Description du projet
De nouvelles approches de l’approximation et de la discrétisation des fonctions
Les problèmes mathématiques de haute dimension impliquent souvent des fonctions complexes et de vastes espaces difficiles à calculer. Pour faciliter la résolution de ces problèmes, il est nécessaire de simplifier les fonctions et de discrétiser les espaces tout en préservant leurs propriétés importantes. Les problèmes se faisant de plus en plus complexes, le besoin de techniques plus efficaces est pressant. Avec le soutien du programme Actions Marie Skłodowska-Curie, le projet HDAD entend améliorer la discrétisation des normes intégrales pour les polynômes algébriques sur des domaines convexes et à l’étendre à tout sous-espace de dimension finie de fonctions continues. Le projet se penchera également sur la manière dont les taux d’approximation polynomiale varient en fonction de la fluidité de la fonction. La recherche intégrera des approches analytiques classiques et des approches probabilistes novatrices, dans le but de produire des résultats révolutionnaires et de faire progresser la carrière du chercheur.
Objectif
Approximation and discretization are two steps of making high dimensional problems more computationally feasible. On the one hand, both the approximation of certain functional classes by simpler functions and the discretization of underlying space while preserving certain important properties are classical problems. On the other hand, new trends and challenges in pure mathematics and applications lead to new approximation and discretization problems.
The main goal of this research is to study certain high dimensional approximation and discretization problems. Firstly, we intend to obtain new innovative results in the problem of integral norms discretization both in the important special case of algebraic polynomials on convex domains and in the general case of any finite dimensional subspace of continuous functions. Secondly, we will study the dependence of the rate of approximation by polynomials on the smoothness properties of functions. While this second problem itself is classical our main aim is to study it in new settings. Finally, both described problems will require the study of various properties of multivariate algebraic polynomials.
The stated goals require the development of a new technique involving a combination of classical analytic and new probabilistic approaches. In order to develop this new technique, the researcher will work under the supervision of Sergey Tikhonov, who is one of the most experienced researchers in the fields of harmonic analysis, approximation, and discretization. While working with the supervisor, the researcher will acquire techniques of classical approximation theory. Then this new obtained techniques will be combined with the researcher's own expertise in probabilistic approaches in functional analysis.
In conclusion, this MSC fellowship will allow the applicant to obtain new important results in various research areas. This will support him as an independent researcher and advance his career opportunities within the EU.
Programme(s)
- HORIZON.1.2 - Marie Skłodowska-Curie Actions (MSCA) Main Programme
Régime de financement
HORIZON-TMA-MSCA-PF-EF - HORIZON TMA MSCA Postdoctoral Fellowships - European FellowshipsCoordinateur
08193 Bellaterra
Espagne