Basic Ideas and Tools for Projection-Based Model Reduction of Parametric Partial Differential Equations

TL;DR

This paper reviews foundational concepts and tools for projection-based model reduction of parametric PDEs, including geometric parametrizations, empirical interpolation, and active subspaces, with illustrative examples.

Contribution

It introduces a comprehensive framework combining reduced basis methods, empirical interpolation, and active subspaces for efficient parametric PDE modeling.

Findings

Effective geometric parametrization techniques explained.

Empirical interpolation method for non-affine dependencies introduced.

Examples demonstrate the practical application of the methodologies.

Abstract

We provide first the functional analysis background required for reduced order modeling and present the underlying concepts of reduced basis model reduction. The projection-based model reduction framework under affinity assumptions, offline-online decomposition and error estimation is introduced. Several tools for geometry parametrizations, such as free form deformation, radial basis function interpolation and inverse distance weighting interpolation are explained. The empirical interpolation method is introduced as a general tool to deal with non-affine parameter dependency and non-linear problems. The discrete and matrix versions of the empirical interpolation are considered as well. Active subspaces properties are discussed to reduce high-dimensional parameter spaces as a pre-processing step. Several examples illustrate the methodologies.