(February 1) Professor Donald Estep: Fast and Reliable Methods for Determining the Evolution of Uncertain Parameters in Differential Equations - A very common problem in science and engineering is the determination of the effects of uncertainty or variation in parameters and data on the output of a nonlinear operator. For example, such variations may describe the effect of experimental error or may arise as part of a sensitivity analysis of the model. The Monte-Carlo Method is a widely used tool for understanding such effects that employs random sampling of the input space in order to produce a pointwise representation of the output. It is a robust and easily implemented tool. Unfortunately, it generally requires sampling the operator very many times at a significant cost. Moreover, it provides no robust measure of the error of information computed from a particular representation. In this paper, we present an alternative approach for ascertaining the effects of variations and uncertainty in parameters in a differential equation that is based on techniques borrowed from a posteriori error analysis for finite element methods. The generalized Green's function is used to describe how variation propagates into the solution around localized points in the parameter space. This information can be used either to create a higher order method or produce an error estimate for information computed from a given representation. In the latter case, this provides the basis for adaptive sampling. Both the higher order method and the adaptive sampling methods are generally orders of magnitude faster than Monte-Carlo methods in a variety of situations.

(February 8) Prof. Alexis S. Mahalov: Global Regularity of the 3D Navier-Stokes Equations with Uniformly Large Initial Vorticity - We prove existence on infinite time intervals of regular solutions to the classical incompressible 3D Navier-Stokes Equations for fully three-dimensional periodic and almost periodic initial data characterized by uniformly large vorticity in R^3 and in bounded cylindrical domains; smoothness assumptions for initial data are the same as in local existence theorems. There are no conditional assumptions on the properties of solutions at later times, nor are the global solutions close to any 2D manifold. The global existence is proven using techniques of fast singular oscillating limits and the Littlewood-Paley dyadic decomposition. The approach is based on the computation of singular limits of rapidly oscillating operators and cancellation of oscillations in the nonlinear interactions for the vorticity field. With nonlinear averaging methods in the context of almost periodic functions, we obtain fully 3D limit resonant Navier-Stokes equations. We establish the global regularity of the latter without any restriction on the size of 3D initial data. With strong convergence theorems, we bootstrap this into the global regularity of the 3D Navier-Stokes Equations with uniformly large initial vorticity. We review applications of our mathematical techniques to numerical analysis of highly oscillatory PDE's arising in geophysical fluid dynamics. Global regularity of the 3D Navier-Stokes Equations of Geophysics is proven for all domain aspect ratios and all small Froude and Rossby numbers.

(February 15) Dr. Kyoung-Sook Moon: Convergence rate of an adaptive algorithm for differential equations - The theory of a posteriori error estimates suitable for adaptive refinement is well established. But the issue of convergence rates of adaptive algorithms is less studied. I will present a simple and general adaptive algorithm applied to ordinary and stochastic differential equations with proven convergence rates. The presentation has two parts: The error approximations used to build error indicators for the adaptive algorithm are based on error expansions with computable leading order terms. The adaptive algorithm, performing successive mesh refinements, either reduces the maximal error indicator by a factor or stops with the error asymptotically bounded by the prescribed accuracy requirement. Furthermore, the algorithm stops using the optimal number of degrees of freedom, up to a problem independent factor.

(February 22) Dr. Soeren Bartels: Approximation of Harmonic Maps -- Gradient Flow Approaches vs. Iterative Minimization - Harmonic maps are stationary points of the Dirichlet energy among functions with values in the unit sphere. Owing to the nonconvex constraint, harmonic maps are non-unique and fail to admit higher regularity properties. Moreover, the constraint prohibits the use of standard tools for the numerical approximation. In this talk we discuss stability and weak convergence of three numerical schemes. The first scheme consists in the minimization of the Dirichlet energy over suitable tangent spaces and a renormalization of the update in each iteration. The second approach penalizes the constraint and leads to a time-dependent Ginzburg-Landau equation. A projection method for the discretization of the harmonic map heat flow is the basis for the third approach. Besides stating sufficient conditions for stability and weak convergence to an exact solution, we indicate generalizations to the approximation of p-harmonic maps. Applications include liquid crystal theory, image processing, and micromagnetics.

Part of this talk is based on joint work with J.W. Barrett, X. Feng, and A. Prohl.

(March 1) Prof. John A. Pelesko: Electrostatic-Elastic Interactions - In 1968, in the context of investigating fundamental questions in electrohydrodynamics, G.I. Taylor studied the electrostatic deflection of elastic membranes. Utilizing soap film as the membrane material and applying a fixed high voltage potential difference between two supported circular membranes, Taylor showed experimentally that at a critical voltage the two membranes snap together and touch. That is, the equilibrium state where the membranes remained separate that existed at smaller voltages either became unstable or failed to exist. This instability is familiar to researchers in the MEMS (microelectromechanical systems) and NEMS (nanoelectromechanical systems) fields where it is known as the "pull-in" instability. In fact, in an interesting historical coincidence H.C. Nathanson and his coworkers studied this instability in the context of a primitive MEMS device at roughly the same time as Taylor was conducting his studies. Nathanson is responsible for the "pull-in" nomenclature and the analysis of a mass-spring model of this effect. Taylor, in conjunction with R.C. Ackerberg developed and numerically analyzed a more accurate membrane based model of electrostatic deflection. Recently, a rigorous analysis of this model was completed. Surprisingly, even this simple model of electrostatic deflection contains a rich solution set exhibiting a bifurcation diagram with infinitely many folds. In this talk, we provide an overview of recent results on the interaction of elastic membranes with electrostatic fields. We discuss a re-creation of the Taylor experiment, some new experimental results and discuss the relevance of this research to MEMS and NEMS systems.

(March 15) Khamron Mekchay: AFEM for the Laplace-Beltrami operator on graphs: a posteriori error estimation and convergence - We develop an adaptive finite element method (AFEM) to approximately solve the Laplace-Beltrami operator on graphs $\Gamma$ in $R^3$ which are described by a $C^1$, or piecewise $C^1$, function on a polygonal domain in $R^2$. We first derive a posteriori error estimators which account for both the approximation of the surface by a polyhedral surface in $R^3$, the geometric error, and the energy error incurred in solving the PDE. We show that both errors are coupled, and that our AFEM decreases their collective contribution, thereby becoming a contraction eventually and yielding convergence. The proof of convergence deals with the fact that the approximate surfaces are no longer nested, as in the flat case, which in turn leads to a novel quasi-orthogonality property in the energy norm. This is joint work with P. Morin (IMAL, Argentina) and R.H. Nochetto.

(March 29) Simon P. Schurr: Universal Duality in Conic Convex Optimization - For a pair of dual convex optimization problems in conic form, we provide simple necessary and sufficient conditions on the ``constraint matrices'' and cone under which a zero duality gap occurs for \textit{every} linear objective function and constraint ``right-hand side''. (The optimal objective function values may be both $+\infty$ or both $-\infty$.) We refer to this property as ``universal duality''. We show that universal duality is generic in both a metric sense and a topological sense, and can be verified by solving a single conic convex program. We illustrate our theory by studying a class of semidefinite programs that appear in control theory and signal processing applications.

This talk is based on joint work with Andr\'e Tits and Dianne O'Leary.

(April 05) Prof. Nick Trefethen: Who invented the great numerical algorithms? - Over coffee one day, a colleague suggested that most of the famous algorithms of numerical analysis had their roots with practitioners, not academics. This intrigued me -- could he be right? I decided to see what I could dig up concerning the origins of thirty of the greatest hits of our field. In this lecture I'll show you what I found.

(April 12) Prof. Mark Embree: A Dozen Cautionary Tales from "Spectra and Pseudospectra" - This past summer Nick Trefethen and I completed "Spectra and Pseudospectra," a book describing the theory and applications of nonnormal matrices and operators. In this talk I will extract a dozen of favorite examples that show how eigenvalues can provide misleading information about system behavior. This tour will begin with Toeplitz matrices and differential operators, then focus on the performance of numerical algorithms (for the solution of linear systems, eigenvalue problems, and differential equations), and conclude with applications in fluid dynamics and population ecology.

(April 19) Prof. Alfred Schmidt: Adaptive Finite Element Simulations for Macroscopic and Mesoscopic Models of Steel - During heat treatment of a steel workpiece and other production processes, fine structures like boundary layers for temperature, concentration, or phase fraction fields appear, while in other parts of the work piece the behaviour of these fields is quite smooth. Nevertheless, it is important to capture these structures during numerical simulations. Local mesh refinements in these regions is needed in order to resolve the behaviour in a sufficient way. On the other hand, these regions of special interest are changing during the process, making it necessary to move also the regions of refined meshes. Adaptive finite element methods present a tool to automatically give criteria for a local mesh refinement, based on the computed solution (and not only on a priori knowledge of an expected behaviour). We will present examples from heat treatment of steel, including phase transitions with transformation induced plasticity. On a mesoscopic scale of grains, similar methods can be used to efficiently and accurately compute phase field models for phase transformations, including stress dependent phase transformations.

(April 26) Dr. Jian Zhong Zhu: Computational modeling of hot tearing and cold crack in metal casting - Hot tearing and cold crack are two of the most detrimental defects occurring in the metal casting process. Therefore, it is practically important to be able to predict such phenomena numerically. Various criteria have been developed in the literature for assessing the susceptibility of hot tearing. Most of them, however, are not suitable for application in numerical simulation of industrial metal casting processes. In this study, we developed a hot tearing indicator based on the criterion of accumulated plastic strain. Combining with an appropriate constitutive model, such indicator is also extended to predict the development of cold crack in the casting parts. Numerical results are presented and compared with experimental results.

(May 05) Prof. Yann Brenier: Derivation of particle, string and membrane motions from Born-Infeld Electromagnetism - We derive classical particle, string and membrane motion equations from a rigorous asymptotic analysis of the Born-Infeld nonlinear electromagnetic theory. The Born-Infeld model is a modification of the classical Maxwell equations, designed in 1934 to cutoff, in a nonlinear fashion, the infinite electrostatic field generated by point charged particles, at a scale of order 10 to the -15 meters. Quickly shadowed by the success of Quantum Electrodynamics in the 40', this model has known a recent revival through the concept of Dirichlet-Branes in String Theory. Our asymptotic analysis starts by adding to the Born-Infeld equations the corresponding energy-momentum conservation laws and write the resulting system as a non-conservative symmetric system of ten first-order evolution PDEs. (Surprisingly enough, the extended system enjoys Galilean invariance, just like classical Fluid Mechanics.) Then, we show that four rescaled versions of the system have smooth solutions existing in the time interval where the corresponding limit problems have smooth solutions. These limits respectively describe the classical linear Maxwell equations (for weak fields), string motions (for large magnetic fields and moderate electric fields), particle motions (for large electromagnetic fields) and membrane motions. This is partly a joint work with Wen-An Yong, from Heidelberg.

(May 06) Prof. Yann Brenier: String integration of some MHD equations - We first review the link between strings and some Magnetohydrodynamics equations. Typical examples are the Born-Infeld system, the Chaplygin gas equations and the shallow water MHD model. They arise in Physics at very different (from subatomic to cosmologic) scales. These models can be exactly integrated in one space dimension by solving the 1D wave equation and using the d'Alembert formula. We show how an elementary "string integrator" can be used to solve these MHD equations through dimensional splitting. A good control of the energy conservation is needed due to the repeted use of Lagrangian to Eulerian grid projections. Numerical simulations in 1 and 2 dimensions will be shown.

(May 10) Prof. Ramani Duraiswami: Computations of Singular and Nearly Singular Integrals in Boundary Integral Methods - For many problems boundary integral (a.k.a. boundary element) methods have long been considered as very promising. These methods can easily handle complex shapes, lead to problems in fewer variables, and further only require meshing of the boundary rather than the entire domain.

Despite these advantages, two issues have impeded the widespread adoption of boundary integral methods. First, these techniques lead to linear systems with dense non-symmetric matrices, for which efficient iterative solvers are not usually available. Second, the computation of the matrix elements requires the computation of various integrals. However the diagonal elements of these matrices require computation of weakly singular, singular or hypersingular integrals. These have to be dealt with separately, and require a complicated analytical apparatus to regularize the singularity. Further they make the development of BEM software difficult as the singular integrals need special separate treatment in the software.

The fast multipole method presents a promising approach to removing the first difficulty. This talk is concerned with the second issue: the treatment of the singular integrals. We are developing a new technique, that appears capable of efficient computation of the integrals that arise from the singular elements. The method appears to be of general applicability for the calculation of all the singular and nearly integrals that arise in different BEM formulations, and further fit naturally in to the FMM accelerated solution of the linear system. Our method only employs regular quadrature formulae. Details of the new method will be presented, along with its tests in the solution of the Laplace and Helmholtz equations via the BEM.

(joint work with Nail Gumerov)