Collocated Reduced-Order Models for High-Order Schemes: Analysis and Application

Abstract

In this talk, I will present Collocated Model Order Reduction (cMOR), a novel hyper-reduction framework specifically designed for high-order discretizations. The method evaluates the governing operator exclusively on a small subset of collocation points, identified via Non-Negative Least Squares (NNLS) sparse quadrature. By combining a restricted POD basis with carefully constructed prolongation and interpolation operators, cMOR enables sparse yet accurate residual evaluation while preserving key structural properties of the underlying scheme.

I will discuss the rigorous theoretical foundations of the approach, including proofs of injectivity, optimality of the interpolation error, conditional stability, and convergence in the Lax–Richtmyer sense. These results establish cMOR as a stable and convergent scheme whose accuracy is controlled by the POD and NNLS approximation errors.

Numerical applications to a parametric 2D shallow water equations dam-break problem and a high-order structure-preserving wave equation demonstrate significant computational speedups (up to two orders of magnitude) with negligible loss of accuracy, even in extrapolation regimes.

Work with Giuliano Carlino, Alessia del Grosso, and Denis Sipp.

Angelo Iollo

Angelo Iollo is an applied mathematician specializing in numerical modeling, reduced-order modeling, and scientific computing for fluid dynamics and aerospace applications. He is a professor at the Institute of Mathematics of Bordeaux at the University of Bordeaux, and he also serves as the head of the MONHADE project, a joint research team of Inria and Onera, where he leads and coordinates research at the intersection of continuum mechanics and scientific computing. He is actively involved in translating applied mathematics into practical business solutions. He helped establish Optimad Engineering, a numerical engineering spin-off originating from the Politecnico di Torino, as well as Nurea, a startup spun out of Inria that provides specialized cardiovascular decision-support software.