×

A parallel-in-time approach for accelerating direct-adjoint studies. (English) Zbl 07500763

Summary: Parallel-in-time methods are developed to accelerate the direct-adjoint looping procedure. Particularly, we utilize the algorithm, previously developed to integrate equations forward in time, to accelerate the direct-adjoint looping that arises from gradient-based optimization. We consider both linear and non-linear governing equations and exploit the linear, time-varying nature of the adjoint equations. Gains in efficiency are seen across all cases, showing that a Paraexp based parallel-in-time approach is feasible for the acceleration of direct-adjoint studies. This signifies a possible approach to further increase the run-time performance for optimization studies that either cannot be parallelized in space or are at their limit of efficiency gains for a parallel-in-space approach. Code demonstrating the algorithms considered in this paper is available.

MSC:

65Mxx Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
65Yxx Computer aspects of numerical algorithms
35Kxx Parabolic equations and parabolic systems
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Bal, G.; Maday, Y., A “parareal” time discretization for non-linear PDE’s with application to the pricing of an American put, (Recent Developments in Domain Decomposition Methods (2002), Springer: Springer Berlin, Heidelberg), 189-202 · Zbl 1022.65096
[2] Bergamaschi, L.; Caliari, M.; Martínez, A.; Vianello, M., Comparing Leja and Krylov approximations of large scale matrix exponentials, (Computational Science. Computational Science, ICCS 2006. Computational Science. Computational Science, ICCS 2006, Lect. Notes in Comp. Sci., vol. 3994 (2006), Springer-Verlag: Springer-Verlag Berlin, Heidelberg), 685-692 · Zbl 1157.65366
[3] Caliari, M., Accurate evaluation of divided differences for polynomial interpolation of exponential propagators, Computing, 80, 2, 189-201 (2007) · Zbl 1120.65009
[4] Clarke, A. T.; Davies, C. J.; Ruprecht, D.; Tobias, S. M., Parallel-in-time integration of kinematic dynamos, J. Comput. Phys. X, 7, Article 100057 pp. (2020)
[5] Clarke, A. T.; Davies, C. J.; Ruprecht, D.; Tobias, S. M.; Oishi, J. S., Performance of parallel-in-time integration for Rayleigh Bénard convection, Comput. Vis. Sci., 23, 1, 10 (2020) · Zbl 07704914
[6] Cox, S. M.; Matthews, P. C., Exponential time differencing for stiff systems, J. Comput. Phys., 176, 2, 430-455 (2002) · Zbl 1005.65069
[7] Eggl, M. F.; Schmid, P. J., A gradient-based framework for maximizing mixing in binary fluids, J. Comput. Phys., 368, 131-153 (2018) · Zbl 1392.76019
[8] Eggl, M. F.; Schmid, P. J., Mixing enhancement in binary fluids using optimised stirring strategies, J. Fluid Mech., 899, A24 (2020) · Zbl 1460.76880
[9] Eggl, M. F.; Schmid, P. J., Shape optimization of stirring rods for mixing binary fluids, IMA J. Appl. Math., 762-789 (2020) · Zbl 1461.76151
[10] Emmett, M.; Minion, M. L., Toward an efficient parallel in time method for partial differential equations, Commun. Appl. Math. Comput. Sci., 7, 105-132 (2012) · Zbl 1248.65106
[11] Falgout, R. D.; Friedhoff, S.; Kolev, Tz. V.; MacLachlan, S. P.; Schroder, J. B., Parallel time integration with multigrid, SIAM J. Sci. Comput., 36, 6, C635-C661 (2014) · Zbl 1310.65115
[12] Farhat, C.; Chandesris, M., Time-decomposed parallel time-integrators: theory and feasibility studies for fluid, structure, and fluid-structure applications, Int. J. Numer. Methods Eng., 58, 9, 1397-1434 (2003) · Zbl 1032.74701
[13] Foures, D. P.G.; Caulfield, C. P.; Schmid, P. J., Optimal mixing in two-dimensional plane Poiseuille flow at finite Péclet number, J. Fluid Mech., 748, 241-277 (2014) · Zbl 1416.76036
[14] Friedhoff, S.; Falgout, R. D.; Kolev, T. V.; MacLachlan, S. P.; Schroder, J. B., A multigrid-in-time algorithm for solving evolution equations in parallel, (Presented at: Sixteenth Copper Mountain Conference on Multigrid Methods. Presented at: Sixteenth Copper Mountain Conference on Multigrid Methods, Copper Mountain, CO, United, States, Mar 17-Mar 22, 2013 (2013))
[15] Gander, M. J., Overlapping Schwarz for linear and nonlinear parabolic problems, (9th International Conference on Domain Decomposition Methods (1996)), 97-104
[16] Gander, M. J., 50 Years of Time Parallel Time Integration, Contributions in Mathematical and Computational Sciences, vol. 9, 69-113 (2014), Springer International Publishing: Springer International Publishing Cham · Zbl 1337.65127
[17] Gander, M. J.; Güttel, S., PARAEXP: a parallel integrator for linear initial-value problems, SIAM J. Sci. Comput., 35, 2, C123-C142 (2013) · Zbl 1266.65123
[18] Gander, M. J.; Neumüller, M., Analysis of a new space-time parallel multigrid algorithm for parabolic problems, SIAM J. Sci. Comput., 38, 4, A2173-A2208 (2016) · Zbl 1342.65225
[19] Gander, M. J.; Güttel, S.; Petcu, M., A nonlinear PARAEXP algorithm, (Domain Decomposition Methods in Science and Engineering XXIV (2018), Springer International Publishing: Springer International Publishing Cham), 261-270 · Zbl 1443.65094
[20] Gander, M. J.; Kwok, F.; Paraopt, J. Salomon, A parareal algorithm for optimality systems, SIAM J. Sci. Comput., 42, 5 (2020), A2773-A2802 · Zbl 1451.49034
[21] Gander, M. J.; Liu, J.; Wu, S.-L.; Yue, X.; Paradiag, T. Zhou, Parallel-in-time algorithms based on the diagonalization technique (2020), URL
[22] Giladi, E.; Keller, H. B., Space-time domain decomposition for parabolic problems, Numer. Math., 93, 2, 279-313 (2002) · Zbl 1019.65076
[23] Götschel, S.; Minion, M. L., Parallel-in-time for parabolic optimal control problems using PFASST, (Domain Decomposition Methods in Science and Engineering XXIV (2018), Springer International Publishing: Springer International Publishing Cham), 363-371 · Zbl 1450.65101
[24] Götschel, S.; Minion, M. L., An efficient parallel-in-time method for optimization with parabolic PDEs, SIAM J. Sci. Comput., 41, 6, C603-C626 (2019) · Zbl 07149707
[25] Griewank, A.; Walther, A., Algorithm 799: revolve: an implementation of checkpointing for the reverse or adjoint mode of computational differentiation, ACM Trans. Math. Softw., 26, 19-45 (2000) · Zbl 1137.65330
[26] Günther, S.; Gauger, N. R.; Schroder, J. B., A non-intrusive parallel-in-time adjoint solver with the XBraid library, Comput. Vis. Sci., 19, 3, 85-95 (2018) · Zbl 07704539
[27] Günther, S.; Gauger, N. R.; Schroder, J. B., A non-intrusive parallel-in-time approach for simultaneous optimization with unsteady PDEs, Optim. Methods Softw., 34, 6, 1306-1321 (2019) · Zbl 1428.35641
[28] Güttel, S.; Kressner, D.; Lund, K., Limited-memory polynomial methods for large-scale matrix functions, GAMM-Mitt., 43, 3, Article e202000019 pp. (2020)
[29] Horton, G.; Vandewalle, S., A space-time multigrid method for parabolic partial differential equations, SIAM J. Sci. Comput., 16, 4, 848-864 (1995) · Zbl 0828.65105
[30] Jameson, A., Aerodynamic design via control theory, J. Sci. Comput., 3, 3, 233-260 (1988) · Zbl 0676.76055
[31] Kallala, H.; Vay, J.-L.; Vincenti, H., A generalized massively parallel ultra-high order FFT-based Maxwell solver, Comput. Phys. Commun., 244, 25-34 (2019) · Zbl 07674828
[32] Kassam, A.-K.; Trefethen, L. N., Fourth-order time-stepping for stiff PDEs, SIAM J. Sci. Comput., 26, 4, 1214-1233 (2005) · Zbl 1077.65105
[33] Kooij, G. L.; Botchev, M. A.; Geurts, B. J., A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations, J. Comput. Appl. Math., 316, 229-246 (2017) · Zbl 1375.65130
[34] Kwok, F., Neumann-Neumann waveform relaxation for the time-dependent heat equation, (Domain Decomposition Methods in Science and Engineering XXI (2014), Springer International Publishing: Springer International Publishing Cham), 189-198 · Zbl 1382.65248
[35] Laizet, S.; Vassilicos, J. C., Direct numerical simulation of fractal-generated turbulence, (Direct and Large-Eddy Simulation VII (2010), Springer: Springer Dordrecht, Netherlands), 17-23
[36] Lelarasmee, E.; Ruehli, A. E.; Sangiovanni-Vincentelli, A. L., The waveform relaxation method for time-domain analysis of large scale integrated circuits, IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., 1, 3, 131-145 (1982)
[37] Lions, J.-L.; Maday, Y.; Turinici, G., Résolution d’edp par un schéma en temps pararéel, C. R. Séances Acad. Sci., Sér. 1 Math., 332, 7, 661-668 (2001) · Zbl 0984.65085
[38] Maday, Y.; Rønquist, E. M., Parallelization in time through tensor-product space-time solvers, C. R. Math., 346, 1, 113-118 (2008) · Zbl 1133.65066
[39] Maday, Y.; Turinici, G., A parareal in time procedure for the control of partial differential equations, C. R. Math., 335, 4, 387-392 (2002) · Zbl 1006.65071
[40] Maday, Y.; Turinici, G., Parallel in time algorithms for quantum control: parareal time discretization scheme, Int. J. Quant. Chem., 93, 3, 223-228 (2003)
[41] Mandal, B. C., A time-dependent Dirichlet-Neumann method for the heat equation, (Domain Decomposition Methods in Science and Engineering XXI (2014), Springer International Publishing: Springer International Publishing Cham), 467-475 · Zbl 1382.65253
[42] Marcotte, F.; Caulfield, C. P., Optimal mixing in two-dimensional stratified plane Poiseuille flow at finite Péclet and Richardson numbers, J. Fluid Mech., 853, 359-385 (2018) · Zbl 1415.76215
[43] Nievergelt, J., Parallel methods for integrating ordinary differential equations, Commun. ACM, 7, 12, 731-733 (1964) · Zbl 0134.32804
[44] Ong, B. W.; Schroder, J. B., Applications of time parallelization, Comput. Vis. Sci., 23, 1, 11 (2020) · Zbl 07704915
[45] Pekurovsky, D., P3DFFT: a framework for parallel computations of Fourier transforms in three dimensions, SIAM J. Sci. Comput., 34, 4, C192-C209 (2012) · Zbl 1253.65205
[46] Pringle, C. T.; Kerswell, R., Using nonlinear transient growth to construct the minimal seed for shear flow turbulence, Phys. Rev. Lett., 105, Article 154502 pp. (2010)
[47] Qadri, U. A.; Magri, L.; Ihme, M.; Schmid, P. J., Optimal ignition placement in diffusion flames by nonlinear adjoint looping, (Ctr. Turb. Res., Proc. of the Summer Program (2016))
[48] Saad, Y., Analysis of some Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal., 29, 1, 209-228 (1992) · Zbl 0749.65030
[49] Schulze, J. C.; Schmid, P. J.; Sesterhenn, J. L., Exponential time integration using Krylov subspaces, Int. J. Numer. Methods Fluids, 60, 6, 591-609 (2009) · Zbl 1163.76038
[50] Skene, C. S., Adjoint based analysis for swirling and reacting flows (2019), Imperial College London, PhD thesis
[51] Skene, C. S.; Eggl, M. F.; Schmid, P. J., csskene/parallel-in-time_direct-adjoint (2020), URL
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.