Journal
SIAM JOURNAL ON SCIENTIFIC COMPUTING
Volume 41, Issue 1, Pages A26-A58Publisher
SIAM PUBLICATIONS
DOI: 10.1137/17M1120531
Keywords
space-time projection; least-squares Petrov-Galerkin projection; residual minimization; model reduction; nonlinear dynamical systems
Categories
Funding
- LDRD program [190968]
- U.S. Department of Energy's National Nuclear Security Administration [DE-NA-0003525]
- U.S. Department of Energy, National Nuclear Security Administration [DE-AC52-07NA27344]
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This work proposes a space-time least-squares Petrov-Galerkin (ST-LSPG) projection method for model reduction of nonlinear dynamical systems. In contrast to typical nonlinear model-reduction methods that first apply (Petrov-)Galerkin projection in the spatial dimension and subsequently apply time integration to numerically resolve the resulting low-dimensional dynamical system, the proposed method applies projection in space and time simultaneously. To accomplish this, the method first introduces a low-dimensional space-time trial subspace, which can be obtained by computing tensor decompositions of state-snapshot data. The method then computes discrete-optimal approximations in this space-time trial subspace by minimizing the residual arising after time discretization over all space and time in a weighted l(2) -norm. This norm can be defined to enable complexity reduction (i.e., hyper-reduction) in time, which leads to space-time collocation and space-time Gauss-Newton with Approximated Tensors (GNAT) variants of the ST-LSPG method. Advantages of the approach relative to typical spatial-projection-based nonlinear model reduction methods such as Galerkin projection and least-squares Petrov-Galerkin projection include a reduction of both the spatial and temporal dimensions of the dynamical system, and a priori error bounds that bound the solution error by the best space-time approximation error and whose stability constants exhibit slower growth in time. Numerical examples performed on model problems in fluid dynamics demonstrate the ability of the method to generate orders-of-magnitude computational savings relative to spatial-projection-based reduced-order models without sacrificing accuracy for a fixed spatio-temporal discretization.
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