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Title: Time-parallel solutions to differential equations via functional optimization

Abstract

We describe an approach to solving a generic time-dependent differential equation (DE) that recasts the problem as a functional optimization one. The techniques employed to solve for the functional minimum, which we relate to the Sobolev Gradient method, allow for large-scale parallelization in time, and therefore potential faster “wall-clock” time computing on machines with significant parallel computing capacity. We are able to come up with numerous different discretizations and approximations for our optimization-derived equations, each of which either puts an existing approach, the Parareal method, in an optimization context, or provides a new time-parallel (TP) scheme with potentially faster convergence to the DE solution. We describe how the approach is particularly effective for solving multiscale DEs and present TP schemes that incorporate two different solution scales. Sample results are provided for three differential equations, solved with TP schemes, and we discuss how the choice of TP scheme can have an orders of magnitude effect on the accuracy or convergence rate.

Authors:
;  [1];  [2]
  1. ERC Inc. (United States)
  2. Air Force Research Laboratory (United States)
Publication Date:
OSTI Identifier:
22769395
Resource Type:
Journal Article
Journal Name:
Computational and Applied Mathematics
Additional Journal Information:
Journal Volume: 37; Journal Issue: 1; Other Information: Copyright (c) 2018 SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional; Article Copyright (c) 2016 SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional (outside the USA); Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0101-8205
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICAL METHODS AND COMPUTING; CONVERGENCE; DIFFERENTIAL EQUATIONS; MATHEMATICAL SOLUTIONS; OPTIMIZATION; TIME DEPENDENCE

Citation Formats

Lederman, C., E-mail: carl.lederman.ctr@us.af.mil, Martin, R., E-mail: robert.martin.81.ctr@us.af.mil, and Cambier, J.-L., E-mail: jean-luc.cambier@us.af.mil. Time-parallel solutions to differential equations via functional optimization. United States: N. p., 2018. Web. doi:10.1007/S40314-016-0319-7.
Lederman, C., E-mail: carl.lederman.ctr@us.af.mil, Martin, R., E-mail: robert.martin.81.ctr@us.af.mil, & Cambier, J.-L., E-mail: jean-luc.cambier@us.af.mil. Time-parallel solutions to differential equations via functional optimization. United States. doi:10.1007/S40314-016-0319-7.
Lederman, C., E-mail: carl.lederman.ctr@us.af.mil, Martin, R., E-mail: robert.martin.81.ctr@us.af.mil, and Cambier, J.-L., E-mail: jean-luc.cambier@us.af.mil. Thu . "Time-parallel solutions to differential equations via functional optimization". United States. doi:10.1007/S40314-016-0319-7.
@article{osti_22769395,
title = {Time-parallel solutions to differential equations via functional optimization},
author = {Lederman, C., E-mail: carl.lederman.ctr@us.af.mil and Martin, R., E-mail: robert.martin.81.ctr@us.af.mil and Cambier, J.-L., E-mail: jean-luc.cambier@us.af.mil},
abstractNote = {We describe an approach to solving a generic time-dependent differential equation (DE) that recasts the problem as a functional optimization one. The techniques employed to solve for the functional minimum, which we relate to the Sobolev Gradient method, allow for large-scale parallelization in time, and therefore potential faster “wall-clock” time computing on machines with significant parallel computing capacity. We are able to come up with numerous different discretizations and approximations for our optimization-derived equations, each of which either puts an existing approach, the Parareal method, in an optimization context, or provides a new time-parallel (TP) scheme with potentially faster convergence to the DE solution. We describe how the approach is particularly effective for solving multiscale DEs and present TP schemes that incorporate two different solution scales. Sample results are provided for three differential equations, solved with TP schemes, and we discuss how the choice of TP scheme can have an orders of magnitude effect on the accuracy or convergence rate.},
doi = {10.1007/S40314-016-0319-7},
journal = {Computational and Applied Mathematics},
issn = {0101-8205},
number = 1,
volume = 37,
place = {United States},
year = {2018},
month = {3}
}