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Title: Decreasing the temporal complexity for nonlinear, implicit reduced-order models by forecasting

Abstract

Implicit numerical integration of nonlinear ODEs requires solving a system of nonlinear algebraic equations at each time step. Each of these systems is often solved by a Newton-like method, which incurs a sequence of linear-system solves. Most model-reduction techniques for nonlinear ODEs exploit knowledge of system's spatial behavior to reduce the computational complexity of each linear-system solve. However, the number of linear-system solves for the reduced-order simulation often remains roughly the same as that for the full-order simulation. We propose exploiting knowledge of the model's temporal behavior to (1) forecast the unknown variable of the reduced-order system of nonlinear equations at future time steps, and (2) use this forecast as an initial guess for the Newton-like solver during the reduced-order-model simulation. To compute the forecast, we propose using the Gappy POD technique. As a result, the goal is to generate an accurate initial guess so that the Newton solver requires many fewer iterations to converge, thereby decreasing the number of linear-system solves in the reduced-order-model simulation.

Authors:
 [1];  [1];  [1]
  1. Sandia National Lab. (SNL-CA), Livermore, CA (United States)
Publication Date:
Research Org.:
Sandia National Lab. (SNL-CA), Livermore, CA (United States); Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1184503
Alternate Identifier(s):
OSTI ID: 1250080
Report Number(s):
SAND-2014-19077J
Journal ID: ISSN 0045-7825; 540696
Grant/Contract Number:  
AC04-94AL85000; AC04-94-AL85000
Resource Type:
Accepted Manuscript
Journal Name:
Computer Methods in Applied Mechanics and Engineering
Additional Journal Information:
Journal Volume: 289; Journal Issue: C; Journal ID: ISSN 0045-7825
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; nonlinear model reduction; Gappy POD; temporal correlation; forecasting; initial guess

Citation Formats

Carlberg, Kevin, Ray, Jaideep, and van Bloemen Waanders, Bart. Decreasing the temporal complexity for nonlinear, implicit reduced-order models by forecasting. United States: N. p., 2015. Web. doi:10.1016/j.cma.2015.02.013.
Carlberg, Kevin, Ray, Jaideep, & van Bloemen Waanders, Bart. Decreasing the temporal complexity for nonlinear, implicit reduced-order models by forecasting. United States. doi:10.1016/j.cma.2015.02.013.
Carlberg, Kevin, Ray, Jaideep, and van Bloemen Waanders, Bart. Sat . "Decreasing the temporal complexity for nonlinear, implicit reduced-order models by forecasting". United States. doi:10.1016/j.cma.2015.02.013. https://www.osti.gov/servlets/purl/1184503.
@article{osti_1184503,
title = {Decreasing the temporal complexity for nonlinear, implicit reduced-order models by forecasting},
author = {Carlberg, Kevin and Ray, Jaideep and van Bloemen Waanders, Bart},
abstractNote = {Implicit numerical integration of nonlinear ODEs requires solving a system of nonlinear algebraic equations at each time step. Each of these systems is often solved by a Newton-like method, which incurs a sequence of linear-system solves. Most model-reduction techniques for nonlinear ODEs exploit knowledge of system's spatial behavior to reduce the computational complexity of each linear-system solve. However, the number of linear-system solves for the reduced-order simulation often remains roughly the same as that for the full-order simulation. We propose exploiting knowledge of the model's temporal behavior to (1) forecast the unknown variable of the reduced-order system of nonlinear equations at future time steps, and (2) use this forecast as an initial guess for the Newton-like solver during the reduced-order-model simulation. To compute the forecast, we propose using the Gappy POD technique. As a result, the goal is to generate an accurate initial guess so that the Newton solver requires many fewer iterations to converge, thereby decreasing the number of linear-system solves in the reduced-order-model simulation.},
doi = {10.1016/j.cma.2015.02.013},
journal = {Computer Methods in Applied Mechanics and Engineering},
number = C,
volume = 289,
place = {United States},
year = {2015},
month = {2}
}

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