Recently, there was a big progress in studying sampling discretization of integral norms of elements of finite dimensional subspaces and collections of such subspaces (universal discretization). This problem is very important in applications because in order to solve a continuous problem numerically one needs to discretize it. However, till recently there was no systematic study of it.
This programme is built on a recent success of the research in sampling discretization and its connections with other areas of mathematics such as spectral properties and operator norms of submatrices, embedding of finite-dimensional subspaces, moments of marginals of high-dimensional distributions, learning theory, and sampling recovery. It concentrates on sampling discretization of integral norms of elements of finite-dimensional subspaces.
An important new ingredient of the discretization problem, desirable in practical applications, will be discussed in detail. It is motivated by applications in sparse approximation. Applying a strategy of sparse m-term approximation with respect to a given dictionary we obtain a collection of all subspaces spanned by at most m elements of the dictionary as a possible source of approximating (representing) elements. Therefore, we would like to build a discretization scheme, which works well for all such subspaces. This kind of discretization is called universal discretization. Recently, it was shown that results on sampling discretization of the square norm allow us to prove general inequalities between error of recovery in the square norm and the Kolmogorov width in the uniform norm. Similar results are obtained for the universal discretization and sparse approximation. We plan to discuss open problems. For instance,
the problem of sampling recovery in integral norms other than the square norm and its connection to
sampling discretization will be discussed.
One of the goals of this programme is to connect together ideas, methods, and results from different areas of research related to problems of discretization and recovery in the case of finite-dimensional subspaces. We plan to bring together experts from classical approximation theory, numerical integration,
probability theory and statistics, functional analysis, and other areas of contemporary research.
The current proposed programme would emphasize several new exciting and promising directions in numerical analysis, aiming to present the most recent results, and to advance the mathematical understanding of the deep interplay between numerical analysis, approximation theory, and probability
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