Kolmogorov widths under holomorphic mappings

TL;DR

This paper extends the understanding of how holomorphic mappings affect the decay rates of Kolmogorov widths of compact sets in Banach spaces, with applications to parametrized PDE solution manifolds.

Contribution

It generalizes Kolmogorov width bounds from linear to holomorphic maps and applies these results to analyze solution manifolds of parametrized PDEs.

Findings

Kolmogorov widths of holomorphic images decay at a rate related to the original widths.

The decay rate of widths decreases by at least one order under holomorphic mappings.

Results inform the efficiency of reduced basis methods in numerical analysis.

Abstract

If $L$ is a bounded linear operator mapping the Banach space $X$ into the Banach space $Y$ and $K$ is a compact set in $X$, then the Kolmogorov widths of the image $L(K)$ do not exceed those of $K$ multiplied by the norm of $L$. We extend this result from linear maps to holomorphic mappings $u$ from $X$ to $Y$ in the following sense: when the $n$ widths of $K$ are $O(n^{-r})$ for some $r\textgreater{}1$, then those of $u(K)$ are $O(n^{-s})$ for any $s \textless{} r-1$, We then use these results to prove various theorems about Kolmogorov widths of manifolds consisting of solutions to certain parametrized PDEs. Results of this type are important in the numerical analysis of reduced bases and other reduced modeling methods, since the best possible performance of such methods is governed by the rate of decay of the Kolmogorov widths of the solution manifold.