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.[-]

The development of a suitable, efficient and accurate numerical method to solve wave problems is encountered in many academic and industrial applications. The Boundary Integral Equation (BIE) technique, whose discretization is known as the Boundary Element Method (BEM), is an appealing alternative to classical domain method because it allows to handle problems defined on the exterior of bounded domains as easily as those defined in the interior, without the introduction of an artificial boundary to truncate the computational domain. Very recently, an Isogeometric Analysis based Boundary Element Method (IgA-BEM) has been proposed in literature for the numerical solution of frequency-domain (Helmholtz) wave problems on 3D domains admitting a multi-patch representation of the boundary surface. While being powerful and applicable to many situations, this approach shares with standard BEMs a disadvantage which can easily become significant in the 3D setting. Indeed, when the required accuracy is increased, it can soon lead to large dense linear systems, whose numerical solution requires huge memory, resulting also in important computational cost. Recently the development of fast H-matrix based direct and iterative solvers for oscillatory kernels, as the Helmholtz one, has been studied. Here, we investigate the effectiveness of the H-matrix technique, along with a suitable GMRES iterative solver, when used in the context of multi-patch IgA-BEM.[-]

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).[-]

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. [-]

Modern machine learning architectures often embed their inputs into a lower-dimensional latent space before generating a final output. A vast set of empirical results---and some emerging theory---predicts that these lower-dimensional codes often are highly structured, capturing lower-dimensional variation in the data. Based on this observation, in this talk I will describe efforts in my group to develop lightweight algorithms that navigate, restructure, and reshape learned latent spaces. Along the way, I will consider a variety of practical problems in machine learning, including low-rank adaptation of large models, regularization to promote local latent structure, and efficient training/evaluation of generative models.[-]

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.[-]