Nonlinear Model Reduction via an Adaptive Weighting of Snapshots

TL;DR

This paper introduces an adaptive, localized model reduction technique for nonlinear parameterized PDEs, improving accuracy and efficiency over traditional global methods by using parameter-dependent weighted snapshots.

Contribution

The paper presents a novel adaptive reduced basis approach that constructs multiple localized subspaces using weighted snapshots, enhancing model accuracy for nonlinear PDEs.

Findings

Achieves higher accuracy with fixed subspace dimension

Demonstrates significant computational speedups

Maintains good accuracy for elliptic and parabolic PDEs

Abstract

In this paper, we propose a new approach to model reduction of parameterized partial differential equations (PDEs) based on the concept of adaptive reduced bases. The presented approach is particularly suited for large-scale nonlinear systems characterized by parameter variations. Instead of using a global basis to construct a global reduced model, the proposed method approximates the original system by multiple lower-dimensional subspaces. Each localized reduced basis is generated by the SVD of a weighted snapshot ensemble; here, each weighting coefficient is a function of the input parameter. Compared with a global model reduction method, such as the classical POD, the adaptive model reduction method could yield a more accurate solution with a fixed subspace dimension. Moreover, we combine the adaptive reduced model with the chord iteration to solve elliptic PDEs in a computationally efficient fashion. The potential of the method for achieving large speedups, while maintaining good accuracy, is demonstrated for both elliptic and parabolic PDEs in a few numerical examples.