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

Journal Article · 12 December 2019 · SIAM Journal on Scientific Computing

DOI: https://doi.org/10.1137/19m1239313 · OSTI ID:1580377

Götschel, Sebastian  ^[1]; Minion, Michael L.  ^[2]

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1. Zuse Inst. Berlin, Berlin (Germany)
2. Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)

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.

• Research Organization: Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)
• Sponsoring Organization: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); German Research Foundation (DFG); Alexander von Humboldt Foundation
• Grant/Contract Number: AC02-05CH11231
• OSTI ID: 1580377
• Journal Information: SIAM Journal on Scientific Computing, Vol. 41, Issue 6; ISSN 1064-8275
• Publisher: SIAMCopyright Statement
• Country of Publication: United States
• Language: English