Registration-based model reduction of parameterized PDEs with spatio-parameter adaptivity

TL;DR

This paper introduces an automated framework combining mesh adaptation, registration, and hyper-reduced models for efficient and reliable simulation of parameterized advection-dominated PDEs, especially in compressible flows.

Contribution

It presents a novel, general approach integrating mesh adaptation, registration, and reduced-order modeling for rapid solutions of complex parameterized PDEs.

Findings

Effective mesh adaptation for parameter ranges

Robust registration method for moving features

Significant speed-up in solution computation

Abstract

We propose an automated nonlinear model reduction and mesh adaptation framework for rapid and reliable solution of parameterized advection-dominated problems, with emphasis on compressible flows. The key features of our approach are threefold: (i) a metric-based mesh adaptation technique to generate an accurate mesh for a range of parameters, (ii) a general (i.e., independent of the underlying equations) registration procedure for the computation of a mapping $\Phi$ that tracks moving features of the solution field, and (iii) an hyper-reduced least-square Petrov-Galerkin reduced-order model for the rapid and reliable estimation of the mapped solution. We discuss a general paradigm -- which mimics the refinement loop considered in mesh adaptation -- to simultaneously construct the high-fidelity and the reduced-order approximations, and we discuss actionable strategies to accelerate the offline phase. We present extensive numerical investigations for a quasi-1D nozzle problem and for a two-dimensional inviscid flow past a Gaussian bump to display the many features of the methodology and to assess the performance for problems with discontinuous solutions.