research-article

Algorithm 997: pySDC—Prototyping Spectral Deferred Corrections

Author:

Robert SpeckAuthors Info & Claims

ACM Transactions on Mathematical Software (TOMS), Volume 45, Issue 3

Article No.: 35, Pages 1 - 23

https://doi.org/10.1145/3310410

Published: 08 August 2019 Publication History

Abstract

In this article, we present the Python framework pySDC for solving collocation problems with spectral deferred correction (SDC) methods and their time-parallel variant PFASST, the parallel full approximation scheme in space and time. pySDC features many implementations of SDC and PFASST, from simple implicit timestepping to high-order implicit-explicit or multi-implicit splitting and multilevel SDCs. The software package comes with many different, preimplemented examples and has seven tutorials to help new users with their first steps. Time parallelism is implemented either in an emulated way for debugging and prototyping or using MPI for benchmarking. The code is fully documented and tested using continuous integration, including most results of previous publications. Here, we describe the structure of the code by taking two different perspectives: those of the user and those of the developer. The first sheds light on the front-end, the examples, and the tutorials, and the second is used to describe the underlying implementation and the data structures. We show three different examples to highlight various aspects of the implementation, the capabilities, and the usage of pySDC. In addition, couplings to the FEniCS framework and PETSc, the latter including spatial parallelism with MPI, are described.

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References

[1]

Satish Balay, Shrirang Abhyankar, Mark F. Adams, Jed Brown, Peter Brune, Kris Buschelman, Lisandro Dalcin, et al. 2018. PETSc Users Manual. Technical Report ANL-95/11—Revision 3.9. Argonne National Laboratory. http://www.mcs.anl.gov/petsc.

[2]

Matthias Bolten, Dieter Moser, and Robert Speck. 2017. A multigrid perspective on the parallel full approximation scheme in space and time. Numerical Linear Algebra with Applications 24, 6 (2017), e2110.

[3]

Matthias Bolten, Dieter Moser, and Robert Speck. 2018. Asymptotic convergence of the parallel full approximation scheme in space and time for linear problems. Numerical Linear Algebra with Applications 25, 6 (2018), e2208.

[4]

Anne Bourlioux, Anita T. Layton, and Michael L. Minion. 2003. High-order multi-implicit spectral deferred correction methods for problems of reactive flow. Journal of Computational Physics 189, 2 (2003), 651--675.

Digital Library

[5]

Elizabeth L. Bouzarth and Michael L. Minion. 2010. A multirate time integrator for regularized Stokeslets. Journal of Computational Physics 229, 11 (2010), 4208--4224.

Digital Library

[6]

Kevin Burrage. 1997. Parallel methods for ODEs. Advances in Computational Mathematics 7 (1997), 1--3.

[7]

Andrew J. Christlieb, Colin B. Macdonald, and Benjamin W. Ong. 2010. Parallel high-order integrators. SIAM Journal on Scientific Computing 32, 2 (2010), 818--835.

Digital Library

[8]

Lisandro D. Dalcin, Rodrigo R. Paz, Pablo A. Kler, and Alejandro Cosimo. 2011. Parallel distributed computing using Python. Advances in Water Resources 34, 9 (2011), 1124--1139.

[9]

Arjen Doelman, Tasso J. Kaper, and Paul A. Zegeling. 1997. Pattern formation in the one-dimensional Gray-Scott model. Nonlinearity 10, 2 (1997), 523.

[10]

Alok Dutt, Leslie Greengard, and Vladimir Rokhlin. 2000. Spectral deferred correction methods for ordinary differential equations. BIT Numerical Mathematics 40, 2 (2000), 241--266.

Digital Library

[11]

Wael R. Elwasif, Samantha S. Foley, David E. Bernholdt, Lee A. Berry, Debasmita Samaddar, David E. Newman, and Raul Sanchez. 2011. A dependency-driven formulation of Parareal: Parallel-in-time solution of PDEs as a many-task application. In Proceedings of the 2011 ACM International Workshop on Many Task Computing on Grids and Supercomputers (MTAGS’11). ACM, New York, NY, 15--24.

Digital Library

[12]

M. Emmett and M. L. Minion. 2012. Toward an efficient parallel in time method for partial differential equations. Communications in Applied Mathematics and Computational Science 7 (2012), 105--132.

[13]

Xinlong Feng, Tao Tang, and Jiang Yang. 2015. Long time numerical simulations for phase-field problems using p-adaptive spectral deferred correction methods. SIAM Journal on Scientific Computing 37, 1 (2015), A271--A294.

Digital Library

[14]

M. Gander and S. Güttel. 2013. PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35, 2 (2013), C123--C142.

Digital Library

[15]

Martin J. Gander. 2015. 50 years of time parallel time integration. In Multiple Shooting and Time Domain Decomposition. Springer.

[16]

Carsten Gräser, Ralf Kornhuber, and Uli Sack. 2013. Time discretizations of anisotropic Allen-Cahn equations. IMA Journal of Numerical Analysis 33, 4 (2013), 1226--1244.

[17]

Carsten Gräser, Uli Sack, and Oliver Sander. 2009. Truncated nonsmooth Newton multigrid methods for convex minimization problems. In Domain Decomposition Methods in Science and Engineering XVIII, M. Bercovier, M. J. Gander, R. Kornhuber, and O. Widlund (Eds.). Springer, Berlin, Germany, 129--136.

[18]

P. Gray and S. K. Scott. 1983. Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Isolas and other forms of multistability. Chemical Engineering Science 38, 1 (1983), 29--43.

[19]

Jingfang Huang, Jun Jia, and Michael Minion. 2006. Accelerating the convergence of spectral deferred correction methods. Journal of Computational Physics 214, 2 (2006), 633--656.

Digital Library

[20]

J. D. Hunter. 2007. Matplotlib: A 2D graphics environment. Computing in Science and Engineering 9, 3 (2007), 90--95.

Digital Library

[21]

Jun Jia, Judith C. Hill, Katherine J. Evans, George I. Fann, and Mark A. Taylor. 2014. A spectral deferred correction method applied to the shallow water equations on a sphere. Monthly Weather Review 141 (2014), 3435--3449.

[22]

Eric Jones, Travis Oliphant, Pearu Peterson, et al. 2018. SciPy Home Page. Retrieved January 30, 2018 from http://www.scipy.org/.

[23]

Jülich Supercomputing Centre. 2016. JURECA: General-purpose supercomputer at Jülich supercomputing centre. Journal of Large-Scale Research Facilities 2, A62 (2016).

[24]

David I. Ketcheson, Kyle T. Mandli, Aron J. Ahmadia, Amal Alghamdi, Manuel Quezada de Luna, Matteo Parsani, Matthew G. Knepley, and Matthew Emmett. 2012. PyClaw: Accessible, extensible, scalable tools for wave propagation problems. SIAM Journal on Scientific Computing 34, 4 (2012), C210--C231.

Digital Library

[25]

Thomas Kluyver, Benjamin Ragan-Kelley, Fernando Pérez, Brian Granger, Matthias Bussonnier, Jonathan Frederic, Kyle Kelley, et al. 2016. Jupyter notebooks—A publishing format for reproducible computational workflows. In Positioning and Power in Academic Publishing: Players, Agents and Agendas, F. Loizides and B. Schmidt (Eds.). IOS Press, 87--90.

[26]

Anita T. Layton and Michael L. Minion. 2004. Conservative multi-implicit spectral deferred correction methods for reacting gas dynamics. Journal of Computational Physics 194, 2 (2004), 697--715.

Digital Library

[27]

J.-L. Lions, Y. Maday, and G. Turinici. 2001. A “Parareal” in time discretization of PDE’s. Comptes Rendus de l’Académie des Sciences—Series I—Mathematics 332 (2001), 661--668.

[28]

LLBL. 2018. PFASST Codes. Retrieved July 30, 2018 from https://pfasst.lbl.gov/codes.

[29]

LLNL. 2018. XBraid. Retrieved July 30, 2018 from https://www.llnl.gov/casc/xbraid.

[30]

Anders Logg, Kent-Andre Mardal, and Garth Wells. 2012. Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book. Springer, Berlin, Germany.

Digital Library

[31]

Anders Logg and Garth N. Wells. 2010. DOLFIN: Automated finite element computing. ACM Transactions on Mathematical Software 37, 2 (2010), 20.

Digital Library

[32]

Sebastian Lührs, Daniel Rohe, Alexander Schnurpfeil, Kay Thust, and Wolfgang Frings. 2016. Flexible and generic workflow management. In Parallel Computing: On the Road to Exascale. Advances in Parallel Computing, Vol. 27. IOS Press, Amsterdam, Netherlands, 431--438.

[33]

Michael L. Minion. 2003. Semi-implicit spectral deferred correction methods for ordinary differential equations. Communications in Mathematical Sciences 1, 3 (2003), 471--500. http://projecteuclid.org/euclid.cms/1250880097

[34]

Michael L. Minion. 2004. Semi-implicit projection methods for incompressible flow based on spectral deferred corrections. Applied Numerical Mathematics 48, 3–4 (2004), 369--387.

Digital Library

[35]

Michael L. Minion. 2010. A hybrid Parareal spectral deferred corrections method. Communications in Applied Mathematics and Computational Science 5, 2 (2010), 265--301.

[36]

Michael L. Minion, Robert Speck, Matthias Bolten, Matthew Emmett, and Daniel Ruprecht. 2015. Interweaving PFASST and parallel multigrid. SIAM Journal on Scientific Computing 37, 5 (2015), S244--S263.

Digital Library

[37]

Andreas Naumann, Daniel Ruprecht, and Joerg Wensch. 2018. Toward transient finite element simulation of thermal deformation of machine tools in real-time. Computational Mechanics 62, 5 (2018), 929--942.

Digital Library

[38]

Tim O’Brien. 2019. Python 3 Statement. Retrieved February 14, 2019 from https://python3statement.org/.

[39]

Benjamin W. Ong, Ronald D. Haynes, and Kyle Ladd. 2016. Algorithm 965: RIDC methods: A family of parallel time integrators. ACM Transactions on Mathematical Software 43, 1 (2016), Article 8, 13 pages.

Digital Library

[40]

Florian Rathgeber, David A. Ham, Lawrence Mitchell, Michael Lange, Fabio Luporini, Andrew T. T. Mcrae, Gheorghe-Teodor Bercea, Graham R. Markall, and Paul H. J. Kelly. 2016. Firedrake: Automating the finite element method by composing abstractions. ACM Transactions on Mathematical Software 43, 3 (2016), Article 24, 27 pages.

Digital Library

[41]

Daniel Ruprecht and Robert Speck. 2016. Spectral deferred corrections with fast-wave slow-wave splitting. SIAM Journal on Scientific Computing 38, 4 (2016), A2535--A2557.

Digital Library

[42]

Martin Schreiber. 2018. SWEET Home Page. Retrieved July 30, 2018 from https://schreiberx.github.io/sweetsite/.

[43]

Robert Speck. 2018a. Github repository for pySDC. Retrieved February 14, 2019 from https://github.com/Parallel-in-Time/pySDC.

[44]

Robert Speck. 2018b. Parallelizing spectral deferred corrections across the method. Computing and Visualization in Science 19, 3-4 (2018), 75--83.

Digital Library

[45]

Robert Speck. 2018c. PyPI site for pySDC. Retrieved February 14, 2019 from https://pypi.python.org/pypi/pySDC.

[46]

Robert Speck. 2019. Parallel-in-Time/pySDC: The 3.0 Release. Retrieved July 10, 2019 from https://zenodo.org/record/2565062#.XSXsZOtJEyU.

[47]

Robert Speck and Daniel Ruprecht. 2017. Toward fault-tolerant parallel-in-time integration with PFASST. Parallel Computing 62 (2017), 20--37.

Digital Library

[48]

Robert Speck, Daniel Ruprecht, Matthew Emmett, Michael Minion, Matthias Bolten, and Rolf Krause. 2015. A multi-level spectral deferred correction method. BIT Numerical Mathematics 55, 3 (2015), 843--867.

Digital Library

[49]

Robert Speck, Daniel Ruprecht, Rolf Krause, Matthew Emmett, Michael L. Minion, Mathias Winkel, and Paul Gibbon. 2012. A massively space-time parallel N-body solver. In Proceedings of the International Conference on High Performance Computing, Networking, Storage, and Analysis (SC’12). IEEE, Los Alamitos, CA, Article 92, 11 pages.

Digital Library

[50]

Robert Speck, Daniel Ruprecht, Michael Minion, Matthew Emmett, and Rolf Krause. 2016. Inexact spectral deferred corrections. In Domain Decomposition Methods in Science and Engineering XXII, T. Dickopf, M. J. Gander, L. Halpern, R. Krause, and L. F. Pavarino (Eds.). 389--396.

[51]

Tao Tang, Hehu Xie, and Xiaobo Yin. 2013. High-order convergence of spectral deferred correction methods on general quadrature nodes. Journal of Scientific Computing 56, 1 (2013), 1--13.

Digital Library

[52]

Travis CI GmbH. 2018. Travis CI: Test and Deploy with Confidence. Retrieved February 14, 2019 from https://travis-ci.org.

[53]

Ulrich Trottenberg, Cornelius W. Oosterlee, and Anton Schuller. 2000. Multigrid. Academic Press.

Digital Library

[54]

P. J. Van Der Houwen and B. P. Sommeijer. 1990. Parallel iteration of high-order Runge-Kutta methods with stepsize control. Journal of Computational and Applied Mathematics 29, 1 (1990), 111--127.

Digital Library

[55]

S. van der Walt, S. C. Colbert, and G. Varoquaux. 2011. The NumPy array: A structure for efficient numerical computation. Computing in Science Engineering 13, 2 (2011), 22--30.

Digital Library

[56]

Martin Weiser. 2014. Faster SDC convergence on non-equidistant grids by DIRK sweeps. BIT Numerical Mathematics 55, 4 (2014), 1219--1241.

Digital Library

[57]

Martin Weiser and Sunayana Ghosh. 2018. Theoretically optimal inexact SDC methods. Communications in Applied Mathematics and Computational Science 13, 1 (2018), 53--86.

[58]

M. Winkel, R. Speck, and D. Ruprecht. 2015. A high-order Boris integrator. Journal of Computational Physics 295 (2015), 456--474.

Digital Library

[59]

Yinhua Xia, Yan Xu, and Chi-Wang Shu. 2007. Efficient time discretization for local discontinuous Galerkin methods. Discrete and Continuous Dynamical Systems—Series B 8, 3 (2007), 677--693.

[60]

J. Zhang and Q. Du. 2009. Numerical studies of discrete approximations to the Allen-Cahn equation in the sharp interface limit. SIAM Journal on Scientific Computing 31, 4 (2009), 3042--3063.

Digital Library

Cited By

Akramov IGötschel SMinion MRuprecht DSpeck R(2024)Spectral Deferred Correction Methods for Second-Order ProblemsSIAM Journal on Scientific Computing10.1137/23M159259646:3(A1690-A1713)Online publication date: 14-May-2024
https://doi.org/10.1137/23M1592596
Kraft RKoraltan SGattringer MBruckner FSuess DAbert C(2024)Parallel-in-time integration of the Landau–Lifshitz–Gilbert equation with the parallel full approximation scheme in space and timeJournal of Magnetism and Magnetic Materials10.1016/j.jmmm.2024.171998597(171998)Online publication date: May-2024
https://doi.org/10.1016/j.jmmm.2024.171998
Čaklović GSpeck RFrank M(2023)A parallel-in-time collocation method using diagonalization: theory and implementation for linear problemsCommunications in Applied Mathematics and Computational Science10.2140/camcos.2023.18.5518:1(55-85)Online publication date: 21-Dec-2023
https://doi.org/10.2140/camcos.2023.18.55
Show More Cited By

Index Terms

Algorithm 997: pySDC—Prototyping Spectral Deferred Corrections
1. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
      1. Ordinary differential equations
      2. Partial differential equations
  2. Mathematical software
    1. Solvers

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Published In

cover image ACM Transactions on Mathematical Software

ACM Transactions on Mathematical Software Volume 45, Issue 3

September 2019

357 pages

ISSN:0098-3500

EISSN:1557-7295

DOI:10.1145/3349340

Editors:
Zhaojun Bai
University of California at Davis, USA
,
Wolfgang Bangerth
Colorado State University, USA

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Copyright © 2019 ACM.

Publication rights licensed to ACM. ACM acknowledges that this contribution was authored or co-authored by an employee, contractor or affiliate of a national government. As such, the Government retains a nonexclusive, royalty-free right to publish or reproduce this article, or to allow others to do so, for Government purposes only.

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New York, NY, United States

Publication History

Published: 08 August 2019

Accepted: 01 February 2019

Received: 01 August 2018

Published in TOMS Volume 45, Issue 3

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Cited By

Akramov IGötschel SMinion MRuprecht DSpeck R(2024)Spectral Deferred Correction Methods for Second-Order ProblemsSIAM Journal on Scientific Computing10.1137/23M159259646:3(A1690-A1713)Online publication date: 14-May-2024
https://doi.org/10.1137/23M1592596
Kraft RKoraltan SGattringer MBruckner FSuess DAbert C(2024)Parallel-in-time integration of the Landau–Lifshitz–Gilbert equation with the parallel full approximation scheme in space and timeJournal of Magnetism and Magnetic Materials10.1016/j.jmmm.2024.171998597(171998)Online publication date: May-2024
https://doi.org/10.1016/j.jmmm.2024.171998
Čaklović GSpeck RFrank M(2023)A parallel-in-time collocation method using diagonalization: theory and implementation for linear problemsCommunications in Applied Mathematics and Computational Science10.2140/camcos.2023.18.5518:1(55-85)Online publication date: 21-Dec-2023
https://doi.org/10.2140/camcos.2023.18.55
Bolten MFriedhoff SHahne J(2023)Task graph-based performance analysis of parallel-in-time methodsParallel Computing10.1016/j.parco.2023.103050118(103050)Online publication date: Nov-2023
https://doi.org/10.1016/j.parco.2023.103050
Hahne JFriedhoff SBolten M(2021)Algorithm 1016ACM Transactions on Mathematical Software10.1145/344697947:2(1-22)Online publication date: 20-Apr-2021
https://dl.acm.org/doi/10.1145/3446979
Speck RKnobloch MLührs SGocht A(2021)Using Performance Analysis Tools for a Parallel-in-Time IntegratorParallel-in-Time Integration Methods10.1007/978-3-030-75933-9_3(51-80)Online publication date: 25-Aug-2021
https://doi.org/10.1007/978-3-030-75933-9_3
Ong BSpiteri R(2020)Deferred Correction Methods for Ordinary Differential EquationsJournal of Scientific Computing10.1007/s10915-020-01235-883:3Online publication date: 11-Jun-2020
https://dl.acm.org/doi/10.1007/s10915-020-01235-8
Schöbel RSpeck R(2020)PFASST-ER: combining the parallel full approximation scheme in space and time with parallelization across the methodComputing and Visualization in Science10.1007/s00791-020-00330-523:1-4Online publication date: 1-Dec-2020
https://dl.acm.org/doi/10.1007/s00791-020-00330-5
Bolten MFriedhoff SHahne J(undefined)Task Graph-Based Performance Analysis of Parallel-in-Time MethodsSSRN Electronic Journal10.2139/ssrn.4201056
https://doi.org/10.2139/ssrn.4201056

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