Wednesday
November 30, 2011 |
Claudia Schillings
, Dept. Math., University of Trier, Germany
On the treatment of uncertainties in aerodynamic design
Time:
16:15-17:15
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Place:
HG E 1.2
Invited by:
C. Schwab
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| Abstract: |
Uncertainties pose problems for the reliability of numerical computations and their results in all technical contexts one can think of. They have the potential to render worthless even highly sophisticated numerical approaches, since their conclusions do not realize in practice due to unavoidable uncertainty in parameter values, initial and boundary conditions, geometry, etc. The proper treatment of these uncertainties within a numerical context is a very important challenge. In this talk, we discuss a novel approach towards aleatory uncertainties for the specific application of optimal aerodynamic design under uncertainties. An appropriate robust formulation of the underlying deterministic problem and efficient approximation techniques of the probability space are investigated. Finally, algorithmic approaches based on multiple-setpoint ideas in combination with one-shot methods are presented as well as numerical results. |
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Wednesday
November 23, 2011 |
Mustafa Khammash
, D-BSSE, ETH Zurich
Stochastic gene expression: mathematical modeling, analysis, and control
Time:
16:15-17:15
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Place:
HG E 1.2
Invited by:
C. Schwab
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| Abstract: |
Mathematical modeling of genetic networks is key to understanding life at the most basic level. One of the challenges to the analysis and synthesis of such networks is that the cellular environment in which they function is abuzz with noise arising from the random nature of biochemical reactions at the molecular level. Cellular noise results in random fluctuations within individual living cells and is a source of variability among genetically identical populations. In this talk, we present a stochastic modeling paradigm for modeling genetic networks based on continuous-time discrete (infinite) state Markov models. We discuss some of the key computational methods for the simulation, approximation, and analysis of such models and how they can be used to enhance our fundamental understanding of stochastic biological phenomena. A quantitive understanding of the stochastic nature of gene expression leads naturally to the development of methods for feedback control of genetic circuits. We describe novel analytical and experimental work demonstrating the feasibility of such control. We end with adiscussion of some open computational problems in the field. |
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Wednesday
October 26, 2011 |
Eric Vanden-Eijnden
, Courant Institute, New York
Dimension reduction, coarse-graining and data assimilation in high-dimensional dynamical systems: modeling and computation
Time:
16:15-17:15
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Place:
HG E 1.2
Invited by:
C. Schwab
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Wednesday
October 12, 2011 |
Steffen Börm
, CAU Kiel, Deutschland
Matrix-Galerkin
Time:
16:15-17:15
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Place:
HG E 1.2
Invited by:
C. Schwab
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| Abstract: |
Certain mathematical problems from control theory or the investigation of stochastic PDEs lead to matrix equations. Storing the $n \times n$ solution matrix in the standard way requires ${\mathcal O}(n^2)$ units of storage, so this representation becomes unusable if $n$ grows large.
In this talk, an alternative approach is discussed: the matrix is approximated in a data-sparse representation requiring only ${\mathcal O}(n k)$ units of storage, where $k$ is a parameter controlling the accuracy of the approximation. A Galerkin approach can be employed to compute the approximation.
The talk outlines the fundamental idea, demonstrates how a number of fundamental properties of the resulting system can be derived, and discusses algorithms for solving this system. |
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Wednesday
September 28, 2011 |
Xavier Claeys
, ISAE Toulouse, France
Multi-trace boundary integral formulation of the first kind for acoustic scattering by composite structures
Time:
16:15-17:15
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Place:
HG E 1.2
Invited by:
R. Hiptmair
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| Abstract: |
We study the scattering of acoustic waves by a penetrable object composed of several adjacent parts with different material properties. Starting from an already well known single trace integral formulation of the first kind for this problem, we derive a new boundary integral formulation of the first kind where all unknowns are doubled on each interface. This formulation is immune to spurious resonnances, and it satisfies a stability property that ensures quasi optimal convergence of conforming Galerkin boundary element methods. Besides the operator of this formulation satisfies a relation similar to the standard Calderon identity. |
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Wednesday
September 21, 2011 |
Tucker Carrington
, Queen's University, Ontario, Canada
Using sparse grids to solve the vibrational Schrödinger equation
Time:
16:15-17:15
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Place:
HG E 1.2
Invited by:
V. Gradinaru
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| Abstract: |
I shall review computational techniques for solving the vibrational Schroedinger equation and present new ideas for using sparse grids to cope with the curse of dimensionality. Mathematicians not interested in solving the Schrödinger equation might appreciate: 1) how to do quadratures with a sparse grid without summing contributions to the integral from each of the grids that compose the sparse grid; 2) how to exploit the structure of nested grids to evaluate matrix-vector produces efficiently. |
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Wednesday
June 1, 2011 |
Carlos Pares
, University of Malaga, Spain
Well-balanced High order methods based on reconstruction of states
Time:
16:15-17:15
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Place:
HG E 1.2
Invited by:
S. Mishra
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| Abstract: |
This talk focuses on the numerical approximation of 1d hyperbolic systems involving source terms. A number of simplified flow models have this form. The goal is to present a general framework for designing high-order well-balanced shock-capturing numerical methods. The emphasis will be put on the well-balanced property: the numerical schemes are required to solve exactly the stationary solutions of the system or at least a certain family among them. The idea is to extend to high order a first-order path-conservative method by using a reconstruction operator. The main difficulty comes from the fact that, in order to have a well-balanced numerical scheme, this operator has also to preserve the stationary solutions of the system. A strategy to overcome this difficulty will be presented and some examples will be shown. |
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Wednesday
May 25, 2011 |
Holger Rauhut
, University of Bonn, Germany
Sparse and low rank recovery
Time:
16:15-17:15
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Place:
HG E 1.2
Invited by:
C. Schwab
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| Abstract: |
Compressive Sensing (sparse recovery) predicts that sparse vectors can be recovered from what was previously believed to be highly incomplete linear measurements. Efficient algorithms such as convex relaxations and greedy algorithms can be used to perform the reconstruction. Remarkably, all good measurement matrices known so far in this context are based on randomness. Recently, it was observed that similar findings also hold for the recovery of low rank matrices from incomplete information, and for the matrix completion problem in particular. Again, convex relaxations and random are crucial ingredients.
The talk gives an introduction and overview on sparse and low rank recovery with emphasis on results due to the speaker.
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Wednesday
May 18, 2011 |
Jinzhi Li
, Seminar for Applied Mathematics, ETH Zurich
Shape calculus in differential forms : Ideas and applications
Time:
16:15-17:15
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Place:
HG E 1.2
Invited by:
R. Hiptmair
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| Abstract: |
We treat Zolesio's velocity method of shape calculus using the formalism of differential forms, in particular, the notion of Lie derivative. This provides a unified and elegant approach to computing even higher order shape derivatives of domain and boundary integrals and skirts the tedious manipulations entailed by classical vector calculus. Hitherto unknown expressions for shape Hessians can be derived with little effort.
The perspective of differential forms perfectly fits second-order boundary value problems. We illustrate its power by deriving the shape derivatives of solutions to second-order elliptic boundary value problems with Dirichlet, Neumann and Robin boundary conditions. A new dual mixed variational approach is employed in the case of Dirichlet boundary conditions. Moreover, applications to acoustic and Maxwell scattering problems will also be addressed. |
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Wednesday
May 11, 2011 |
Katharina Kormann
, University of Uppsala, Sweden
An adaptive Magnus-Lanczos-spectral element solver for the time-dependent Schrödinger equation
Time:
16:15-17:15
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Place:
HG E 1.2
Invited by:
V. Gradinaru
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| Abstract: |
In this talk, we present a discretization of the time-dependent Schrödinger equation based on a Magnus-Lanczos time integrator and high-order Gauss-Lobatto elements in space. A truncated Galerkin orthogonality is used to obtain duality-based a posteriori error estimates that treat the temporal and the spatial error separately. Based on this theory, an adaptive solver for the Schrödinger equation is devised.
Since high-order elements are used, the memory consumption of a sparse matrix implementation of the spatial operator is prohibitive. Instead, we present an implementation based on cell-based stencils. In this way, we can also exploit the structure of the tensor-product operator to reduce the number of computations per matrix-vector product. We demonstrate the performance of the algorithm for the example of matter-field interaction. |
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Monday
May 9, 2011 |
Arnulf Jentzen
, Princeton University, USA
On the global Lipschitz assumption in computational stochastics
Time:
15:30-16:15
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Place:
HG F 33.1
Invited by:
C. Schwab
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| Abstract: |
Stochastic differential equations are often simulated with the Monte Carlo Euler method. Convergence of this method is well understood in the case of globally Lipschitz continuous coefficients of the stochastic differential equation. The important case of superlinearly growing coefficients, however, remained an open question for a long time now. The main difficulty is that numerically weak convergence fails to hold in many cases of superlinearly growing coefficients. In this talk we overcome this difficulty and establish convergence of the Monte Carlo Euler method for a large class of stochastic differential equations whose drift functions have at most polynomial growth. |
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Monday
May 9, 2011 |
Ludwig Gauckler
, University of Tübingen, Germany
Long-time analysis of Hamiltonian partial differential equations and their discretizations
Time:
13:30-14:15
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Place:
HG F 33.1
Invited by:
C. Schwab
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| Abstract: |
The long-time behaviour of numerical methods for ODEs is in many aspects well understood. For example the energy, a conserved quantity of a Hamiltonian differential equation, is nearly conserved along a symplectic discretization of a Hamiltonian ODE. The situation is much less clear in the case of PDEs.
In this talk we discuss long-time near-conservation properties of numerical discretizations of Hamiltonian PDEs, for example nonlinear Schroedinger equations, in a weakly nonlinear regime. The considered numerical methods are based on a splitting integrator in time and a spectral method in space. It is shown that energy and also actions are nearly conserved over long times along such a numerical solution, uniformly in the discretization parameters. This result is obtained by analysing a modulated Fourier expansion of the numerical solution, that will be introduced in the talk. As a preparatory work for such a numerical analysis, but also as an interesting problem in its own right, we study the influence of a nonlinear perturbation on the exact solution of a Hamiltonian PDE using modulated Fourier expansions as well. |
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Monday
May 9, 2011 |
Armin Lechleiter
, Ecole Polytechnique, Palaiseau, France
Time and frequency domain wave imaging
Time:
10:15-11:00
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Place:
HG F 33.1
Invited by:
C. Schwab
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| Abstract: |
A wave hitting an object creates a scattered wave. The shape of the scattering object can be mathematically characterized and numerically computed from (partial) measurements of scattered waves using so-called sampling methods. These methods compute an indicator function of the object's support on a grid of sampling points to produce an image of the scatterer. In contrast to, e.g., non-linear optimization techniques, sampling methods do not need to solve direct scattering problems to obtain information on the scattering object. Sampling methods for inverse scattering problems have been introduced in recent years in the frequency domain, that is, for time-harmonic waves. In practice (e.g., in ultrasound applications) one often measures time-domain signals. Working with Fourier- transformed data at a single frequency means to potentially throw away information, and it introduces the new problem to choose that single frequency. Multi-frequency approaches usually introduce the need to synthesize several single frequency reconstructions. In this talk, we consider time-domain sampling methods that naturally incorporate multiple frequencies, and that share many of the good features of their frequency domain counterparts.
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Monday
May 9, 2011 |
Philippe Grohs
, ETH Zurich
Multiscale analysis beyond wavelets
Time:
08:15-09:00
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Place:
HG F 33.1
Invited by:
C. Schwab
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| Abstract: |
Since the wavelet-boom in the 1980's, researchers have come a long way in exploiting the capabilities of multiscale methods with impressive results in both pure and applied mathematics. However, by now also the inherent limitations of wavelets are quite well-understood.
Examples include the incapability to deal with data taking values in nonlinear manifolds, the incapability to deal with high dimensional data with singularities along subsurfaces (think of edges in images), or the incapability to discretize hyperbolic PDEs in a stable fashion.
Of these three topics I will focus on the problem of processing manifold-valued data. Such data arises in several modern applications such as stress/strain measurements and diffusion tensor MRI in medical imaging (where the data points are elements of the symmetric space of symmetric positive definite matrices) or kinematics (where the data points are elements of the Lie group of Euclidean motions).
Due to the inherent nonlinearity, for these data types conventional methods of signal processing break down.
I will describe new strategies capable of processing manifold valued data, and I will also talk about the analysis of such nonlinear multiscale methods.
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Thursday
May 5, 2011 |
Osterferien bis 15. April
Time:
00:00-17:00
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Wednesday
May 4, 2011 |
Annalisa Buffa
, University of Pavia, Italy
Advances in Isogeometric analysis: the blessing of regularity
Time:
16:15-17:15
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Place:
HG E 1.2
Invited by:
R. Hiptmair
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Wednesday
April 27, 2011 |
Dave Hewett
, University of Reading, England
Novel boundary element methods for high frequency scattering problems
Time:
16:15-17:15
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Place:
HG E 1.2
Invited by:
R. Hiptmair
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| Abstract: |
Traditional numerical methods for time-harmonic acoustic scattering problems become prohibitively expensive in the high-frequency regime where the scatterer is large compared to the wavelength of the incident wave. By enriching the approximation space with oscillatory basis functions, chosen to efficiently capture the high-frequency asymptotic behaviour of the solution, it is sometimes possible to dramatically reduce the number of degrees of freedom required, thereby making tractable problems which are currently beyond the capability of traditional methods. In this talk we focus in particular on the problem of scattering by polygons in two dimensions. We propose and analyse, with rigorous error bounds, a hybrid boundary element method (BEM) for a class of non-convex polygonal scatterers, which requires only O(log f) degrees of freedom to maintain a fixed accuracy as the frequency f tends to infinity. This appears to be the first effective hybrid BEM for a class of non-convex obstacles. We also discuss possible extensions to transmission problems and three dimensional scattering problems. |
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Wednesday
March 30, 2011 |
Manuel Castro
, University of Malaga, Spain
Simulating shallow flows on GPUs: Numerical schemes, implementation and applications
Time:
16:15-17:15
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Place:
HG E 1.2
Invited by:
S. Mishra
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Wednesday
March 16, 2011 |
Ronald Hoppe
, University of Augsburg, Germany, and University of Houston, USA
Optimal diffeomorphic matching with applications in biomedical imaging
Time:
16:15-17:15
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Place:
HG E 1.2
Invited by:
D. Kressner
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| Abstract: |
We are concerned with optimal matching of dynamically deformable curves and surfaces R3 with applications in biomedical imaging. In particular, we will focus on diffeomorphic matching which amounts to the solution of an optimization problem featuring a regularized disparity cost functional subject to a dynamical system in terms of a time-dependent family of diffeomorphisms in R3 describing the temporal deformation of the curve or surface under consideration. As an application in biomedical imaging, we will consider the optimal matching of snapshots from the mitral valve apparatus of the human heart extracted from echocardiographical data.
The presented results are based on joint work with R. Azencott, R. Glowinski, J. He, A. Ja joo, Y. Li, A. Martynenko (all UofH), and S. Ben Zekry, MD, S.A. Little, MD, W.A. Zoghbi, MD (all The Methodist Hospital Research Institute, Houston). |
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Wednesday
March 9, 2011 |
Per Christian Hansen
, Technical University of Denmark, Lyngby, Denmark
AIR tools -- A MATLAB package of algebraic iterative reconstruction methods
Time:
16:15-17:15
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Place:
HG E 1.2
Invited by:
D. Kressner
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| Abstract: |
We present a MATLAB package AIR Tools with implementations of several Algebraic Iterative Reconstruction methods for discretizations of inverse problems. Two classes of methods are implemented: Simultaneous Iterative Reconstruction Techniques (SIRT) and Algebraic Reconstruction Techniques (ART). In addition we provide a few simplified test problems from medical and seismic tomography.
For each iterative method, a number of strategies are available for choosing the relaxation parameter and the stopping rule. The relaxation parameter can be fixed, or chosen adaptively in each iteration; in the former case we provide the possibility for choosing the parameter by means of "training," i.e., finding the optimal parameter for a given test problem. The stopping rules provided are the discrepancy principle, the monotone error rule, and the NCP criterion; for the first two methods ``training'' can be used to find the optimal discrepancy parameter.
In addition to giving an overview of the package, we will present some of the underlying theory related to the semi-convergence, the parameter-choice methods, and the stopping criteria. |
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Wednesday
March 2, 2011 |
Thorsten Hohage
, University of Göttingen, Germany
Hardy space infinite elements for Maxwell's equations
Time:
16:15-17:15
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Place:
HG E 1.2
Invited by:
R. Hiptmair
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| Abstract: |
If partial differential equations on infinite domains are solved by finite elements, the infinite domain is split into a bounded computational domain in which standard finite elements are used, and unbounded exterior domain which requires special methods. In this talk we discuss so-called Hardy space infinite elements for the solution of time-harmonic electromagnetic scattering and resonance problems. They are based on the pole condition as radiation condition which requires the Laplace transform of the solution in radial direction to have a holomorphic extension to the lower part of the complex plane for each point on the (star-shaped) coupling boundary. It turns out that the restrictions of these holomorphically extended Laplace transforms belong to the corresponding L^2 based Hardy space. The pole condition is equivalent to the standard Silver Mueller radiation condition (or Sommerfeld in the scalar case). After the Laplace transform, incoming and outgoing solutions belong to orthogonal subspaces, and the radiation condition can be imposed by a Galerkin ansatz in a transformed variational formulation of the problem. More precisely, we discuss the construction of an exact sequence of Hardy space infinite element spaces using tensor products of chain complexes on the coupling boundary and in radial direction.
Hardy space infinite element methods fit naturally into the finite element framework and exhibit super-algebraic convergence with the number of degress of freedom in the Hardy space. Moreover, they are particularly well suited for the solution of resonance problems since they preserve the eigenvalue structure of these problems. |
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Wednesday
February 23, 2011 |
Olivier Le Maitre
, LIMSI, Paris, France
A Galerkin method for parametric uncertainty propagation in hyperbolic systems
Time:
16:15-17:15
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Place:
HG E 1.2
Invited by:
C. Schwab
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| Abstract: |
We present a Galerkin method for the propagation of parametric uncertainties in systems of conservation laws. The method is based on a probabilistic treatment of the uncertainties, yielding a stochastic system of equations assumed hyperbolic almost surely. For the resolution of this system, we use a Galerkin technique with a stochastic discretization involving the expansion of the solution on a basis of orthonormal (uncorrelated) stochastic functionals. The Galerkin projection of the stochastic problem results in a large system of deterministic equations for the expansion coefficients of the solution, with a structure similar to conservation laws. We first study the properties of the Galerkin system and show, in particular, conditions ensuring its hyperbolic character. |
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