Simone Brugiapaglia (Concordia University)
Title: Classical and emerging ideas in sparse high-dimensional approximation.
Abstract:ÌýApproximating functions of many variables from limited samples is a key task in modern computational mathematics and data science. This task is made intrinsically difficult by the so-called curse of dimensionality, a term introduced by R.E. Bellman in the 1970s that refers to computational challenges arising in high dimensions. The objective of this seminar is to introduce classical and emerging mathematical ideas in high-dimensional approximation.
We will illustrate the rudiments of sparse polynomial approximation theory, motivating our study with applications to parametric differential equations. Specifically, we will present methods for computing sparse polynomial approximations of holomorphic functions of many variables from limited Monte Carlo samples, focusing on techniques based on least squares and compressed sensing and showing under what circumstances they are provably able to alleviate the curse of dimensionality. Time permitting, we will also discuss how these ideas come into play in current research areas such as deep learning approximation theory and spectral methods for high-dimensional PDEs. This seminar is mainly based on the book "Sparse Polynomial Approximation of High-Dimensional Functions", co-authored by the speaker with B. Adcock and C.G. Webster, and published for SIAM in 2022 ().
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