Manifold Approximations via Transported Subspaces: Model reduction for transport-dominated problems

TL;DR

This paper introduces a nonlinear model reduction technique for transport-dominated problems that combines transport dynamics with local linear approximations, enabling efficient online computations and significant speedups.

Contribution

It proposes a novel nonlinear reduction method using transported subspaces that effectively captures transport phenomena and improves computational efficiency over traditional linear methods.

Findings

Achieves orders of magnitude speedups compared to linear reduced models.

Effectively models hyperbolic conservation laws with transport dynamics.

Demonstrates efficiency on problems like Burgers' equation and heterogeneous media transport.

Abstract

This work presents a method for constructing online-efficient reduced models of large-scale systems governed by parametrized nonlinear scalar conservation laws. The solution manifolds induced by transport-dominated problems such as hyperbolic conservation laws typically exhibit nonlinear structures, which means that traditional model reduction methods based on linear approximations are inefficient when applied to these problems. In contrast, the approach introduced in this work derives reduced approximations that are nonlinear by explicitly composing global transport dynamics with locally linear approximations of the solution manifolds. A time-stepping scheme evolves the nonlinear reduced models by transporting local approximation spaces along the characteristic curves of the governing equations. The proposed computational procedure allows an offline/online decomposition and is online-efficient in the sense that the complexity of accurately time-stepping the nonlinear reduced model is independent of that of the full model. Numerical experiments with transport through heterogeneous media and the Burgers' equation show orders of magnitude speedups of the proposed nonlinear reduced models based on transported subspaces compared to traditional linear reduced models and full models.