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HEIndividual fellowship2023–2025

HDAD · High Dimensional Approximation and Discretization

Horizon Europe — Marie Skłodowska-Curie Actions

Duration
2023-09-01 → 2025-08-31
EU contribution
€165,313
Participants
1
Scheme
HORIZON-TMA-MSCA-PF-EF

Lines connect the coordinator with its partners.

Results in brief

High Dimensional Approximation and Discretization

The project addresses two fundamental problems in modern approximation theory. The first concerns the sampling discretization of integral norms, a central question that investigates how integral norms on function classes can be approximated by evaluating the functions at a fixed and relatively small set of points. In many applications, particularly in high-dimensional settings, the computation of integral norms is highly complex and demands substantial analytical and computational resources. Consequently, discretization problems are closely related to numerical integration and information-based complexity. However, unlike classical problems of numerical integration, sampling discretization problems belong to the domain of nonlinear approximation theory. One of the main objectives in this area is to determine the smallest integer m for which the initial measure can be replaced by a discrete uniform measure supported on m points, ensuring that the corresponding continuous and discrete norms remain comparable. The second problem focuses on the properties of integral norms of polynomials on convex domains and more general settings. In particular, for a given measure, Markov-Bernstein-type inequalities provide estimates for the integral L^p-norm of the gradient of a polynomial in terms of the L^p-norm of the polynomial itself. These inequalities serve as classical tools for studying polynomial approximation in high-dimensional spaces through reverse Jackson-type theorems and play an important role in the discretization of integral norms on spaces of polynomials.

Data: CORDIS, © European Union

Project objective

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.

Original text from CORDIS.

Participants

  • Consorci Centre de Recerca Matematica · BellaterraCoordinatorSpain

Links

Data: CORDIS, © European Union