2023/07/20 by Nicholas Mueller, Mueller, Nicholas, Santiago Badia +1 · 2 citations
Engineering · Mathematics · Physics and Astronomy · #65D05 (secondary) #65M22 (primary) #Algorithm #Applied mathematics #Backward Euler method #Basis (linear algebra) #Basis function #Computational fluid dynamics #Computer science #Dimensionality reduction #Discretization #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Finite element method #Galerkin method #Interpolation (computer graphics) #Krylov subspace #Mathematical analysis #Mathematical optimization #Mathematics #Model Reduction and Neural Networks #Model order reduction #Numerical Analysis (math.NA) #Numerical methods for differential equations #Parameterized complexity #Partial differential equation #Projection (relational algebra) #Reduction (mathematics) #Speedup
paper · pdf · doi:10.48550/arxiv.2307.10605
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2023/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This work proposes novel techniques for the efficient numerical simulation of parameterized, unsteady partial differential equations. Projection-based reduced order models (ROMs) such as the reduced basis method employ a (Petrov-)Galerkin projection onto a linear low-dimensional subspace. In unsteady applications, space-time reduced basis (ST-RB) methods have been developed to achieve a dimension reduction both in space and time, eliminating the computational burden of time marching schemes. However, nonaffine parameterizations dilute any computational speedup achievable by traditional ROMs. Computational efficiency can be recovered by linearizing the nonaffine operators via hyper-reduction, such as the empirical interpolation method in matrix form. In this work, we implement new hyper-reduction techniques explicitly tailored to deal with unsteady problems and embed them in a ST-RB framework. For each of the proposed methods, we develop a posteriori error bounds. We run numerical tests to compare the performance of the proposed ROMs against high-fidelity simulations, in which we combine the finite element method for space discretization on 3D geometries and the Backward Euler time integrator. In particular, we consider a heat equation and an unsteady Stokes equation. The numerical experiments demonstrate the accuracy and computational efficiency our methods retain with respect to the high-fidelity simulations.