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(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.