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Introduction to the Parareal Algorithm
Ma. Cristina Bargo
Laboratoire Jacques-Louis Lions
Universite Pierre et Marie Curie
University of the Philippines Diliman
Math 296
January 10, 2011
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Classical Discretization
The Problem
Find u such that
(1)
tu = A (t,u) , t (t0, T]
u = u0, t = t0
where A : R V V (V a Hilbert space) and t0 0.
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Classical Discretization
The Problem
Find u such that
(1)
tu = A (t,u) , t (t0, T]
u = u0, t = t0
where A : R V V (V a Hilbert space) and t0 0.
The Discrete Problem
Let t0 = T0 < T1 < < TN = T with Ti+1 Ti = T
(uniform). Let Ui be an approximation ofu(Ti) solution to (1) att = Ti. Then U0 = u0 and i = 0, . . . , N 1
Ui+1 = F(Ti+1; Ti, Ui)
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Classical Discretization
Example
u(t) =
1 55 1
u(t), t (0, 1.5]
with
u(0) =
10
T = 0.1
F: One-step implicit Euler scheme
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Classical DiscretizationExact Solution
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Classical Discretizationt = T0
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Classical Discretizationt = T1
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Classical Discretizationt = T2
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Classical Discretizationt = T3
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Classical Discretizationt = T4
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Classical Discretizationt = T5
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Classical Discretizationt = T6
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Classical Discretizationt = T7
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Classical Discretizationt = T8
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Classical Discretizationt = T9
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Classical Discretizationt = T10
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Cl i l Di i i
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Classical Discretizationt = T11
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Cl i l Di i i
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Classical Discretizationt = T12
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Cl i l Di i i
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Classical Discretizationt = T13
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Cl i l Di ti ti
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Classical Discretizationt = T14
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Cl i l Di ti ti
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Classical Discretizationt = T15
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Classical Discretization
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Classical Discretization
Remarks:
more accurate results using small time steps
computationally costly due to sequential nature of the scheme
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Classical Discretization
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Classical Discretization
Remarks:
more accurate results using small time steps
computationally costly due to sequential nature of the scheme
Going back to example...
T = 0.1
F: 15-step implicit Euler scheme, t = 6.67 103
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Classical Discretization
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Classical DiscretizationExact Solution
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Classical Discretization
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Classical Discretizationt = T0
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Classical Discretization
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Classical Discretizationt = t
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Classical Discretization
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Classical Discretizationt = 2t
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Classical Discretization
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Classical Discretizationt = 3t
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Classical Discretization
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Classical Discretizationt = 4t
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Classical Discretization
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Classical Discretizationt = 5t
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Classical Discretization
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Classical Discretizationt = 6t
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t = 7t
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Classical Discretization
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t = 8t
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Classical Discretization
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t = 9t
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Classical Discretization
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t = 10t
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Classical Discretization
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t = 11t
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Classical Discretization
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t = 12t
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t = 13t
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Classical Discretization
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t = 14t
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Classical Discretization
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t = 15t = T1
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Classical Discretization
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t = T15
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Classical Discretization
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Goal: Find a fast algorithm without sacrificing accuracy
high order schemes (ex. Crank-Nicholson, Runge-Kutta, etc.)
use multiple processors and solve the problem in parallel
parallel in the spatial domainparallel in the systemparallel in the methodparallel in the time domain
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The Parareal Algorithm
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PARAREAL - parallel in real time
Lions, Maday and Turinici [1], 2001Resolution dEDP par un schema en temps parareel
Bal and Maday [2], 2002A parareal time discretization for nonlinear PDEs withapplication to the pricing of an American put
Baffico et al [3], 2002Parallel-in-time molecular-dynamics simulations
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The Parareal Algorithm
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Notations:
Fine solver F(Ti+1; Ti, Ui) - accurate but computationallycostly
Coarse solver G(Ti+1; Ti, Ui) - less accurate than F butcomputationally cheap
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The Parareal Algorithm
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Notations:
Fine solver F(Ti+1; Ti, Ui) - accurate but computationallycostly
Coarse solver G(Ti+1; Ti, Ui) - less accurate than F butcomputationally cheap
U00 = u0 andU
0n+1 = G(Tn+1; Tn,U
0n)
For k = 1, 2, , Uk0 = u0 and
Uk+1n+1 = G(Tn+1; Tn,U
k+1n )
+ F(Tn+1; Tn,Uk
n) G(Tn+1; Tn,Uk
n)
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The Parareal Algorithm
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Uk+1n+1 = G(Tn+1; Tn,U
k+1n ) + F(Tn+1; Tn,U
k
n) G(Tn+1; Tn,Uk
n)
F(Tn+1; Tn,Ukn) G(Tn+1; Tn,U
kn): corrector (parallel)
G(Tn+1; Tn,Uk+1n ): predictor (sequential)
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The Parareal Algorithm
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Uk+1n+1 = G(Tn+1; Tn,U
k+1n ) + F(Tn+1; Tn,U
k
n) G(Tn+1; Tn,Uk
n)
F(Tn+1; Tn,Ukn) G(Tn+1; Tn,U
kn): corrector (parallel)
G(Tn+1; Tn,Uk+1n ): predictor (sequential)
Going back to example...
T = 0.1
G: one-step implicit Euler schemeF: 15-step implicit Euler scheme
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The Parareal AlgorithmThe Exact Solution
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The Exact Solution
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The Parareal AlgorithmIteration 0
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Iteration 0
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The Parareal AlgorithmIteration 1: t = Ti
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Iteration 1: t = Ti
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The Parareal AlgorithmIteration 1: t = Ti + t
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Iteration 1: t Ti + t
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The Parareal AlgorithmIteration 1: t = Ti + 2t
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i +
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The Parareal AlgorithmIteration 1: t = Ti + 3t
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The Parareal AlgorithmIteration 1: t = Ti + 4t
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The Parareal AlgorithmIteration 1: t = Ti + 5t
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The Parareal AlgorithmIteration 1: t = Ti + 6t
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The Parareal AlgorithmIteration 1: t = Ti + 7t
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The Parareal AlgorithmIteration 1: t = Ti + 8t
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The Parareal AlgorithmIteration 1: t = Ti + 9t
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The Parareal AlgorithmIteration 1: t = Ti + 10t
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The Parareal AlgorithmIteration 1: t = Ti + 11t
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The Parareal AlgorithmIteration 1: t = Ti + 12t
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The Parareal AlgorithmIteration 1: t = Ti + 13t
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The Parareal AlgorithmIteration 1: t = Ti + 14t
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The Parareal AlgorithmIteration 1: End of Parallel Part
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The Parareal AlgorithmIteration 1: t = T1
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The Parareal AlgorithmIteration 1: t = T1
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The Parareal AlgorithmIteration 1: t = T2
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The Parareal AlgorithmIteration 1: t = T2
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The Parareal AlgorithmIteration 1: t = T3
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The Parareal AlgorithmIteration 1: t = T3
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The Parareal AlgorithmIteration 1: t = T4
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The Parareal AlgorithmIteration 1: t = T5
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The Parareal AlgorithmIteration 1: t = T6
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The Parareal AlgorithmIteration 1: t = T7
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The Parareal AlgorithmIteration 1: t = T8
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The Parareal AlgorithmIteration 1: t = T10
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The Parareal AlgorithmIteration 1: t = T11
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The Parareal AlgorithmIteration 1: t = T12
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The Parareal AlgorithmIteration 1: t = T13
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The Parareal AlgorithmIteration 1: t = T14
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The Parareal AlgorithmIteration 1: t = T15
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The Parareal AlgorithmIteration 2
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The Parareal AlgorithmIteration 3
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The Parareal AlgorithmIteration 5
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The Parareal AlgorithmFine scheme vs. Parareal algorithm
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ANR PITAC
Past Work: http://www.ljll.math.upmc.fr/PITAC/
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p j p
theoretical: convergence and stability ([4] and [5])applications
molecular dynamic simulations [3]American put option pricing [2]Navier-Stokes equations [6]
control problems [7]Schrodinger equation [8]stochastic ordinary differential equations [9]
To do:
general results for convergence and stability
improve speedup and efficiency
use on large problems
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J.-L. Lions, Y. Maday, and G. Turinici, Resolution dEDP parun schema en temps parareel, C. R. Acad. Sci. Paris Ser. IMath., vol. 332, no. 7, pp. 661668, 2001.
G Bal and Y Maday A parareal time discretization for
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G. Bal and Y. Maday, A parareal time discretization for
nonlinear PDEs with application to the pricing of an americanput, in Recent Developments in Domain DecompositionMethods (L. Pavarino and A. Toselli, eds.), vol. 23 of LectureNotes in Computational Science and Engineering,pp. 189202, Berlin: Springer-Verlag, 2002.
L. Baffico, S. Bernard, Y. Maday, G. Turinici, and G. Zerah,Parallel-in-time molecular-dynamics simulations, Phys. Rev.E, vol. 66, p. 057701, Nov 2002.
M. Gander and S. Vandewalle, On the superlinear and linear
convergence of the parareal algorithm, in DomainDecomposition Methods in Science and Engineering XVI (T. J.Barth, M. Griebel, D. E. Keyes, R. M. Nieminen, D. Roose,T. Schlick, O. B. Widlund, and D. E. Keyes, eds.), vol. 55 of
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Lecture Notes in Computational Science and Engineering,pp. 291298, Springer Berlin Heidelberg, 2007.
G. Bal, On the convergence and the stability of the pararealalgorithm to solve partial differential equations, in Domain
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Decomposition Methods in Science and Engineering (T. J.
Barth, M. Griebel, D. E. Keyes, R. M. Nieminen, D. Roose,T. Schlick, T. J. Barth, M. Griebel, D. E. Keyes, R. M.Nieminen, D. Roose, T. Schlick, R. Kornhuber, R. Hoppe,J. Priaux, O. Pironneau, O. Widlund, and J. Xu, eds.), vol. 40of Lecture Notes in Computational Science and Engineering,pp. 425432, Springer Berlin Heidelberg, 2005.
P. F. Fischer, F. Hecht, and Y. Maday, A parareal in timesemi-implicit approximation of the Navier-Stokes equations,in Domain Decomposition Methods in Science and
Engineering, vol. 40 of Lecture Notes in ComputationalScience and Engineering, pp. 433440, Berlin: Springer, 2005.
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Y. Maday and G. Turinici, A parareal in time procedure forthe control of partial differential equations, Comptes RendusMathematique, vol. 335, no. 4, pp. 387392, 2002.
Y. Maday and G. Turinici, Parallel in time algorithms for
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Y. Maday and G. Turinici, Parallel in time algorithms for
quantum control: Parareal time discretization scheme, Int. J.Quantum Chem., vol. 93, no. 3, pp. 223228, 2003.
G. Bal, Parallelization in time of (stochastic) ordinarydifferential equations. Submitted, available athttp://www.columbia.edu/ gb2030/PAPERS/paralleltime.pdf
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