Computational and Applied Mathematics

, Volume 37, Issue 1, pp 27–51 | Cite as

Time-parallel solutions to differential equations via functional optimization

  • C. LedermanEmail author
  • R. Martin
  • J.-L. Cambier


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.


Time-parallel computing Differential equations Functional optimization 

Mathematics Subject Classification

65Y05 Numerical analysis-Computer aspects of numerical algorithms-Parallel computation 34-04 Ordinary differential equations-Explicit machine computation and programs 



The authors wish to acknowledge the support of the Air Force Office of Scientific Research (AFOSR), Grant No. 12RZ06COR (PM: Dr. F. Fahroo) for this work. We would also like to thank Dr. Justin Koo of AFRL at Edwards AFB for many fruitful discussions.


  1. Amodio P, Brugnano L (2009) Parallel solution in time of ODEs: some achievements and perspectives. Appl Numer Math 59:424–435MathSciNetCrossRefzbMATHGoogle Scholar
  2. Conte SD, deBoor C (1972) Elementary numerical analysis. McGraw-Hill, New YorkGoogle Scholar
  3. Engblom S (2009) Parallel in Time Simulation of Multiscale Stochastic Chemical Kinetics. Multiscale Model Sim 8:46–68MathSciNetCrossRefzbMATHGoogle Scholar
  4. Engquist W (2003) The heterogeneous multi scale methods. Commun Math Sci 1:87–132MathSciNetCrossRefzbMATHGoogle Scholar
  5. Farhat C, Cortial J (2008) A time-parallel implicit method for accelerating the solution of nonlinear structural dynamics problems. Int J Num Methods Eng 1–25Google Scholar
  6. Fox C (1987) An introduction to the calculus of variations. Courier Dover PubGoogle Scholar
  7. Frantziskonis G, Muralidharan K, Deymier P, Simunovic S, Nukala P, Pannala S (2009) Time-parallel multiscale/multiphysics framework. J Comput Phys 228:8085–8092MathSciNetCrossRefzbMATHGoogle Scholar
  8. Gear CW (1987) Parallel methods for ordinary differential equations. Calcolo 25(1–2):1–20MathSciNetzbMATHGoogle Scholar
  9. Gear CW, Kevrekidis IG (2003) Projective methods for stiff differential equations: problems with gaps in their eigenvalue spectrum. SIAM J Sci Comp 24:109–1110MathSciNetCrossRefzbMATHGoogle Scholar
  10. Harden C, Peterson J (2006) Combining the parareal algorithm and reduced order modeling for time dependent partial differential equations. (Harden, C. and Peterson, J. on-line post) Google Scholar
  11. Iserles A (1996) A first course in the numerical analysis of differential equations, 2nd edn. Cambridge University Press, CambridgeGoogle Scholar
  12. Lederman C, Joshi A, Dinov I, Vese L, Toga A, Van Horn JD (2011b) The generation of tetrahedral mesh models for neuroanatomical MRI 55:153–164Google Scholar
  13. Lederman C, Vese L, Chien A (2011a) Registration for 3D morphological comparison of brain aneurysm growth. Adv Visual Comput 6938:392–399Google Scholar
  14. Lin T, Dinov I, Toga A, Vese L (2010) Nonlinear elasticity registration and sobolev gradients. Biomed Image Regist 6204:269–280Google Scholar
  15. Lions J-L, Maday Y, Turinici G (2001) A parareal in time discretization of pde’s. CR Acad Sci Paris Serie I 332:661–668CrossRefzbMATHGoogle Scholar
  16. Mahavier WT (2011b) Solving boundary value problems numerically using steepest descent in Sobolev spaces. Missouri J Math Sci 11:19–32Google Scholar
  17. Majid A, Sial S (2011) Approximate solutions to Poisson-Boltzmann systems with Sobolev gradients. J Comput Phys 230:5732–5738MathSciNetCrossRefzbMATHGoogle Scholar
  18. Martin G (1993) A waveform relaxation algorithm with overlapping splitting for reaction diffusion equations. Numer Linear Algebra Appl 1:1–7Google Scholar
  19. Mujeeb D, Neuberger JW, Sial S (2008) Recursive form of sobolev gradient method for ODEs on long intervals. Int J Comput Math 85:1727–1740MathSciNetCrossRefzbMATHGoogle Scholar
  20. Neuberger J (1997) Sobolev gradients and differential equations. Lecture Notes in Mathematics 1670. SpringerGoogle Scholar
  21. Renka RJ (2004) Constructing fair curves and surfaces with a Sobolev gradient method. Comput Aided Geom Des 21(2):137–149MathSciNetCrossRefzbMATHGoogle Scholar
  22. Renka RJ (2009) Image segmentation with a sobolev gradient method. Nonlinear Anal 71:774–780MathSciNetCrossRefzbMATHGoogle Scholar
  23. Samaddar D, Newman DE, Sanchez R (2010) Parallelization in time of numerical simulations of fully-developed plasma turbulence using the parareal algorithm. J Comput Phys 229:6558CrossRefzbMATHGoogle Scholar
  24. Sand J, Burrage K (1998) A Jacobi waveform relaxation method for ODEs. SIAM J Sci Comput 20(2):534–552. ISSN 1064-8275Google Scholar
  25. Stone HS (1975) Parallel tridiagonal equation solvers. ACM Trans Math Softw l(4):289–307Google Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional (outside the USA)  2016

Authors and Affiliations

  1. 1.ERC Inc.Edwards AFBUSA
  2. 2.Air Force Research LaboratoryEdwards AFBUSA

Personalised recommendations