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

Please always quote using this URN: urn:nbn:de:0297-zib-71490
accepted for publication on 2019-09-18
  • 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 adjoint equations simultaneously. 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 non-linear reaction-diffusion optimal control problem demonstrate the parallel speedup and efficiency of different approaches.

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Author:Sebastian GötschelORCiD, Michael L. Minion
Document Type:ZIB-Report
Tag:PDE-constrained optimization; PFASST; parallel-in-time methods
MSC-Classification:65-XX NUMERICAL ANALYSIS / 65Kxx Mathematical programming, optimization and variational techniques / 65K10 Optimization and variational techniques [See also 49Mxx, 93B40]
65-XX NUMERICAL ANALYSIS / 65Mxx Partial differential equations, initial value and time-dependent initial- boundary value problems / 65M55 Multigrid methods; domain decomposition
65-XX NUMERICAL ANALYSIS / 65Mxx Partial differential equations, initial value and time-dependent initial- boundary value problems / 65M70 Spectral, collocation and related methods
65-XX NUMERICAL ANALYSIS / 65Yxx Computer aspects of numerical algorithms / 65Y05 Parallel computation
Date of first Publication:2019/01/28
Series (Serial Number):ZIB-Report (19-04)
ISSN:1438-0064