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Title: An Efficient Parallel-in-Time Method for Optimization with Parabolic PDEs

Abstract

To solve optimization problems with parabolic PDE constraints, often methods working on the reduced objective functional are used. They are computationally expensive due to the necessity of solving both the state equation and a backward-in-time adjoint equation to evaluate the reduced gradient in each iteration of the optimization method. In this study, we investigate the use of the parallel-in-time method PFASST in the setting of PDE-constrained optimization. In order to develop an efficient fully time-parallel algorithm, we discuss different options for applying PFASST to adjoint gradient computation, including the possibility of doing PFASST iterations on both the state and the adjoint equations simultaneously. Here, we also explore the additional gains in efficiency from reusing information from previous optimization iterations when solving each equation. Numerical results for both a linear and a nonlinear reaction-diffusion optimal control problem demonstrate the parallel speedup and efficiency of different approaches.

Authors:
ORCiD logo [1]; ORCiD logo [2]
  1. Zuse Inst. Berlin, Berlin (Germany)
  2. Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
Publication Date:
Research Org.:
Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); German Research Foundation (DFG); Alexander von Humboldt Foundation
OSTI Identifier:
1580377
Grant/Contract Number:  
AC02-05CH11231
Resource Type:
Accepted Manuscript
Journal Name:
SIAM Journal on Scientific Computing
Additional Journal Information:
Journal Volume: 41; Journal Issue: 6; Journal ID: ISSN 1064-8275
Publisher:
SIAM
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; PDE-constrained optimization; parallel-in-time methods; PFASST

Citation Formats

Götschel, Sebastian, and Minion, Michael L. An Efficient Parallel-in-Time Method for Optimization with Parabolic PDEs. United States: N. p., 2019. Web. https://doi.org/10.1137/19m1239313.
Götschel, Sebastian, & Minion, Michael L. An Efficient Parallel-in-Time Method for Optimization with Parabolic PDEs. United States. https://doi.org/10.1137/19m1239313
Götschel, Sebastian, and Minion, Michael L. Thu . "An Efficient Parallel-in-Time Method for Optimization with Parabolic PDEs". United States. https://doi.org/10.1137/19m1239313. https://www.osti.gov/servlets/purl/1580377.
@article{osti_1580377,
title = {An Efficient Parallel-in-Time Method for Optimization with Parabolic PDEs},
author = {Götschel, Sebastian and Minion, Michael L.},
abstractNote = {To solve optimization problems with parabolic PDE constraints, often methods working on the reduced objective functional are used. They are computationally expensive due to the necessity of solving both the state equation and a backward-in-time adjoint equation to evaluate the reduced gradient in each iteration of the optimization method. In this study, we investigate the use of the parallel-in-time method PFASST in the setting of PDE-constrained optimization. In order to develop an efficient fully time-parallel algorithm, we discuss different options for applying PFASST to adjoint gradient computation, including the possibility of doing PFASST iterations on both the state and the adjoint equations simultaneously. Here, we also explore the additional gains in efficiency from reusing information from previous optimization iterations when solving each equation. Numerical results for both a linear and a nonlinear reaction-diffusion optimal control problem demonstrate the parallel speedup and efficiency of different approaches.},
doi = {10.1137/19m1239313},
journal = {SIAM Journal on Scientific Computing},
number = 6,
volume = 41,
place = {United States},
year = {2019},
month = {12}
}

Journal Article:
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