Sampling discretization, cubature formulas and quantitative approximation in multidimensional settings

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The primary goal of my research is to investigate sampling discretization, cubature formulas  and their connections with quantitative approximation theory in higher dimensional settings.  Sampling discretization refers to the process of transferring continuous objects into their discrete counterparts through function values at a fixed finite set of points, whereas cubature refers to a method for numerically approximating a multidimensional integral of a function through a  weighted sum of function values on a finite set of points, which is called a cubature formula. Discretization is an important step in making a continuous problem computationally feasible, while cubature formulas have been playing crucial roles in discretization and practical evaluation of high dimensional integrals. To ensure the problems are accessible by computers, the first step towards discretization is usually the process of approximating continuous operators and higher dimensional function spaces by lower dimensional counterparts (e.g. multivariate polynomials, splines, neural networks). In this program, I will focus on quantitative estimates of errors that inevitably arise in the process of approximation and discretization.    Many questions of utmost importance to discretization, cubature formulas and approximation theory in higher dimensions remain wide open. The questions under investigation in this program include (i)  sampling discretizations of Lq norms of functions from a high-dimensional subspace; (ii) discrepancy estimates and cubature formulas in high-dimensional function spaces; (iii) interplay between energy minimization and optimal cubature formulas; (iv) connections between locally supported positive definite functions on spheres and related domains; (v) new constructions of well-distributed point sets (low-discrepancy, cubature, energy-minimizing, lattices); and (vi) dimension-free estimates for approximation on high-dimensional domains.     My research will use a new technique, which combines powerful probabilistic techniques, based on chaining and large deviation inequalities, with various deep results in multidimensional approximation theory (e.g., estimates of entropy numbers and N-widths, various polynomial inequalities,  direct and inverse dimension-free  estimates in polynomial approximation). I also expect that the theory of classical orthogonal polynomial expansions, especially spherical harmonic analysis,  will be a powerful  tool in my research.    This program is connected to computational mathematics (i.e., the methods of numerical integration), probability, statistics, artificial intelligence and other areas of mathematics. The scientific outputs of this program are expected to impact several areas of mathematics, enriching and cross-fertilizing them with new results, ideas, and methods.