(February 3) Dr. Kyoung-Sook Moon: Adaptive Monte Carlo Algorithm for Stochastic Differential Equations - I will present an adaptive Monte Carlo algorithm for weak approximations of stochastic differential equations. The goal is to compute an expected value of a given function depending on a stochastic process. Based on an a posteriori error expansion, the adaptive algorithm is proven to stop with asymptotically optimal number of final time steps and the approximation error is asymptotically bounded by the specified error tolerance as the tolerance parameter tends to zero. Finally, I will show numerical results from computations of barrier options in financial mathematics. These results show that the adaptive algorithm achieve the time discretization error of order N^{-1} with N adaptive time steps, while the error is of order N^{-1/2} for a method with N uniform time steps.

(February 10) Dr. Francisco Pena: Some Contributions to the Modeling of Ferro-alloys Production - Our motivation is to model the production of ferro-alloys in reduction furnaces. The main part are the electrodes. Heat is generated in them by the Joule and by the electric arc at the bottom. A sui table model is the system of equations composed by the Maxwell equation in low frequency regime (eddy currents) coupled with the transient heat equation with phase change. This model is related to the so called ``thermistor'' problems. The main difficulties are the belonging of the Joule effect to the non-reflexive space $L^1$ and the Stefan problem in the heat equation. We will show a result about existence of weak solution for a simpler case, when enthalpy is uni-valued.

(February 17) Dr. R. Bruce Kellogg: Corner Singularities in the Two Dimensional Compressible Navier-Stokes System - The compressible Navier-Stokes system, and linearized versions of this system, are not elliptic. When a steady state boundary value problem for this system is posed on a polygonal domain, one may ask for the behavior of the solution near a vertex of the polygon. This talk will discuss some results of J. R. Kweon and myself, as well as some unresolved questions, on this problem.

(February 24) Dr. Thomas Russell: Oh no, not the wiggles again! A revisit of an old problem and a new approach. - Lumping is often used to control oscillations in schemes that use the method of weighted residuals. Standard lumping procedures add diffusion indiscriminately, resulting in excessive numerical diffusion in solutions. Here it is shown that the mass matrix can be selectively lumped (SLUMPED), with an optimal amount of diffusion added to each row of the mass matrix. The amount of diffusion added is calculated from the right-hand-side vector. The optimal amount of diffusion is found in 4 steps. First, the monotonicity problem is recast in the form of a maximum principle. Second, for a 2x2 element matrix, the amount of diffusion is calculated for an arbitrary right-hand side so that the solution obeys a maximum principle. Third, the result is generalized for larger matrices. Finally, the result is recast to meet the monotonicity requirement. The result is an equation giving the amount of diffusion to be added in terms of a given right-hand-side vector.

Selective lumping is shown to be effective for both an Eulerian-Lagrangian Localized Adjoint Method (ELLAM) solution of the transport equation and a finite element solution of the heat equation. In both cases, solutions were monotonic and contained less numerical diffusion than in standard lumping schemes. The SLUMPING method is general and can be applied to any numerical approximation based on the method of weighted residuals.

(joint work with Philip J. Binning)

(March 2) Dr. Bing Song: A Fast Algorithm for Variational Level Set Segmentation - We develop a fast method to solve Variational Level Set Segmentation. This direct method improves the computational speed dramatically. It differs from previous methods in that we do not need to solve the Euler-Lagrange equation of the underlying variational problem. Instead, we calculate the energy directly and check if the energy is decreased when we change a point inside the level set to outside or vice versa. We analyze the algorithm and prove that under most initial conditions, we only need one sweep over the pixels to converge to the correct solution for 2-phase images. Another advantage of this method is that the gradient of the functional is not needed. This enables it to be applied to broader range of optimization problems.

(March 4) Vanessa Lopez: Periodic Solutions of Chaotic Partial Differential Equations - We consider the problem of finding relative time-periodic solutions of chaotic partial differential equations with symmetries. Relative time-periodic solutions are solutions that are periodic in time, up to a transformation by an element of the equations' symmetry group. As a model problem we work with the 1D complex Ginzburg-Landau equation (CGLE), which is a standard example of an evolution equation that exhibits chaotic behavior. The problem of finding relative time-periodic solutions numerically is reduced to one of finding solutions to a system of nonlinear algebraic equations, obtained after applying a spectral-Galerkin discretization in space and time to the CGLE. The discretization is designed to include as an unknown the group element that defines a relative time-periodic solution. Using this approach, we found a large collection of distinct relative time-periodic solutions in a chaotic region of the CGLE. These solutions, all of which have broad temporal and spatial spectra, were previously unknown. There is a great deal of variety in their Lyapunov spectra and spatio-temporal profiles. Moreover, none bear resemblance to the time-periodic solutions of the CGLE studied previously. We also briefly discuss the problem of finding relative time-periodic solutions to the Navier-Stokes equations and present some preliminary work done towards its solution.

(March 9) Dr. Andreas Prohl: Analysis and Numerics of Total Variation Flow and Regularized Mumford-Shah Flow - Automatic denoising and segmentation are principle tasks in image processing. Mathematically, the goal is to detect/preserve essential geometric structures (edges, domains,...) during the filtering process to extract relevant image information. The first part of the talk addresses analysis of the total variation flow for initial data $u_0 \in L^2(\Omega)$ by relating it to the prescribed mean curvature flow; then, a numerical analysis for a fully discrete realization is presented which leads to proper scaling laws to balance regularization and discretization effects. The second part of the talk addresses analysis and numerics of the gradient flow for a $\Gamma$-convergent approximation by Ambrosio and Tortorelli of the Mumford-Shah functional. I present a numerical scheme which guarantees convergence of the whole sequence $\{ (\, u_h, \varphi_h\, )\}_h$ towards weak solutions of the limiting problem; in particular, this analysis is based on explicit characterization of a (possible) singularity set. Computational experiments for both scenarios are presented to illustrate solution's behavior of the discussed problems. --- This is joint work with X.~Feng (Univ. of Tennessee).

(March 16) Dr. Haesun Park: Dimension Reduction for Undersampled Problems in Pattern Analysis - Dimension reduction is imperative for efficient handling of data represented in a very high dimensional space. Linear Discriminant Analysis (LDA) has been successfully applied to dimension reduction problems for many decades. However, in undersampled problems where the number of data items is smaller than the dimension of the data space, it is difficult to apply LDA due to the potential singularity of the scatter matrices caused by high dimensionality.

In this talk, we examine an optimization criterion that mathematically models many commonly used dimension reduction methods such as LDA, Principal Component Analysis (PCA), and Latent Semantic Indexing (LSI). Then we present efficient new dimension reduction algorithms such as generalized LDA based on the generalized singular value decomposition (LDA/GSVD) and the Orthogonal Centroid Method (OCM), which are applicable regardless of the relative dimensions of the data matrices. By utilizing the GSVD, we establish a relationship between support vector machines (SVMs) and generalized linear discriminant analysis. The LDA/GSVD and OCM can easily be extended to nonlinear dimension reduction methods by using kernel functions. Also, when LDA/GSVD is reformulated as a a Mean Squared Error (MSE) problem, a highly efficient incremental dimension reduction method can be derived. Our substantial experimental results in text classification, facial recognition, and fingerprint classification demonstrate the effectiveness of the proposed methods.

(March 30) Dr. John D. Anderson: History of the Evolution of the Navier-Stokes Equations And Computational Fluid Dynamics: A Marriage Made in Heaven? - The Navier-Stokes equations are a result of intellectual thought that begins with early Greek science and mathematics. A brief history of the intellectual evolution of the Navier-Stokes equations is discussed. By the middle of the 19th century, the modern form of these equations was well in hand, but their general solution was not. To the present, no general analytical solution of the Navier-Stokes equations exist. The dust of ages was shaken off these equations, however, when practical computational fluid dynamics (CFD) evolved rather quickly in the middle of the 20th century, and allowed the numerical solution of the Navier-Stokes equations for a host of practical applications. The history of the evolution of CFD is discussed, and its impact on fluid dynamics in general, and the solution of the Navier-Stokes equations in particular, is highlighted.

(April 6) Dr. Peter Monk: A discontinuous Galerkin method for linear symmetric hyperbolic systems in inhomogeneous media - The Discontinuous Galerkin (DG) method provides a powerful tool for approximating hyperbolic problems. We shall present a new space-time DG method for linear time dependent hyperbolic problems written as a symmetric system (including the wave equation and Maxwell's equations). The main features of the scheme are that it can handle inhomogeneous media, and can be time-stepped by solving a sequence of small linear systems resulting from applying the method on small collections of space-time elements. We show that the method is stable provided the space-time grid is appropriately constructed (this corresponds to the usual time-step restriction for explicit methods, but applied locally) and give an error analysis of the scheme. We shall also provide some simple numerical tests of the algorithm applied to the wave equation in two space dimensions (plus time).

(April 13) Dr. Georgios Akrivis: Linearly implicit methods for nonlinear parabolic equations - We discuss the discretization of nonlinear parabolic equations

u'(t) + Au(t) = B(t,u(t))

in a Hilbert space setting, by combinations of rational implicit and explicit multistep methods. The resulting schemes are linearly implicit and include as particular cases implicit--explicit multistep schemes as well as the combination of implicit Runge--Kutta schemes and extrapolation. An optimal condition for the stability constant is derived under which the schemes are locally stable. Optimal order error estimates are established.

The results are extended to equations of the form

Lu'(t) + Au(t) = B(t,u(t))

By modifying the backward differentiation formulae, we arrive at linearly implicit schemes with improved stability properties.

(April 20) Dr. Tobias von Petersdorff: Fast numerical methods for parabolic problems in high dimensions - Fast numerical methods for parabolic problems in high dimensions

We consider a parabolic initial value problem in d space dimensions. One application is the pricing of options for a basket of d assets where d can be larger than 20. The initial data are assumed to be rough (H^epsilon), therefore the solution has singular behavior at t=0.

Standard space discretization with mesh size h and time discretization with M time steps leads to a work of O(M h^-d) which is impractical for d>3. We propose a space discretization with wavelet based sparse grid spaces requiring only O(h^-1) degrees of freedom. For the time discretization we use an hp discontinuous Galerkin method with exponential convergence. The resulting method has a total work of O(h^-1 log(h)^c). We give a complete error analysis of space discretisation, time discretisation and of the iterative solution of the resulting linear systems. Numerical results for dimension up to 25 confirm the theoretical estimates.

This is joint work with C. Schwab (ETH Zurich).

(April 27) Dr. Marco Verani: On the control of output linear functionals - The purpose of this talk is to explore the question of accurate approximation of output linear functionals of linear elliptic PDEs, through the use of adjoint problems and duality arguments. We present a possible paradigm for an adaptive algorithm designed to produce specially tuned grids for the computation of the functional value within a given tolerance. Such an algorithm stems from an equivalent saddle point formulation of the initial problem. In particular we make use of an adaptive Uzawa algorithm formulated in infinite dimensions [Dahmen et al., Nochetto et al.]

(joint work with S. Micheletti and S. Perotto)

(May 6) Dr. Benoît Perthame: Selection, mutation: adaptive dynamics - Living systems are subject to constant evolution. The environment, which we consider as a nutriment available for all the population, allows individuals to develop and thus to select the `best adapted trait' in the population. On the other hand the population can develop with small variance on the trait (mutations). We will give a mathematical model of such dynamics and show that an asymptotic method allows us to describe the `best adapted trait' and eventually to compute bifurcations which lead to the cohabitation of two different traits.

(May 7) Dr. Benoît Perthame: Mathematical models for cell motion - Several transport-diffusion systems arise as simple models in chemotaxis (motion of bacterias or amebia interacting through a chemical signal) and in angiogenesis (development of capillary blood vessels from an exhogeneous chemoattractive signal by solid tumors). These systems describe the evolution of a density (of cells or blood vessels) coupled with the evolution equation for a chemical substance, through a nonlinear transport term depending on the gradient of the chemoattracting substance. Such systems are successful in recovering various qualitative behavior (chemotactic collapse, ring dynamics). Endothelial (i.e. cells forming blood vessels) have a tendency to form different patterns, initiating the vessels shape. Then hyperbolic models seem better adapted to describe this kind of network formation.

We will present these models, their main mathematical properties (quantitative and qualitative), numerical simulations and, for bacteria E. Coli, we will give a microscopic picture based on a kinetic modelling of the interaction (nonlinear scattering equation). We show that such models can have global solutions that converge in finite time to the Keller-Segel model, as a scaling parameter vanishes. This point of view has also the advantage of unifying all the models.

(May 11) Dr. Jian-Guo Liu: Accurate, Stable and Efficient Navier-Stokes Solvers Based on Explicit Treatment of the Pressure Term - In this talk, I will present numerical schemes for the incompressible Navier-Stokes equations based on a primitive variable formulation in which the incompressibility constraint has been replaced by a pressure Poisson equation. The pressure is treated explicitly in time, completely decoupling the computation of the momentum and kinematics equations. The result is a class of extremely efficient Navier-Stokes solvers. Full time accuracy achieved for all flow variables. The key to the schemes is a Neumann boundary condition for the pressure Poisson equation which enforces the incompressibility condition for the velocity field. Irrespective of explicit or implicit time discretization of the viscous term in the momentum equation, the explicit time discretization of the pressure term does not affect the time step constraint. Indeed, we show some unconditional stability properties of the this new formulation for the Stokes equation with explicit treatment of the pressure term and first or second order implicit treatment of the viscous term. Systematic numerical experiments for the full Navier-Stokes equations indicate that a second order implicit time discretization of the viscous term, with the pressure and convective terms treated explicitly, is stable under the standard CFL condition. Additionally, various numerical examples are presented, including both implicit and explicit time discretizations, using spectral and finite difference spatial discretizations, demonstrating the accuracy, flexibility and efficiency of this class of schemes. In particular, a Galerkin formulation is presented requiring only C⁰ elements to implement.