Multilevel Weighted Least Squares Polynomial Approximation – Sören Wolfers, KAUST

Explore multilevel weighted least squares polynomial approximation in this 37-minute lecture by Sören Wolfers from KAUST, presented at the Alan Turing Institute. Delve into the mathematical foundations of approximating high-dimensional functions from limited information, addressing the curse of dimensionality. Learn about modern approaches that bypass this issue through structural assumptions like low intrinsic dimensionality and partial separability. Discover the single-level and multilevel approaches to inexact evaluations, convergence analysis, and numerical examples. Examine the application to stationary diffusion equations with random coefficient fields, including Smolyak decomposition and adaptive algorithms. Gain insights into this rich theory that has developed over the past decade, bridging multivariate approximation theory, high-dimensional integration, and non-parametric regression.

Intro
Polynomial least squares approximation
Accuracy - Summary
Accuracy - References
Sampling from optimal density
Single level approach to inexact evaluations Ides Apply least squares approximation to freed to Problem: Good approximation requires both a large subspace
Multilevel approach to inexact evaluations
Multilevel convergence analysis
Numerical example Stationary diffusion equation with random coefficient field
Setup
Curse of dimensionality
Smolyak decomposition
Decay of mixed differences
Adaptive algorithm
Special case multilevel polynomial approximation