Approximation Theory of Integral Discretization on High Dimensional Domains
DescriptionTo make a continuous problem computationally feasible, it is often desired to approximate an integration by a discrete summation. Then it is natural to ask how to choose the sampling points and what decay of the error with the growing number of sampling points is. Investigation in this direction includes cubeture formulas, Marcinkiewcz-Zygmund inequalities and sampling discretization. Recently, a systematic study of these problems has begun and gained attraction in different areas of research, ranging over supervised learning theory, numerical integration and discrepancy theory.Modern problems are often formulated on various regular domains, such as spheres, cubes, balls, and simplexes, driven by applications in engineering and data analysis. At the current stage, however, in contrast to the situation in low dimensional domains,many fundamental problems are in demand. Actually, only with classical smoothness assumptions (such as Lipschitz or Besov regularity) on the underlying functions, approximation rates deteriorate severely with the dimensionality. This is the so-calledcurse of dimensionality. The purpose of this project is to investigate integral discretization on unit spheres for a certain class of functions without suffering the effect of dimensionality. Precisely, we aim at building a good finite set of points on amultidimensional domain and investigating the error analysis of integral discretization by using function values on the points. Then we shall apply the expected results to study approximation on spheres by ReLU neural networks.
|Effective start/end date||1/01/21 → …|