### Barycentric Interpolation

## Abstract

I further discuss how to extend this idea to bivariate data, where it leads to the concept of generalized barycentric coordinates and to an efficient method for interpolating data given at the vertices of an arbitrary polygon.

### The exponentially convergent trapezoid rule

## Abstract

It is well known that the trapezoid rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with powerful algorithms all across scientific computing, including double exponential and Gauss quadrature, computation of inverse Laplace transforms, special functions, computational complex analysis, the computation of functions of matrices and operators, rational approximation, and the solution of partial differential equations.

This talk represents joint work with Andre Weideman of the University of Stellenbosch.

### Regularity theory of degenerate elliptic equations in nondivergence form with applications to homogenization

## Abstract

We will present a regularity result for degenerate elliptic equations in nondivergence form.

In joint work with Charlie Smart, we extend the regularity theory of Caffarelli to equations with possibly unbounded ellipticity-- provided that the ellipticity satisfies an averaging condition. As an application we obtain a stochastic homogenization result for such equations (and a new estimate for the effective coefficients) as well as an invariance principle for random diffusions in random environments. The degenerate equations homogenize to uniformly elliptic equations, and we give an estimate of the ellipticity.

### A Lagrangian approach for nonhomogeneous incompressible fluids

## Abstract

density. The aim is to prove existence and uniqueness results in the case of a discontinuous

initial density (typically we are interested in discontinuity along an interface).

In the first part of the talk, by making use of Fourier analysis techniques, we establish the existence of global-in-time unique solutions in a critical

functional framework, under some smallness condition over the initial data,

In the second part, we use another approach to avoid the smallness condition over the nonhomogeneity : as a matter of fact, one may consider any density bounded

and bounded away from zero and still get a unique solution. The velocity is required to have subcritical regularity, though.

In all the talk, the Lagrangian formulation for describing the flow plays a key role in the analysis.

### Introduction to tensor numerical methods in higher dimensions

## Abstract

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The recent quantized-TT (QTT) approximation method is proven to provide the logarithmic data-compression on a wide class of functions and operators. Furthermore, QTT-approximation makes it possible to represent multi-dimensional steady-state and dynamical equations in quantized tensor spaces with the log-volume complexity scaling in the full-grid size, $O(d \log n)$, instead of $O(n^d)$.

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We show how the grid-based tensor approximation in quantized tensor spaces applies to super-compressed representation of functions and operators (super-fast convolution and FFT, spectrally close preconditioners) as well to hard problems arising in electronic structure calculations, such as multi-dimensional convolution, and two-electron integrals factorization in the framework of Hartree-Fock calculations. The QTT method also applies to the time-dependent molecular Schr{\"o}dinger, Fokker-Planck and chemical master equations.

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Numerical tests are presented indicating the efficiency of tensor methods in approximation of functions, operators and PDEs in many dimensions.

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### Optimization meets Statistics: Fast global convergence for high-dimensional statistical recovery

## Abstract

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Particular examples include $\ell_1$-based methods for sparse vectors and matrices, nuclear norm for low-rank matrices, and various combinations thereof for matrix decomposition and robust PCA. In this talk, we describe an interesting connection between computational and statistical efficiency, in particular showing that the same conditions that guarantee that an estimator has good statistical error can also be used to certify fast convergence of first-order optimization methods up to statistical precision.

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Joint work with Alekh Agarwahl and Sahand Negahban Pre-print (to appear in Annals of Statistics)

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http://www.eecs.berkeley.edu/~wainwrig/Papers/AgaNegWai12b_SparseOptFul…

### High frequency acoustic scattering by screens: computation and analysis

## Abstract

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This is joint work with Dave Hewett, Steve Langdon, and Ashley Twigger, all at Reading.

### Pointwise convergence of the feasibility violation for Moreau-Yosida regularized optimal control problems

## Abstract

And applications to problems involving pointwise constraints on the gradient of the state on non smooth polygonal domains

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In this talk we are concerned with an analysis of Moreau-Yosida regularization of pointwise state constrained optimal control problems. As recent analysis has already revealed the convergence of the primal variables is dominated by the reduction of the feasibility violation in the maximum norm.

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We will use a new method to derive convergence of the feasibility violation in the maximum norm giving improved the known convergence rates.

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Finally we will employ these techniques to analyze optimal control problems governed by elliptic PDEs with pointwise constraints on the gradient of the state on non smooth polygonal domains. For these problems, standard analysis, fails because the control to state mapping does not yield sufficient regularity for the states to be continuously differentiable on the closure of the domain. Nonetheless, these problems are well posed. In particular, the results of the first part will be used to derive convergence rates for the primal variables of the regularized problem.