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Title: Space-time least-squares Petrov-Galerkin projection in nonlinear model reduction.

Abstract

Our 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 de ned to enable complexity reduction (i.e., hyper-reduction) in time, which leads to space-time collocation and space-time 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: (1) a reduction of both the spatial and temporal dimensions of the dynamical system, (2) the removal of spurious temporal modes (e.g., unstable growth) from the state space, and (3) error bounds that exhibit slower growth in time. Numerical examples performed on model problemsmore » 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.« less

Authors:
 [1];  [2]
  1. Sandia National Laboratories (SNL-CA), Livermore, CA (United States). Extreme-scale Data Science and Analytics Dept.; Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
  2. Sandia National Laboratories (SNL-CA), Livermore, CA (United States). Extreme-scale Data Science and Analytics Dept.
Publication Date:
Research Org.:
Sandia National Lab. (SNL-CA), Livermore, CA (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1429634
Report Number(s):
SAND-2017-10430J
657316
DOE Contract Number:  
AC04-94AL85000; NA-0003525
Resource Type:
Program Document
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICS AND COMPUTING

Citation Formats

Choi, Youngsoo, and Carlberg, Kevin Thomas. Space-time least-squares Petrov-Galerkin projection in nonlinear model reduction.. United States: N. p., 2017. Web.
Choi, Youngsoo, & Carlberg, Kevin Thomas. Space-time least-squares Petrov-Galerkin projection in nonlinear model reduction.. United States.
Choi, Youngsoo, and Carlberg, Kevin Thomas. Fri . "Space-time least-squares Petrov-Galerkin projection in nonlinear model reduction.". United States.
@article{osti_1429634,
title = {Space-time least-squares Petrov-Galerkin projection in nonlinear model reduction.},
author = {Choi, Youngsoo and Carlberg, Kevin Thomas},
abstractNote = {Our 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 ℓ2-norm. This norm can be de ned to enable complexity reduction (i.e., hyper-reduction) in time, which leads to space-time collocation and space-time 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: (1) a reduction of both the spatial and temporal dimensions of the dynamical system, (2) the removal of spurious temporal modes (e.g., unstable growth) from the state space, and (3) error bounds that 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.},
doi = {},
journal = {},
number = ,
volume = ,
place = {United States},
year = {2017},
month = {9}
}

Program Document:
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