Introduction to the Parareal Algorithm

8/8/2019 Introduction to the Parareal Algorithm

1/93

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

MC Bargo Intro to Parareal Algorithm
http://find/http://goback/


2/93

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.



3/93


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)



4/93


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



5/93

Classical DiscretizationExact Solution



6/93

Classical Discretizationt = T0



7/93




8/93




9/93




10/93




11/93




12/93




13/93




14/93




15/93




16/93



Cl i l Di i i


17/93



Cl i l Di i i


18/93



Cl i l Di i i


19/93



Cl i l Di ti ti


20/93



Cl i l Di ti ti


21/93





22/93


Remarks:

more accurate results using small time steps

computationally costly due to sequential nature of the scheme




23/93


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




24/93

Classical DiscretizationExact Solution




25/93





26/93

Classical Discretizationt = t




27/93

Classical Discretizationt = 2t




28/93





29/93





30/93





31/93





32/93

t = 7t




33/93

t = 8t




34/93

t = 9t




35/93

t = 10t




36/93

t = 11t




37/93

t = 12t




38/93

t = 13t




39/93

t = 14t




40/93

t = 15t = T1




41/93

t = T15




42/93

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


The Parareal Algorithm


43/93

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




44/93

Notations:

Fine solver F(Ti+1; Ti, Ui) - accurate but computationallycostly

Coarse solver G(Ti+1; Ti, Ui) - less accurate than F butcomputationally cheap




45/93

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)




46/93


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)




47/93


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


The Parareal AlgorithmThe Exact Solution


48/93

The Exact Solution


The Parareal AlgorithmIteration 0


49/93

Iteration 0


The Parareal AlgorithmIteration 1: t = Ti


50/93

Iteration 1: t = Ti


The Parareal AlgorithmIteration 1: t = Ti + t


51/93

Iteration 1: t Ti + t


The Parareal AlgorithmIteration 1: t = Ti + 2t


52/93

i +




53/93




54/93




55/93




56/93




57/93




58/93




59/93




60/93




61/93




62/93




63/93




64/93



65/93

The Parareal AlgorithmIteration 1: End of Parallel Part


66/93


The Parareal AlgorithmIteration 1: t = T1


67/93




68/93




69/93




70/93




71/93




72/93




73/93




74/93




75/93




76/93




77/93



78/93



79/93




80/93




81/93




82/93




83/93




84/93




85/93




86/93




87/93




88/93


The Parareal AlgorithmFine scheme vs. Parareal algorithm


89/93


ANR PITAC

Past Work: http://www.ljll.math.upmc.fr/PITAC/
http://find/http://goback/http://www.ljll.math.upmc.fr/PITAC/http://www.ljll.math.upmc.fr/PITAC/


90/93

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


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
http://www.ljll.math.upmc.fr/PITAC/http://www.ljll.math.upmc.fr/PITAC/http://find/http://goback/


91/93

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


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


92/93

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.


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


93/93

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


Documents

Introduction to the Parareal Algorithm