The aim of this talk is to present a new variation formulation of the time-dependent many-body electronic Schrödinger equation with Coulombic singularities. More precisely, its solution can actually be expressed as the solution of a global space-time quadratic minimization problem that proves to be useful for several tasks:
1) first, it is amenable to Galerkin time-space discretization schemes, using an appropriate least-square formulation
2) it enables to yield a new variational principle for the construction dynamical low-rank approximations, that is different from the classical Dirac-Frenkel variational principle
3) it enables to obtain fully certified a posteriori error estimators between the exact solution and approximate solutions.
The present analysis can be applied to the electronic many-body time-dependent Schrödinger equation with an arbitrary number of electrons and interaction potentials with Coulomb singularities.[-]

Ingrid Daubechies' construction of orthonormal wavelet bases with compact support published in 1988 started a general interest to employ these functions also for the numerical solution of partial differential equations (PDEs). Concentrating on linear elliptic and parabolic PDEs, I will start from theoretical topics such as the well-posedness of the problem in appropriate function spaces and regularity of solutions and will then address quality and optimality of approximations and related concepts from approximation the- ory. We will see that wavelet bases can serve as a basic ingredient, both for the theory as well as for algorithmic realizations. Particularly for situations where solutions exhibit singularities, wavelet concepts enable adaptive appproximations for which convergence and optimal algorithmic complexity can be established. I will describe corresponding implementations based on biorthogonal spline-wavelets.
Moreover, wavelet-related concepts have triggered new developments for efficiently solving complex systems of PDEs, as they arise from optimization problems with PDEs.[-]

The unitary extension principle (UEP) by Ron & Shen yields a convenient way of constructing tight wavelet frames in L2(R). Since its publication in 1997 several generalizations and reformulations have been obtained, and it has been proved that the UEP has important applications within image processing. In the talk we will present a recent extension of the UEP to the setting of generalized shift-invariant systems on R (or more generally, on any locally compact abelian group). For example, this generalization immediately leads to a discrete version of the UEP.
(The results are joint work with Say Song Goh).[-]

In this talk I will discuss recent joint work with Mike McCourt (SigOpt, San Francisco) that has led to progress on the numerically stable computation of certain quantities of interest when working with positive definite kernels to solve scattered data interpolation (or kriging) problems.
In particular, I will draw upon insights from both numerical analysis and modeling with Gaussian processes which will allow us to connect quantities such as, e.g., (deterministic) error estimates in terms of the power function with the kriging variance. This provides new kernel parametrization criteria as well as new ways to compute known criteria such as MLE. Some numerical examples will illustrate the effectiveness of this approach.[-]

Multigrid is an iterative method for solving large linear systems of equations whose Toeplitz system matrix is positive definite. One of the crucial steps of any Multigrid method is based on multivariate subdivision. We derive sufficient conditions for convergence and optimality of Multigrid in terms of trigonometric polynomials associated with the corresponding subdivision schemes.
(This is a joint work with Marco Donatelli, Lucia Romani and Valentina Turati).[-]

The Bézier representation of homogenous polynomials has little and not the usual geometric meaning if we consider the graph of these polynomials over the sphere. However the graph can be seen as a rational surface and has an ordinary rational Bézier representation. As I will show, both Bézier representations are closely related. Further I consider rational spline constructions for spherical surfaces and other closed manifolds with a projective or hyperbolic structure.[-]

An ubiquitous problem in applied science is the recovery of physical phenomenons, represented by multivariate functions, from uncomplete measurements. These measurements typically have the form of pointwise data, but could also be obtained by linear functional. Most often, recovery techniques are based on some form of approximation by finite dimensional space that should accurately capture the unknown multivariate function. The first part of the course will review fundamental tools from approximation theory that describe how well relevant classes of multivariate functions can be described by such finite dimensional spaces. The notion of (linear or nonlinear) n-width will be developped, in relation with reduced modeling strategies that allow to construct near-optimal approximation spaces for classes of parametrized PDE's. Functions of many variables that are subject to the curse of dimensionality, will also be discussed. The second part of the course will review two recovery strategies from uncomplete measurements: weighted least-squares and parametrized background data-weak methods. An emphasis will be put on the derivation of sample distributions of minimal size for ensuring optimal convergence estimates.[-]

we discuss classification problems in high dimension. We study classification problems using three classical notions: complexity of decision boundary, noise, and margin. We demonstrate that under suitable conditions on the decision boundary, classification problems can be very efficiently approximated, even in high dimensions. If a margin condition is assumed, then arbitrary fast approximation rates can be achieved, despite the problem being high-dimensional and discontinuous. We extend the approximation results ta learning results and show close ta optimal learning rates for empirical risk minimization in high dimensional classification. [-]