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(September 4) Prof. José Manuel Cascón Barbero: Quasi-Optimal Convergence Rates for AFEM - We analyze the simplest and most standard adaptive finite element method (AFEM), with any polynomial degree, for general second order linear, symmetric elliptic operators. As it is customary in practice, AFEM computes the standard residual estimator, marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property. We prove that AFEM is a contraction for the sum of energy error and scaled error estimator, between two consecutive adaptive loops. This geometric decay is instrumental to derive optimal cardinality of AFEM. We show that AFEM yields a decay rate of energy error plus oscillation in terms of number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.

Finally, we extend this result to AFEM based on other families of error estimators. This requires some minor but important changes in the algorithm: the marking has to be made according to the sum of error estimator and oscillation and the refinement has to satisfy an interior node property.

This is joint work with C. Kreuzer, R.H. Nochetto, and K. G. Siebert.

(September 11) Prof. Constantin Bacuta: Schur Complements on Hilbert Spaces and Saddle Point Systems - For any continuous bilinear form defined on a pair of Hilbert spaces satisfying the Babusca-Brezzi compatibility condition, symmetric Schur complement operators can be defined on each of the two Hilbert spaces. Using the spectral properties of the Schur operators, we prove that for any symmetric saddle point problem, the inexact Uzawa algorithm converges provided that the inexact process for inverting the residual at each step has the relative error smaller than 0.333333333333. Our analysis, combined with standard techniques of discretization and a posteriori error estimates, could lead to new and efficient adaptive algorithms for solving saddle point systems.

(September 18) Prof. Ludmil Zikatanov: Uniform preconditioning for nearly singular elliptic problems - We discuss convergence results for general successive subspace correction methods as iterative methods for solving and preconditioning nearly singular systems of equations. The goal is to provide parameter independent estimates under appropriate assumptions on the subspace solvers and space decompositions. The main result is based on the assumption that any component in the kernel of the singular part of the system can be decomposed into a sum of local (in each subspace) kernel components. This assumption also covers the case of ``hidden'' nearly singular behavior due to decreasing mesh size in the systems resulting from finite element discretizations of second order elliptic problems. To illustrate the abstract convergence framework, we show how these tools can be applied to analyze multigrid methods for H(div) and H(curl) systems.

This is a joint work with Jinchao Xu (Penn State), Young Ju Lee (UCLA) and Jnbiao Wu (Beijing University).

(September 25) Dr. Nail A. Gumerov: Terascale on the Desktop: Fast Multipole Methods on Graphical Processors - Graphics Processors (GPUs) provide access to significant computational processing resources at low costs. They contain a large number of processing units with access to local and shared memory, and achieve significant speedups vis--vis CPUs on problems that can be mapped to their SPMD architecture. Many applications in molecular dynamics, astrophysics and other areas require the O(N^2) computation of mutual Coulombic potentials and forces among N particles. The FMM provides a hierarchical approximate algorithm, to compute these quantities to a specified error �at O(NlogN) cost and memory. More generally FMM like algorithms are used to accelerate matrix vector products in applications such as the solution of integral equations, radial basis function interpolation/evaluation and machine learning. I will discuss both the GPU architecture for scientific computing, and the modifications necessary to the FMM for this architecture. On an NVIDIA 8800 GTX installed on a PC, our FMM algorithm achieves timings that if computed using an O(N^2) algorithm correspond to speeds of 25-45 Tflops (for achieved L2 errors of ~ 10-6 - 2�0-4).

(joint work with Ramani Duraiswami)

(October 2) Prof. Howard Elman: Fast Iterative Solution of Models of Incompressible Flow - We discuss new efficient algorithms for computing the numerical solution of the incompressible Navier-Stokes equations. We show that preconditioning algorithms that take advantage of the structure of the linearized equations can be combined with Krylov subspace methods to produce algorithms that are optimal with respect to discretization mesh size, largely insensitive to Reynolds numbers, and easily adapted to handle both steady and evolutionary problems. We also show the relation between these approaches and traditional methods derived from operator splittings, and we demonstrate the performance of the new methods in some practical settings.

(October 9) Prof. Adam Oberman: Fully nonlinear second order elliptic Partial Differential Equations: numerics and game interpretations - The theory of viscosity solutions gives powerful existence, uniqueness and stability results for first and second order degenerate elliptic partial differential equations. The approximation theory developed by Barles and Sougandis in the early nineties gave conditions for the convergence of numerical schemes.

While there has been a lot of work on first order equations, there has been very little work on genuinely nonlinear or degenerate second order equations. This despite the fact that many of these equations are of the subject of current research and applications. For example: level set motion by mean curvature, the Infinity Laplacian, the PDE for the value function in Stochastic Control, the Monge-Ampere equation.

In this talk, we build convergent schemes for the aforementioned equations, and also for some less well-known or newer equations, including the Pucci Equations and a new PDE for the convex envelope.

Another subject of recent work has been finding non-generic stochastic control or game interpretations of the PDEs. While it has been known for some time that this is always possible, there have been recent interpretations for the motion by mean curvature (by Kohn and Serfaty) and for the Infinity Laplacian (by Peres-Schramm-Scheffield-Wilson). These interpretations are linked to simple numerical approximation schemes which we build.

We will also discuss a general framework for building these types of schemes, and extensions to higher order accurate methods.

The majority of the work discussed can be found on the author's web page.

(October 16) Prof. John Osborn: Quadrature for Meshless Methods - It is well-known that creating effective quadrature schemes for Meshless Methods (MM) is an important problem (see, e.g., A stabilized conforming nodal integration for Galerkin mesh free methods, J.-S. Chen, C.-T. Wu, S. Yoon, and Y. You, Int. J. Numer. Meth. Engng. 2001; 50:435{466). In this talk we discuss quadrature schemes for MM of order one (MMs that reproduce linear functions). We consider the Neumann Problem and derive an estimate for the energy norm error between the exact solution and the quadrature approximate solution in terms of the mesh parameter and quantities that measure the relative errors in the stiffness matrix, in the lower order term, and in the right-hand side vector, respectively, due to the quadrature. The major hypothesis in the estimate is that the quadrature stiffness matrix has zero row sums, a hypothesis that can be easily achieved by a simple correction of the diagonal elements. The talk is based on joint work with Ivo Babuska, Uday Banerjee, and Helen Li.

(October 23) Dr. Shawn Walker: Modeling and Simulating the Fluid Dynamics of Electrowetting On Dielectric (EWOD) with Contact Line Friction - Electrowetting On Dielectric (EWOD) refers to a parallel-plate device that moves fluid droplets through electrically actuated surface tension effects. These devices have potential applications in biomedical `Lab-On-A-Chip' devices (automated DNA testing, cell separation) and controlled micro-fluidic transport (e.g. mixing and concentration control). The fluid dynamics are modeled using Hele-Shaw type equations (in 2-D) with a focus on including the relevant boundary phenomena. Specifically, we model contact line pinning as a static (Coulombic) friction effect and effectively becomes an inequality constraint for the motion of the liquid-gas interface that accounts for the "sticking" effect of the interface. The model is presented in a variational framework and is discretized using Finite Elements. The curvature/surface tension is discretized in a semi-implicit way for accuracy using an explicit representation of the interface. Simulations are presented and compared to experimental videos of EWOD driven droplets. These experiments exhibit droplet pinching and merging events and are reasonably captured by our simulations.

(October 30) Prof. Doron Levy: Numerical Methods for Nonlinear Dispersive Equations - In 1993 Rosenau and Hyman introduced a family of nonlinear dispersive equations with compactly supported soliton solutions, the so-called "compactons". These models are particularly interesting due to the local nature of their solutions, which serves as a caricature of a wide range of phenomena in nature. The non-smooth interfaces of the compactons and the strong nonlinearity of the equation present significant theoretical and numerical challenges.

In this talk we will discuss two approaches for numerically approximating compacton solutions. First, we will present a particle method that is based on an extension of the diffusion-velocity method of Degond and Mustieles to the dispersive framework. This is a joint work with A. Chertock.

Our second approach is based on deriving local discontinuous Galerkin methods for approximating the solutions of more general nonlinear dispersive problems. These methods are designed as to satisfy stability properties that correspond to the conservation laws of the approximated PDEs. This is a joint work with J. Yan and C.-W. Shu.

(November 6) Prof. Jean Luc Guermond: Nonlinear approximation in L¹ - A finite element-based approximation technique for solving nonlinear first-order PDEs will be presented. The method is based on minimizing a functional comprising the L¹-norm of the residual plus possible entropy terms depending on the nature of the PDE. Convergence results will be stated for stationary Hamilton-Jacobi equations in one and two space dimensions. Algorithms for computing approximate minimizers with optimal complexity will be described.

(November 15) Prof. Albert Cohen: Near optimal recovery of arbitrary signals from incomplete measurements - Compressed sensing is a recent concept in signal and image processing where one seeks to minimize the number of measurements to be taken from signals or images while still retaining the information necessary to approximate them well. The ideas have their origins in certain abstract results from functional analysis and approximation theory but were recently brought into the forefront by the work of Candes-Romberg-Tao, and Donoho who constructed concrete algorithms and showed their promise in application. There remain several fundamental questions on both the theoretical and practical side of compressed sensing. This talk is primarily concerned about one of these issues revolving around just how well compressed sensing can approximate a given signal from a given budget of fixed linear measurements, as compared to adaptive linear measurements. More precisely, we consider discrete N-dimensional signals x with N>>1, allocate n<<N linear measurements of x, and we describe the range of k for which these measurements encode enough information to recover x to the accuracy of best k-term approximation. We also consider the problem of having such accuracy only with high probability.

(November 16) Prof. Albert Cohen: Adaptive Approximation by Greedy Algorithms - This talk will discuss computational algorithms that deal with the following general task : given a function f and a dictionary of functions D in a Hilbert space, extract a linear combination of N functions of D which approximates f at best. We shall review the convergence properties of existing algorithms. This work is motivated by applications as various as data compression, adaptive numerical simulation of PDE's, statistical learning theory.

(November 20) Prof. Shi Jin: Computations of Multivalued Solutions in Nonlinear PDEs - Many physical problems arising from high frequency waves, dispersive waves or Hamiltonian systems require the computations of multivalued solutions which cannot be described by the viscosity methods. In this talk I will review several recent numerical methods for such problems, including the moment methods, kinetic equations and the level set method. Applications to the semiclassical Schroedinger equation and Euler-Poisson equations with applications to modulated electron beams in Klystrons will be discussed.

(December 4) Prof. George Hsiao: Boundary Element Methods for the Last 30 Years - Variational methods for boundary integral equations deal with weak formulations of the equations. Boundary element methods are numerical schemes for seeking approximate weak solutions of the corresponding boundary variational equations in finite-dimensional subspaces of the Sobolev spaces with special basis functions, the so-called boundary elements. This lecture gives an overview of the method from both theoretical and numerical point of view. It summaries the main results obtained by the author and his collaborators over the last 30 years. Fundamental theory and various applications will be illustrated through simple examples. Some numerical experiments in elasticity as well as in fluid mechanics will be included to demonstrate the efficiency of the methods.

(December 11) Prof. Manuel Triglio: Some open problems in numerical relativity - The problem of describing the collision of two compact objects such as black holes and/or neutron stars in General Relativity is of great interest. This is so because these processes are expected to be one of the main sources of gravitational waves to be measured by laser interferometer gravitational wave observatories such as LIGO. From the simulation point of view, the modeling of these scenarios presents a number of challenges. First, physically interesting evolutions are quite long, and many numerical schemes tend to acummulate rather large errors overy so many cycles. Second, one typically needs to solve the equations in complex geometries, which actually change over time. Finally, the Einstein equations have constraints, with associated subtleties when imposing boundary conditions. I will describe current approaches and some open mathematical/numerical/scientific computing problems which are of great interest in realistic applications. These include boundary conditions, well posedness of the associated initial-boundary value problem, multi-domain high order and spectral methods, including the possibility of shocks, numerical stability and parallelization strategies.