Computational Mathematics

Titel Stochastic integration in UMD Banach spaces
Referent:in, Affiliation Prof. Dr. Jan van Neerven, Delft University of Technology
Datum, Zeit 12. Mai 2014, 16:15-17:15
Ort HG D 1.2
Abstract In this talk I present an overview of the joint work with Mark Veraar and Lutz Weis which has led to the construction of a stochastic integral with respect to Brownian motion for stochastic processes taking values in a UMD Banach space. What makes this integral particularly useful is that two-sided estimates can be shown for the L^p-norm of this integral which provide a natural extension of the Ito isometry and Burkholder's inequality. We show how this integral can be used to prove existence, uniqueness and regularity for semilinear stochastic PDEs.
Stochastic integration in UMD Banach spacesread_more
HG D 1.2
28. Mai 2014
16:00-17:00 		Dr. Markus Bachmayr
RWTH Aachen
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Zurich Colloquium in Applied and Computational Mathematics

Titel Adaptive Low-Rank Methods for High-Dimensional Second-Order Elliptic Problems
Referent:in, Affiliation Dr. Markus Bachmayr, RWTH Aachen
Datum, Zeit 28. Mai 2014, 16:00-17:00
Ort HG E 1.2
Abstract We consider the application of subspace-based tensor formats to high-dimensional operator equations on Hilbert spaces, and combine such tensor representations with adaptive basis expansions of the arising lower-dimensional components. This leads to highly nonlinear approximation problems. We review the general framework of [1], where we have analyzed an iterative method which is not tied to a fixed background discretization and under standard assumptions can be guaranteed to converge to the solution of the continuous problem. Furthermore, under additional low-rank representation sparsity assumptions, the scheme constructs an approximate solution using a number of arithmetic operations that is optimal up to fixed logarithmic terms. Here, the major difficulty lies in obtaining meaningful bounds for the tensor ranks of iterates. In this talk, we focus on problems posed on function spaces for which the inner product does not induce a cross norm, e.g., problems on Sobolev spaces such as second-order elliptic PDEs on product domains. We discuss the additional issues that arise in this case (which are connected to general obstructions with preconditioning in the context of low-rank expansions), outline how one can again obtain a method whose complexity can be analyzed, and show some first numerical experiments. The presented results are joint work with Wolfgang Dahmen. _____ [1] M. Bachmayr, W. Dahmen: Adaptive Near-Optimal Rank Tensor Approximation for High-Dimensional Operator Equations, Foundations of Computational Mathematics, 2014, DOI 10.1007/s10208-013-9187-3.
Adaptive Low-Rank Methods for High-Dimensional Second-Order Elliptic Problemsread_more
HG E 1.2

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Organisatoren:innen: Philipp Grohs, Ralf Hiptmair, Arnulf Jentzen, Siddhartha Mishra, Christoph Schwab

Archiv: FS 25 HS 24 FS 24 HS 23 FS 23 HS 22 FS 22 HS 21 FS 21 HS 20 FS 20 HS 19 FS 19 HS 18 FS 18 HS 17 FS 17 HS 16 FS 16 HS 15 FS 15 HS 14 FS 14 HS 13 FS 13 HS 12 FS 12 HS 11 FS 11 HS 10 FS 10 HS 09

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