Space-time least-squares Petrov-Galerkin projection for nonlinear model\n reduction

2017/03/13 by Youngsoo Choi, Kevin Carlberg, Choi, Youngsoo +1 · 9 citations
Engineering · Physics and Astronomy · #65L05 #65L06 #65L60 #65M15 #65M22 #68U20 #FOS: Mathematics #Fluid Dynamics and Vibration Analysis #Lattice Boltzmann Simulation Studies #Model Reduction and Neural Networks #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.1703.04560

openalex publication_date 2017/03/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

This work proposes a space-time least-squares Petrov-Galerkin (ST-LSPG)\nprojection method for model reduction of nonlinear dynamical systems. In\ncontrast to typical nonlinear model-reduction methods that first apply\n(Petrov-)Galerkin projection in the spatial dimension and subsequently apply\ntime integration to numerically resolve the resulting low-dimensional dynamical\nsystem, the proposed method applies projection in space and time\nsimultaneously. To accomplish this, the method first introduces a\nlow-dimensional space-time trial subspace, which can be obtained by computing\ntensor decompositions of state-snapshot data. The method then computes\ndiscrete-optimal approximations in this space-time trial subspace by minimizing\nthe residual arising after time discretization over all space and time in a\nweighted \ℓ²-norm. This norm can be defined to enable complexity reduction\n(i.e., hyper-reduction) in time, which leads to space-time collocation and\nspace-time GNAT variants of the ST-LSPG method. Advantages of the approach\nrelative to typical spatial-projection-based nonlinear model reduction methods\nsuch as Galerkin projection and least-squares Petrov-Galerkin projection\ninclude: (1) a reduction of both the spatial and temporal dimensions of the\ndynamical system, (2) the removal of spurious temporal modes (e.g., unstable\ngrowth) from the state space, and (3) error bounds that exhibit slower growth\nin time. Numerical examples performed on model problems in fluid dynamics\ndemonstrate the ability of the method to generate orders-of-magnitude\ncomputational savings relative to spatial-projection-based reduced-order models\nwithout sacrificing accuracy.\n

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