(September 7) Prof. Ronald DeVore:
Estimators for supervised learning -
The following regression problem occurs in various settings of
supervised learning. We are given a domain $X\subset \R^d$ and
an interval $Y=[-M,M]\subset \R$. There is a probability measure
$\rho$ defined on $Z:=X\times Y$ which is unknown to us. We see
information about $\rho$ from samples
$\cZ:=\{(x_i,y_i)\}_{i=1}^m$ which are drawn at random with
respect to $\rho$. We are interested in learning the function
$f_\rho(x):=\int_Yy\, d\rho(y|x)$ which is the expected value of
$y$ given $x$. One way of constructing an estimator for $f_\rho$ is to choose a
space of functions $\cH$ (called the hypothesis space in learning
theory) and find the function $f_\cZ$ from $\cH$ which minimizes
the empirical least squares error. Cucker and Smale have shown
how to estimate the performance of such an estimator by using the
Kolmogorov entropy of the set $\cH$ in $L_\infty(X)$. They then
build estimators based on a priori knowledge that $f_\rho$ is in
a ball $B(W)$ of a smoothness class $W$, they choose $\cH$
depending on this knowledge and then obtain the estimator
$f_\cZ$. This talk will center on several ways to improve on these
estimators. We show that using tools from approximation theory
such as linear or nonlinear widths then we can construct
estimators where the right side of \eref{experror} is replaced by
$cm^{-\frac{r}{2r+d}}$ and that this decay rate is optimal.
Secondly, the class $W^r(L_\infty(X))$ can be replaced by much
larger classes such as the Sobolev classes $W^r(L_p(X))$, $p>d$.
Perhaps most importantly we show how to construct estimators that
do not require any a priori assumptions on $f_\rho$ but still
perform optimaly for a large range of smoothness classes. These
latter estimators are built using nonlinear methods of
approximation such as adaptive partitioning and $n$ term wavelet
approximation. This work is in collaboration with Peter Binev, Albert Cohen,
Wolfgang Dahmen, Gerard Kerkyacharian, Dominique Picard, and
Vladimir Temlyakov. (September 14) Khamron Meckhay:
Convergence of adaptive FEM for second-order linear elliptic PDE -
We prove convergence of adaptive finite element method (AFEM) for general
second order linear elliptic PDE, thereby extending the result of Morin,
Nochetto, Seibert [SINUM 2000, SIGEST 2002]. The proof relies on
quasi-orthogonality, which accounts for the bilinear form not being a scalar
product, together with novel error and oscillation reduction estimates,
which now do not couple. We show that AFEM is a contraction for the sum of
energy error plus oscillation. Numerical experiments, including oscillatory
coefficients and convection-diffusion PDE, illustrate the theory and yield
optimal meshes. (September 21) Dr. Anthony J. Kearsley:
An infeasible point method for solving a
class of multidimensional scaling problems -
The term Multidimensional Scaling (MDS) is used to describe a
collection of techniques for constructing configurations of
points from information about inter-point distances. Many of
these techniques were originally developed for psychometric
applications. MDS is now widely used to visualize multivariate
data sets. Important contemporary applications include the
problem of inferring the 3-dimensional structure of a molecule
from information about its inter-atomic distances (for example
some Nuclear Magnetic Resonance (N MR) output). MDS can be
formulated as a collection of optimization problems, most of
which require numerical solution. Most well-known and frequently
used MDS algorithms are actually first-order (gradient) methods
for solving specific optimization problems. Some researchers have
argued that second-order methods are inappropriate for MDS. This
talk presents a second-order method based on a reformulation of
these problems through a smooth approximation of the objective
functions and an expansion of the number of variables that couple
through the constraints. This formulation can be solved, in turn,
using constrained optimization algorithms for nonlinear
programming problems. We will conclude the talk with a simple
example of reconstructing a molecular conformation from NMR data
and inter-atomic distance constraints. (September 28) Dr. Raul Tempone:
Convergence rates for an adaptive dual weighted residual finite
element algorithm
-
Basic convergence rates are established for an adaptive algorithm that
aims to approximate functionals of solutions to second order elliptic
partial differential equations in bounded domains.
The results are based on the dual weighted residual error
representation, applied to tensor products of linear finite elements on
quadrilateral meshes.
In contrast to the usual aim to derive an a posteriori error estimate,
this work derives, as the mesh size tends to zero, a uniformly
convergent error expansion for the error density, with computable
leading order term.
The main result proves that the adaptive algorithm based on successive
subdivisions of elements either reduces the maximal error indicator with
a factor or stops with the error asymptotically bounded by the tolerance
using the optimal number of elements, up to a problem independent
factor.
Numerical experiments show that the algorithm is useful in practice also
for relative large tolerances, much larger than the small tolerances
needed to theoretically guarantee that the algorithm works well. (October 5) Prof. Andreas Veeser:
Convergent adaptive finite elements for the elliptic obstacle problem
-
We consider the minimization of an `inhomogeneous Dirichlet energy' in
the set of functions above a piecewise affine obstacle. In order to
approximate the minimum (point and value), we propose an adaptive
algorithm that relies on minima with respect to admissibile linear
finite element functions and on a new a~posteriori estimator for the
error in the minimum value. It is proven that the generated sequence
of approximate minima converges to the exact one. Furthermore, our
numerical results indicate that the convergence rate coincides with
the one of nonlinear approximation. This is a joint work with Kunibert G. Siebert (Universit"at Augsburg). (October 12) Prof. Eitan Tadmor:
Twenty examples of entropy stable schemes
-
We provide a general overview of entropy stability theory for
difference approximations in the context of quasilinear balance laws and
related time-dependent problems governed by additional dissipative and
dispersive forcing terms. As our main toll we use a comparison principle, comparing the entropy
production of a given scheme against properly chosen entropy-conservative
schemes. To this end, we introduce closed-form expressions for new
(families) of new entropy-conservative schemes, keeping the 'perfect
differencing' property of the underlying differential form. In particular,
entropy stability is enforced on rarefactions while keeping sharp resolution
of shock discontinuities. We employ this framework for a host of first- and
second-order accurate schemes. This approach yields precise haracterizations
of entropy stable semi-discrete schemes for both scalar problems and
multi-dimensional systems of equations. We extend these results to the fully
discrete case, where the question of stability is settled under optimal CFL
conditions using a complementary approach based on homotopy arguments.
(October 19) Dr. Edward J. Garboczi:
Computing the Linear Elastic Properties of Random Materials
-
Many of the materials whose mechanical properties we care about
are random materials at some level. Simple composites made up of
mono-sized inclusions randomly dispersed in a uniform matrix are
only random at one length scale, the inclusion size. Some random
materials are not inclusion-matrix type composites or are
multi-phase or multi-length scale materials, which makes the job
of computing their elastic properties even harder. And in many
materials, non-linear properties like ultimate strength are of
more technological importance than linear elastic properties.
This talk will give a brief history of attempts at computing the
linear elastic properties of random materials, as well as several
examples of how I currently carry out this kind of research with
scalar and parallel finite element algorithms.
(October 26) Prof. Benjamin Shapiro:
Results and Challenges in Control of Micro Fluidic Systems
-
his talk will address our efforts to integrate research in system design
and feedback control with the rapid progress being made in micro-fluidic
systems. I will touch on results and challenges in controlling 3
micro-fluidic systems. The first is the Electro-Wetting-On-Dielectric
(EWOD) system developed at UCLA by CJ Kim. Here a grid of electrodes is
used to locally change surface tension forces on liquid droplets: by
choosing the electrode firing sequence it is possible to move, split,
join, and mix liquids in the droplets. The second system is a
micro-fluidic "no-laser tweezer" system that can be used to steer many
particles at once. I will show how we use flow control to create an
underlying, time-varying fluid flow that carries all the particles at
once along their desired trajectories. The third system we are just
beginning to consider: using laser tweezers to control vesicle shapes. (November 2) Prof. Albert Cohen:
Adaptive multiscale methods for transport equations - theory
and algorithms
-
In this talk, we shall first describe the approximation
theoretic background which supports the analysis of adaptive
methods in numerical simulations. We shall then focus on
non-linear transport PDE's and propose a multiscale approach
towards adaptivity, which will be illustrated by numerical experiments
for conservation laws (Burgers and Euler) and kinetic
equations (Vlassov-Poisson) (November 9) Prof. Dianne P. O'Leary:
The Linear Algebra of Quantum Computing
-
Conventional computer circuits perform logic operations on bits of
data through a sequence of gates. Quantum computers transform data
by multiplication by a unitary matrix, using a sequence of gates
determined by a factorization of that matrix into a product
of allowable elementary factors. This talk describes how to
use matrix factorization to design quantum circuits with optimal
order gate counts for computing with qubits (0-1 logic) and qudits
(multilevel logic). This is joint work with Stephen Bullock and
Gavin Brennan at NIST. (November 16) Prof. Weizhu Bao:
Efficient and stable numerical methods for
the generalized and vector Zakharov system
-
In this talk, we present efficient and stable
numerical methods for the generalized Zakharov
system (GZS) describing the propagation of Langmuir waves
in plasma. The key point in designing the methods is based on
a time-splitting discretization of a Schroedinger-type
equation in GZS, and to discretize a nonlinear wave-type
equation by pseudospectral method for spatial derivatives,
and then solving the ordinary differential equations
in phase space analytically under appropriate chosen
transmission conditions between different time intervals
or applying Crank-Nicolson/leap-frog for linear/nonlinear
terms for time derivatives. The methods are explicit,
unconditionally stable, of spectral-order accuracy in
space and second-order accuracy in time. Moreover,
they are time reversible and time transverse invariant
if GZS is, conserve the wave energy as that in GZS,
give exact results for the plane-wave solution and
possesses `optimal' meshing strategy in `subsonic limit'
regime. Extensive numerical tests are presented for plane waves,
solitary-wave collisions in 1D of GZS and 3D dynamics of GZS
to demonstrate efficiency and high resolution of the numerical
methods. Finally the methods are extended to vector Zakharov system
for multi-component plasma and Maxwell-Dirac system (MD)
for time-evolution of fast (relativistic) electrons and
positrons within self-consistent generated electromagnetic fields. (November 18) Professor Michael Vogelius:
Non-linear Elliptic Boundary Value Problems Related to
Corrosion Modeling
-
I shall discuss the solution structure and the
blow-up phenomena associated with two dimensional boundary
value problems of the form \Delta u = 0 in \Omega,\frac{\partial u }{\partial {\bf n} }
=Du+\lambda f(u)\hbox{ on } \partial \Omega, for certain f that are odd, non-decreasing, and with f'(0)=1.
Special emphasis will be placed on functions f of an exponential
character: in particular f(u) = sinh(u)=(e^u-e^{-u})/2. As will be shown, the solution structure is extremely different depending
on whether \lambda < 0 (generically: finitely many solutions) or
\lambda > 0 (generically: infinitely many solutions). Non-trivial solutions
in general blow up as \lambda \rightarrow 0, but again: the nature
of the blow-up for exponential f is completely different depending
on whether \lambda \rightarrow 0_- or \lambda \rightarrow 0_+.
Some of the character of the blow-up for
positive \lambda is reminiscent of
phenomena associated with the Ginzburg-Landau equations. Our general analysis has been significantly aided by the discovery
of surprisingly simple explicit solutions and by significant
computational experimentation. I shall discuss both of these
aspects in some detail. (November 19) Professor Michael Vogelius :
Electromagnetic Imaging for Small Inhomogeneities -
Electromagnetic Imaging in this context
refers to the identification
of internal characteristics of a medium based on boundary
(or near-field) measurements of the electric and/or magnetic fields.
After a brief review of some of the main mathematical
results in Electromagnetic- and Impedance Imaging (from the last 20 years,
or so) I shall proceed to discuss some very recent, extremely
efficient representation formulas that lead to a surprisingly accurate
identification of the size, and the location of relatively small
inhomogeneities. These representation formulas take into account polarization effects,
and they may be derived by variational techniques
related to H- (or \Gamma-) convergence. The magnitude of
the polarization effects
may be estimated in ways that are very reminiscent of effective
media bounds (of the Hashin-Shtrikman type). A precise assessment
of the polarization effects is very important for highly
accurate size estimates. Finally, these representation formulas lend themselves very naturally
to the application of reconstruction methods of a linear sampling- or
MUSIC (MUltiple SIgnal Chararcterization) character. On this matter
I shall discuss some general ideas, and implementation
issues, as well as provide examples of computational reconstructions.
(November 30) Long Chen:
Optimal Anisotropic Error Estimates and Applications to
Convection Dominated Problems -
In this talk, we first present an interpolation error
estimate in $L^p$ norm ($1\leq p\leq \infty$) for finite element
simplicial meshes in any spatial dimensions and then discuss its
applications to convection dominated problems. We show that an
asymptotically optimal error estimate can be obtained under near
optimal meshes. A sufficient condition for a mesh to be nearly optimal
is that it is quasi-uniform under a new metric defined by a modified
Hessian matrix of the function to be interpolated. We further show
such estimates are in fact asymptotically sharp for strictly convex
functions. To illustrate the useful of optimal meshes, we give an exact
gradient recovery formula and briefly discuss some interesting and
related problems in the computational geometry, such as sphere covering
and mesh smoothing. The above interpolation error estimate is useful for approximating
functions with anisotropic singularity. Thus it can be applied to
convection diffusion problem with small diffusion parameter $\epsilon$,
of which solutions usually present boundary layers or interior layers.
We are interested in when and how discretization errors may be governed
by interpolation errors. We show, theoretically and numerically, that
the discretization error of the standard FEM is sensitive to the
perturbation of the grid points in the region where the solution is
smooth. We have carefully designed a special streamline diffusion
finite element method whose discretization error is shown to be
uniformly governed by the interpolation error in maximum norm. For
problems in multidimensions, we shall discuss some practical issues in
the algorithms especially the homotopy with respect to the parameter
$\epsilon$. Our overarching goal is to develop and analyze
discretization schemes for one-parameter family PDEs, which is stable
and accurate uniformly with respect to the parameter.
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