(September 9) Sren Bartels: Numerical Analysis of Some Non-Convex Variational Problems - Non-convex variational problems arise in the mathematical modeling of phase transitions in crystalline solids, in particular in a mathematical description of shape memory alloys. Non-convexity of the minimization problem typically implies non-existence of classical solutions and enforces numerical schemes to develop oscillations on mesh dependent scales. The first part of the talk is based on joint work with A. Prohl (ETH Zurich) and gives a precise characterization of those oscillations. A numerical analysis of a well posed extended (relaxed) formulation of the original problem in the scalar case is presented in the second part of the talk. This part states a posteriori error estimates for the adaptive approximation of measure valued solutions and discusses efficient solution strategies. The third part of the talk focuses on aspects of the effective simulation of non-convex vectorial variational problems and includes approximation results for upper and lower bounds of quasiconvex envelopes.

(September 16) Marcus Calhoun-Lopez: Numerical Solutions of Hyperbolic Conservation Laws: Incorporating Multi-Resolution Viscosity Methods into the Finite Element Framework - It is well known that the classic Galerkin finite-element method is unstable when applied to hyperbolic conservation laws, such as the Euler equations for compressible flow. Adding a diffusion term to the equations stabilizes the method but sacrifices too much accuracy to be of any practical use. An elegant solution devised by Eitan Tadmor for spectral methods is to add diffusion only to the high frequency modes of the solution, which stabilizes the method without the sacrifice of accuracy. We incorporate this idea into the finite-element framework by using hierarchical functions as a multi-frequency basis. The result is a new finite element method for solving hyperbolic conservation laws. For this method, we are able to prove convergence for a one-dimensional scalar conservation law. Numerical results are presented for one- and two-dimensional hyperbolic conservation laws.

(September 23) John Zweck: Euclidean invariant computation of salient closed contours in images - Vision is by far the most highly developed sense in humans. Much of our knowledge of the physical world is based on visual information, and a large portion of the brain is devoted to visual information processing. One of the basic tasks of human and computer vision systems is to complete the boundaries of partially occluded objects. In this talk I will discuss a biologically plausible algorithm to complete salient closed contours in images.

Our work is different from previous work in three ways. First, as in primary visual cortex the input image consists of spots not edges. Second, our computation is based on a distribution of closed contours characterized by a random process. Third, although it is implemented in a discrete neural network our computation is Euclidean invariant in the continuum.

From a numerical point of view, our algorithm was based on a novel numerical solution of a Fokker-Planck partial differential equation on the space of positions and directions in the plane, together with some numerical linear algebra, and a new twist on the Shannon sampling theorem.

This work is joint with Lance Williams.

(September 30) Mark Ainsworth: Adaptive hp-Finite Element Methods for Maxwell's Equations - Recently, there has been a dramatic increase in the use of high order finite element methods for the approximation of Maxwell's equations. We shall discuss some of our own work in this area. In particular, we shall present families of hierarchic basis functions for the Galerkin discretisation of the space $H({\rm curl};\Omega)$ that naturally arises in the variational formulation of Maxwell equations. The conditioning and dispersive behaviour of the elements is discussed along with approximation theory. Numerical examples are shown which demonstrate the accuracy and efficiency of the methods for computing solutions of the time-harmonic Maxwell's equations.

(October 7) Richard Falk: Finite Element Approximation Theory Using Families of Reference Elements - A standard method for the derivation of error estimates for the approximation of smooth functions by families of finite elements indexed by a mesh-size parameter h is to use a scaling argument based upon the introduction of a single reference element together with the Bramble-Hilbert lemma. There are a number of situations, however, when this method does not apply or depends on some additional special estimates to make it applicable. In this talk, we consider a modification of this approach, using a family of reference elements, that has wider applicability. Several applications of this modified approach are discussed.

(October 14) Christoph Schwab: Finite Elements for Elliptic Problems with Stochastic Data - We consider the Finite Element Solution of elliptic problems with random coefficients which are spatially inhomogeneous with finite second moments. We express these as Wiener Polynomial Chaos (PC) expansion of Karhunen Loeve (KL) type in infinitely many random variables.

We give a log-linear complexity algorithm to compute KL expansions for general two-point correlation functions of the data. New decay estimates for KL Eigenvalues and for pointwise convergence of KL expansions are presented in terms of the smoothness of the two-point correlation functions of the data.

Truncation of the KL expansion after M terms leads to a deterministic elliptic problem in M stochastic variables and d physical dimensions. Analytic dependence of the random solution on the M stochastic variables is proved. Precise estimates of domains of analyticity show how that high complexity due to large M can be broken.

A polynomial basis is given for which the O(p^M) deterministic problems can be solved independently and in parallel, reducing the computational complexity to that of a MC method, but with exponential convergence in the number of samples, thereby allowing to handle also random fields with slowly convergent KL expansions.

A-priori error analysis of the fully discrete scheme is provided. Numerical experiments in physical dimension 2 and 3 with stochastic dimension M up to 30 taking minutes of CPU on the ETH BEOWOLF CLUSTER ASGARD are presented.

This is joint work with R.A. Todor and P. Frauenfelder at ETHZ.

(November 04) Raul Tempone: Adaptive Weak approximation of Stochastic Differential Equations with Jumps - We develop adaptive time stepping methods, based on the Monte Carlo Euler method, for weak approximation of jump diffusion driven stochastic differential equations. The main result is new expansions of the computational error, with computable leading order term in a posteriori form, based on stochastic flows and discrete dual backward problems. The expansions lead to efficient and accurate computation of error estimates. We describe adaptive algorithms for either stochastic time steps or quasi-deterministic time steps and present numerical examples illustrating their behavior.

(November 13) Tom Hou: Efficient Numerical Computations of Stochastic Partial Differential Equations - Stochastic PDE's with solutions depending on multiple scales play fundamental and important roles in many problems. Numerical simulations become an important strategy in practical applications. Monte Carlo methods are used widely, but convergence is usually slow and accuracy is poor. Here we propose a strategy, which uses Wiener Chaos expansions, to design efficient numerical methods for stochastic PDE's. One of the main advantages of this approach is that it separates randomness from the problems. We can calculate all statistical properties of the solutions by solving the induced deterministic coefficient equations. This approach does not use random number generating in the computations. Thus, well developed numerical techniques can be used to solve the coefficient equations. We apply this approach to solve the randomly forced Bousinesq equations and Burgers' equation. We demonstrate that our approach is more efficient than the traditional Monte-Carlo method in achieving a comparable accuracy. Moreover, the method gives better control over the computational errors. Finally, we have performed careful long time computations for the randomly forced Bousinesq equations to study the mixing property of the unstably stratefied fluid flow.

(November 14) Tom Hou: Multiscale Modeling and Computation of Flow in Heterogeneous Media - Many problems of fundamental and practical importance contain multiple scale solutions. Composite materials, flow and transport in porous media, and turbulent flow are examples of this type. Direct numerical simulations of these multiscale problems are extremely difficult due to the range of length scales in the underlying physical problems. In this talk, I will give an overview of the multiscale finite element method and describe some of its applications, including composite materials, wave propagation in random media, convection enhanced diffusion, flow and transport in heterogeneous porous media. It is important to point out that the multiscale finite element method is designed for problems with many or continuous spectrum of scales without scale separation. Further, we introduce a new multiscale analysis for convection dominated 3-D incompressible flow with multiscale solutions. The main idea is to construct semi-analytic multiscale solutions locally in space and time, and use them to construct the coarse grid approximation to the global multiscale solution. Our multiscale analysis provides an important guideline in designing a systematic multiscale method for computing incompressible flow with multiscale solutions.

(November 18) Mario Ohlberger: Error estimates for finite volume approximations of non-linear conservation laws on bounded domains - In 1979 Bardos, Leroux and Nedelec gave a characterization of solutions of non-linear conservation laws on bounded domains in the class BV using the vanishing viscosity method. In 1992 Otto gave a uniqueness proof for entropy solutions on bounded domains in the L^infinity class. Based on the Otto formulation we derive an a posteriori error bound for finite volume approximations which is at least of order h^1/6 for meshes with uniform mesh size h. (joint work with Julien Vovelle)

(November 25) Madhu Nayakkankuppam: Solving Large-Scale Semidefinite Programs in Parallel - Semidefinite programming (SDP), which may be described as linear programming in the space of symmetric matrices, has been one of the most intensively researched topics in optimization in the past decade. In this talk, we first survey some of the many applications of SDP, which span numerical linear algebra, combinatorial optimization, control theory, structural design, and statistics, just to name a few. Then we discuss subgradient bundle methods, and how they may be effectively parallelized to solve large-scale problems. We present significant computational results on some of the largest SDP's solved to date.

(December 2) Junping Wang: Computational Simulation of Surface and Subsurface Fluid Flow - Simulation of surface and subsurface flow has significant impact on the economic and urban development of the society. An ideal model for the simulation is expected to handle fluid bodies and simulating regions of arbitrary shape. The modeling system usually combines an overland flow model, a channel flow model, an infiltration model, a sewer flow model (in urban areas), and a model linking surface flow and sewer flow.

Most of the existing computational models are based on approximate or greatly simplified forms of the Saint Venant equations and the subsurface flow equation obtained from the Darcy's law. Our study offers a systematic understanding of the coupling between the surface and subsurface flows. This talk will introduce the modeling equations and their numerical approximations associated with the coupled system of surface and subsurface flows. A modified version of the Saint Venant equation will be presented and approximated by finite element and finite volume methods combined with the method of characteristics. A new discretization scheme for the Stokes equation will be discussed if time permits.

(December 9) Eric de Sturler: Preconditioning Generalized Saddle-Point Problems Using Cheap Approximations to the Schur Complement - Recently, for saddle point type problems we have analyzed block diagonal preconditioners derived from the work by Murphy, Golub, and Wathen [SISC 21, 2000], and we have derived constraint preconditioners from these. This gives interesting links between the two types of preconditioners. For the latter we proved better clustering ofthe eigenvalues, and we demonstrated for one class of optimization problems that this translates in significantly faster convergence. However, in these results we used the exact inverse of a 'Schur complement' type matrix derived from the splitting that defines our preconditioner. Although this worked well for the application discussed, it may be expensive for other problems. Moreover, we did not address systems arising in stabilized finite element discretizations that have a nonzero (2,2) block. We will address those issues in this presentation. This is joint work with Joerg Liesen (TU Berlin) and Chris Siefert (UIUC).