(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.
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