The aim of this tutorial on "Numerical methods for high-dimensional problems" is to give an overview of the different numerical strategies which were developped recently in different application fields to deal with computations involving a large number of variables. Actually, standard numerical methods often fail in this case, since the size of the resulting discretized problems scales exponentially with respect to the number of variates due to the so-called "curse of dimensionality" [1]. To circumvent this problem, a wide variety of methods have been developped recently in different application fields and we particularly aim at enhancing the potential breakthroughs such methods can bring to the field of ab initio electronic structure calculations.

Indeed, for large molecular systems, even in the framework of the Born-Oppenheimer approximation, the resolution of the electronic problem requires to compute the solution of the full many-body Schrödinger equation. The groundstate electronic wavefunction of such systems is an antisymmetric function defined over R^{3N} where N is the number of electrons of the molecule under consideration. The computation of this groundstate is often impossible in practice when N is too large. A full zoology of methods and models [2] have been developped in order to circumvent the curse of dimensionality inherent to ab initio simulations for electronic structure calculations. Density Functional Theory [3] and Hartree-Fock [4] models are among the first solutions which were proposed to deal with this difficulty. Later, post Hartree-Fock methods, such as Multi Configuration Self-Consistent Field (MCSCF) [5], Configuration Interaction (CI) [6], Coupled Clusters (CC) [7], Moller-Plesset (MP) [8], and more recently Density Matrix Renormalization Group (DMRG), tensor network states (TNS) [9] and Green's function [10] methods have also been developped to tackle with this problem. The study of entanglement in many-body systems [11] is also a way to get access to physical information about a given molecule, which may be composed of a large number of atoms, while avoiding the curse of dimensionality.

In this tutorial, we wish to invite people from other commmunities, who do not necessarily work in the field of quantum chemistry computations, but who encounter similar difficulties inherent to the high-dimensional character of the problems they have to solve. In mechanical engineering, uncertainty quantification, optimization and inverse problems, other approaches have recently been developped in order to find practical ways to solve problems involving a significant number of variables. Specific numerical methods are used in these contexts to circumvent the curse of dimensionality and produce accurate reduced-order models, such as sparse grids [12,13], reduced bases [14,15,16], Proper Orthogonal Decomposition (POD) [17], Proper Generalized Decomposition (PGD) [18, 19, 20] and special tensor formats such as Hierarchical Tensor Train (HTT) or Quantized Tensor Train (QTT) [21,22].

The wide variety of all these different approaches, which come from very different backgrounds, may make it very intricate for someone beginning in these fields to have a complete overview of the existing methods.

We propose therefore to organize an intensive 4.5 day tutorial on "numerical methods for high-dimensional problems" open to physicists, mechanists and mathematicians, students and researchers, who are interested in these issues. We would like that this tutorial offers the possibility to draw a picture as complete as possible of the different methods used in different application fields and mathematics to circumvent the curse of dimensionality. Our aim is to bring together people from very different communities, but who share the common point that they have to deal with the resolution of very high-dimensional PDEs in different contexts, in order to present comprehensive introduction to their methods. We hope that the diverse backgrounds of the lecturers will allow for the cross-fertilization of ideas coming from different fields and that discussions between the participants, researchers and students, tackling these kinds of problems with very different point of views might lead to new original ideas in any of these fields, especially for electronic structure computations. We expect to have around 60 participants.

So far, we would like to organize the lectures of the tutorial in thematic days as follows:

-Monday: quantum chemistry;

-Tuesday: mechanics;

-Wednesday: numerical analysis;

-Thursday and Friday: optimization, inverse problems and uncertainty quantification.

Long keynote introductory lectures (9 times 1h30) will be alternated with shorter talks (34 times 45 min). The full programm and more information can be found here.

A poster session will be organized for participants. A roundtable with Thomas Leurent from the Akselos company about "high-dimensional problems and industry" will also be organized.