Nonlinear model reduction for transport-dominated problems

TL;DR

This paper reviews nonlinear model reduction techniques that effectively handle transport-dominated problems with wave-like phenomena, focusing on nonlinear parametrizations, reduced dynamics, and online solvers.

Contribution

It categorizes existing nonlinear model reduction methods into transformation-based, adaptive, and residual minimization approaches, providing a comprehensive survey.

Findings

Effective nonlinear reduction methods for transport-dominated problems

Categorization of approaches into three main classes

Insights into the organization of nonlinear model reduction techniques

Abstract

This article surveys nonlinear model reduction methods that remain effective in regimes where linear reduced-space approximations are intrinsically inefficient, such as transport-dominated problems with wave-like phenomena and moving coherent structures, which are commonly associated with the Kolmogorov barrier. The article organizes nonlinear model reduction techniques around three key elements -- nonlinear parametrizations, reduced dynamics, and online solvers -- and categorizes existing approaches into transformation-based methods, online adaptive techniques, and formulations that combine generic nonlinear parametrizations with instantaneous residual minimization.