Zurich Colloquium in Applied and Computational Mathematics

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Archive 2009

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Speaker

Title

Location

2 September 2009
16:15-17:15

R. Krause
Institute of Computational Science, University of Lugano

Event Details

Speaker invited by

P. Arbenz

Abstract

Often, the mathematical modeling of physical, biological and chemical problems gives rise to non-convex minimization problems, which only can be solved numerically. For instance, the deformation of soft tissue in biomechanics leads to a non-convex minimization problem which is connected to a non-linear partial diferential equation, usually given on a complex geometry. Multiscale or multilevel methods are well known to provide solution methods of optimal complexity for linear elliptic partial differential equations. However, for the problem classes considered her, they suffer from two drawbacks: first, geometric multigrid methods are based on a hierarchy of nested meshes, which for complex geometries might be hard to create; second, transferring the optimality of multigrid methods to strongly non-linear and possibly constrained problems is far from trivial, since corrections on coarser grids might not accellerate but even spoil the convergence of the multilevel iteration process. In this talk, we present some new approaches for dealing with these difficulties: First, for the purpose of developing multilevel methods based on non-nested meshes, we present a pseudo-L^2 projection allowing for an efficient information transfer between non-nested meshes. Stability and approximation properties of the new operator are discussed. Employing these findings within a rather sophisticated construction of an appropriate space hierarchy, we show how to construct multilevel methods for complex geometries. In the second part of the talk, we will present a recursive multilevel trust region strategy (RMTR), which allows for the efficient solution of strongly non-linear problems on a multilevel-hierarchy. Convergence results as well as illustrative numerical experiments from the field of mechanics will be given.

Multiscale solution methods for real world applications - from complicated geometries to strongly nonlinear problems

HG G 19.1

9 September 2009
16:15-17:15

S. Liao
Shanghai Jiao Tong University, China

Event Details

Speaker invited by

M. Gutknecht

Abstract

A general approach to get convergent series solutions of nonlinear partial differential equations (PDE) is described. This approach is based on homotopy in topology, and thus is independent of any small/large physical parameters. Besides, it provides us with great freedom to choose good enough base functions. More importantly, it provides us with a convenient way to guarantee the convergence of series solutions of nonlinear PDEs. A few examples are given to show its validity.

A general approach to get convergent series solutions of nonlinear partial differential equations

HG G 19.1

16 September 2009
16:15-17:15

D. Pavlov
EPFL Lausanne

Event Details

Speaker invited by

R. Hiptmair

Abstract

The geometric nature of Euler fluids has been clearly identified and extensively studied in mathematics. However computational approaches to fluid mechanics, mostly derived from a numerical-analytic point of view, are rarely designed with structure preservation in mind, and often suffer from spurious numerical artifacts. In contrast, we geometrically derive discrete equations of motion for fluid dynamics from first principles. Our approach uses a finite-dimensional Lie group to discretize the group of volume-preserving diffeomorphisms, and the discrete Euler equations are derived from a variational principle with non-holonomic constraints. The resulting discrete equations of motion induce a structure-preserving time integrator with good long-term energy behavior, for which an exact discrete Kelvin circulation theorem holds. Possible extensions of our method to magnetohydrodynamics, viscous flows, optimal transport and a link to Brenier's generalized flows.

Structure-preserving discretization of incompressible fluids

HG E 1.1

23 September 2009
15:15-16:15

T. Jahnke
University of Karlsruhe, Germany

Event Details

Speaker invited by	C. Schwab

Hybrid models and numerical methods for high-dimensional master equations

HG E 1.1

7 October 2009
16:15-17:15

H. Harbrecht
University of Bonn, Germany

Event Details

Speaker invited by

C. Schwab

Abstract

In this talk we present computable bounds to estimate the distance of finite element approximations to the solution of the Poisson equation. If the finite element approximation is a Galerkin solution, the derived error estimator coincides with the standard element and edge based estimator. If Galerkin orthogonality is not satisfied, the discrete residual additionally appears in terms of the BPX preconditioner. A consequence of the present analysis is the proof of the reliability and efficiency of hierarchical error estimation.

On error estimation in finite element methods without having Galerkin orthogonality

HG E 1.1

21 October 2009
16:15-17:15

N. Beerenwinkel
ETH Zürich

Event Details

Speaker invited by

C. Schwab

Abstract

The diversity of virus populations within single infected hosts presents a major difficulty for the natural immune response, vaccine design, and antiviral drug therapy. Recently developed deep sequencing technologies can be used for quantifying this diversity by direct sequencing of the mixed virus population. We present statistical and computational methods for the analysis of such sequence data. Inference of the underlying virus population from observed reads is based on a non-parametric Bayesian clustering approach and involves error correction, reconstruction of a minimal set of haplotypes that explain the data, and estimation of haplotype frequencies. We demonstrate our approach by analyzing simulated data and real HIV data.

Assessing genetic diversity by deep sequencing of mixed samples

HG E 1.1

26 October 2009
15:15-16:15

A. Bentata
Université Pierre et Marie Curie, Paris, France

Event Details

Speaker invited by

C. Schwab

Abstract

We give conditions under which the flow of marginal distributions of a discontinuous semimartingale X can be matched by a Markov process whose infinitesimal generator is expressed in terms of the local characteristics of X. Our results extend previous results of Gyongy (1986) to discontinuous semimartingales. Our results allows to derive a forward partial integro-differential equation for option prices in a large class of (non-Markovian) semimartingale models with jumps. This equation generalizes the classical result of Dupire (1994) to the case of jump processes. We discuss applications to time-changed Levy processes and index options.

Mimicking the marginal distributions of a semimartingale

HG E 1.1

28 October 2009
16:15-17:15

G. Hagedorn
Virginia Tech, Blacksburg, USA

Event Details

Speaker invited by

V. Gradinaru

Abstract

Essentially everything that is known about molecules is obtained from Born-Oppenheimer Approximations. We begin with a mathematical review of the standard time-independent Born-Oppenheimer theory. Then we discuss modifications that are specifically designed to describe vibrational motions of molecules that contain symmetric or non-symmetric hydrogen bonds. The modified approximations are those proposed in recent joint work with Alain Joye.

A review of the standard Born-Oppenheimer theory for molecules, and modifications concerning hydrogen bonding

HG E 1.1

4 November 2009
12:15-13:15

N. Paragios
Ecole Centrale de Paris, INRIA, France

Event Details

Speaker invited by

D. Kressner and M. Pollefeys

Abstract

Computer aided human perception aims at developing intelligent algorithms towards understanding visual cues coming from images, video, or other means of gathering visual information. Such a process often consists of three stages, initially the problem of perception is parameterized through a mathematical model where the estimation of its parameters will lead to visual understanding. Then, the model is associated with the available observations through the definition of an objective function and last, this function is optimized using computational methods. The main challenges that one has to address in this context is the curses of dimensionality, non-linearity, non-convexity and modularity. In simple words, even the simplest possible perception problem could involve too many parameters where the association between the data and them is not straightforward and is done through non-convex functions. In this talk, we will present a generic mathematical framework that exploits recent advances in discrete optimization to address computational visual perception. In particular we will present a class of algorithms that do exploit efficient linear programming, duality for low and higher order models. The challenging problems of inter/intra-modal deformable registration and 3D reconstruction using shape grammars will be considered to demonstrate the potentials of this framework.

Discrete MRFs in computer vision and medical imaging: Deformable image fusion & 3D reconstruction with shape grammars

HG D 5.2

18 November 2009
16:15-17:15

C. Klingenberg
University of Würzburg, Germany

Event Details

Speaker invited by

S. Mishra

Abstract

We present a relaxation system for hydro and magnetohydrodynamics from which one can derive approximate Riemann solvers. The solvers satisfy discrete entropy inequalities, and preserve positivity of density and internal energy. Next we consider their practical implementation, and derive explicit wave speed estimates satisfying the stability conditions. We put this into an astrophysical application by comparing our new positive and entropy stable approximate Riemann solver with state-of the-art algorithms for astrophysical fluid dynamics, the Prometheus code. The new Riemann solver increases the computational speed without loss of accuracy. These 3-dimensional turbulence simulations are part of a plan to develop, implement, and apply a new numerical scheme for modeling turbulent, multiphase astrophysical flows. The method combines the capabilities of adaptive mesh refinement and large eddy simulations; we refer to it as Fluid mEchanics with Adaptively Refined Large Eddy SimulationS or FEARLESS. We shall present advances in this ongoing project.

New relaxation solvers for Hydro- and Magnetohydrodynamics applied to astrophysical turbulence simulations

HG E 1.1

10 December 2009
09:15-10:15

W.-J. Beyn
University of Bielefeld, Germany

Event Details

Speaker invited by

D. Kressner

Abstract

We consider nonlinear time dependent reaction diffusion systems on unbounded domains, the solutions of which show specific spatio-temporal patterns. For nonlinearities of so called excitable type, typical examples of patterns are travelling waves in one, rigidly rotating or meandering spiral waves in two, and scroll waves in three space dimensions. In the first part of the talk we present a result on stability with asymptotic phase for localized two-dimensional rotating waves (joint work with Jens Lorenz, Albuquerque). We discuss in some detail the underlying spectral assumptions for the linearized operator, which has both essential spectrum and isolated eigenvalues, some of them lying on the imaginary axis. It will be shown that the result applies to spinning solitons in the complex Ginzburg Landau equations. In the second part we present the freezing method which transforms the given time-dependent PDE into a PDAE (Partial Differential Algebraic Equation). Solving this PDAE numerically allows to determine a moving coordinate frame in which the aforementioned patterns become stationary. The method generalizes to evolution equations that are equivariant with respect to the action of a generally noncompact Lie group. In some cases we can show that moving patterns which are stable with asymptotic phase for the original PDE become asymptotically stable in the classical Lyapunov sense for the PDAE.

Nonlinear stability of rotating patterns and the freezing method

HG G 19.1

16 December 2009
16:15-17:15

M. Schmuck
MIT, USA

Event Details

Speaker invited by

C. Schwab

Abstract

We introduce a macroscopic model which allows to describe the essential electrokinetic phenomena as electro-phoresis and -osmosis. Then, we present the basic analytical results for the Navier-Stokes-Nernst-Planck-Poisson system. Next, we propose and analyze two convergent Finite element discretizations which preserve all characteristic properties of weak solutions in the discrete setting. We begin with a scheme based on perturbation and then show how we can improve the properties and consistency of the scheme by using a suitable truncation. In the last part of the talk, we derive eective macroscopic properties of a porous solid-electrolyte composite. This leads to a huge dimensional reduction by upscaling the microstructure. The results are gained by the two-scale convergence method. To motivate the original ideas of homogenization, we also introduce the formal multiple-scale method.

Modeling, analysis and numerics of the Navier-Stokes-Nernst-Planck-Poisson system

HG E 1.1

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