Filtering and Domain Decomposition Techniques for Intrusive and Non-intrusive Reduced Order Models of Convection-Dominated Problems

Abstract

Galerkin reduced order models (ROMs) offer significant computational savings for the simulation of partial differential equations, yet they often exhibit spurious oscillations and loss of physical fidelity in convection-dominated and multiscale regimes. This dissertation develops a framework for correcting the intrinsic spatial under-resolution of Galerkin ROMs through complementary regularization and hybridization strategies. Large eddy simulation-inspired filtering approaches are introduced, including the Ladyzhenskaya ROM and the approximate deconvolution Leray ROM (ADL-ROM), which incorporate spatial filtering to model the effects of truncated scales and enhance stability while maintaining efficiency. Rigorous numerical analysis is provided for both models, establishing stability and convergence results that strengthen their theoretical foundations. To address strongly localized high-gradient features, the OpInf-Schwarz ROM combines operator inference with a Schwarz domain decomposition framework, restricting full order modeling to localized regions while retaining ROM efficiency elsewhere. Finally, I apply spatial filtering concepts directly to OpInf to stabilize non-intrusive ROMs for convection-dominated systems.