Approximation Bounds for Model Reduction on Polynomially Mapped Manifolds

TL;DR

This paper establishes new approximation bounds for reduced order models on polynomially mapped manifolds, linking the bounds to the polynomial degree and the Kolmogorov n-width of the original problem.

Contribution

It introduces the concept of Kolmogorov (n, p)-width, providing a lower bound for ROM errors on polynomially mapped manifolds, extending classical MOR theory.

Findings

Approximation bounds depend on polynomial degree p and Kolmogorov n-width.

Introduces Kolmogorov (n, p)-width as a new theoretical measure.

Provides lower bounds for errors in polynomially mapped manifold ROMs.

Abstract

For projection-based linear-subspace model order reduction (MOR), it is well known that the Kolmogorov n-width describes the best-possible error for a reduced order model (ROM) of size n. In this paper, we provide approximation bounds for ROMs on polynomially mapped manifolds. In particular, we show that the approximation bounds depend on the polynomial degree p of the mapping function as well as on the linear Kolmogorov n-width for the underlying problem. This results in a Kolmogorov (n, p)-width, which describes a lower bound for the best-possible error for a ROM on polynomially mapped manifolds of polynomial degree p and reduced size n.