parareal algorithm

8/8/2019 Parareal Algorithm

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Parareal Algorithm

Ma. Cristina Bargo

Laboratoire Jacques-Louis Lions

Universit Pierre et Marie Curie

University of the Philippines Diliman

CEMRACS 2009

August 7, 2009

MC Bargo (UPMC + UPD) Parareal Algorithm CEMRACS 2009 1 / 15
http://goforward/http://find/http://goback/


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Outline

1 History

2 The Algorithm

3 Some Properties

4 Work Done on Parareal Algorithm

5 Simple Implementation



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History



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History

Lions, Maday and Turinici [4] (2001) parallel in real time parareal



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History


Bal and Maday [2] (2002) equivalent to [4] for linear problems better results for nonlinear problems



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History


Bal and Maday [2] (2002) equivalent to [4] for linear problems better results for nonlinear problems

Baffico et al [1] (2002) most common formulation



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The Problem

Find u such that

(1)

tu+ A (t,u) = 0, t > t0

u = u0, t = t0

where A : R V V (V a Hilbert space) and t0 0.



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The Problem

Find u such that

(1)

tu+ A (t,u) = 0, t > t0

u = u0, t = t0


u(t + ) = E(t + ,t,v), where v = u(t) and > 0, with

> 0, > 0, E(t + + , t + , E(t + ,t,v)) = E(t + + ,t,v)



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The Problem

Find u such that

(1)

tu+ A (t,u) = 0, t > t0

u = u0, t = t0


u(t + ) = E(t + ,t,v), where v = u(t) and > 0, with

> 0, > 0, E(t + + , t + , E(t + ,t,v)) = E(t + + ,t,v)

Let t0 = T0 < T1 < < TN = T, and Tn = Tn Tn1. Then

n > 0, u(Tn) = E(Tn, T0,u0) = E(Tn, Tn1,u(Tn1))



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The Tools

Notations :



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The Tools

Notations :

Fine solver F(t2, t1,u1)



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The Tools

Notations :

Fine solver F(t2, t1,u1) approximation of the solution u(t2) to problem (1) with initial

condition u(t1) = u1

can be a classical discretization scheme with small timestep t



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The Tools

Notations :



can be a classical discretization scheme with small timestep tCoarse solver G(t2, t1,u1)



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The Tools

Notations :



can be a classical discretization scheme with small timestep tCoarse solver G(t2, t1,u1) another approximation to u(t2), less accurate than F(t2, t1,u1) but

cheaper to solve can be another discretization scheme with a larger timestep T

(T >> t)



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The Algorithm



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The Algorithm

Iteration 0 :



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The Algorithm

Iteration 0 : U

00 = u0 (the initial condition in problem (1))

U0n+1 = G(tn+1, tn,U

0n

) (the coarse solver)


h l h


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The Algorithm

Iteration 0 : U



0n


Iteration 1, 2,


Th Al h


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The Algorithm

Iteration 0 : U



0n


Iteration 1, 2, Parallel : F(tn+1, tn,Ukn) for n = 0, 1, 2, , n 1


Th Al i h


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The Algorithm

Iteration 0 : U



0n


Iteration 1, 2, Parallel : F(tn+1, tn,Ukn) for n = 0, 1, 2, , n 1 Serial : Uk0 = u0 and

(2) Uk+1n+1 = G(tn+1, tn,U

k+1n

) + F(tn+1, tn,Uk

n) G(tn+1, tn,U

k

n)


S P i


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Some Properties


S P ti


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Some Properties

For k , then Ukn Un where

Un+1 = F(tn+1, t0,U0) = F(tn+1, tn,Un)


S P ti


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Some Properties


Un+1 = F(tn+1, t0,U0) = F(tn+1, tn,Un)

n = 0, 1, 2, , N we can show that Unn = Un


Some Properties


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Some Properties


Un+1 = F(tn+1, t0,U0) = F(tn+1, tn,Un)


converges much faster (Maday, Rnquist and Staff, 2006 [5])


Some Properties


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Some Properties


Un+1 = F(tn+1, t0,U0) = F(tn+1, tn,Un)


converges much faster (Maday, Rnquist and Staff, 2006 [5])

stability results (Staff and Rnquist, 2005 [8])


Work Done on Parareal Algorithm


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pricing of an American put (Bal and Maday, 2002 [2])




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pricing of an American put (Bal and Maday, 2002 [2])molecular dynamics simulations (Baffico et al, 2002 [1]) - fine schemeuses the full model, coarse scheme is based on a simpler model of theoriginal




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control problems (Maday and Turinici, 2002 [6])




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control problems (Maday and Turinici, 2002 [6])combined with domain-decomposition methods (Maday and Turinici,2005 [7])




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Navier-Stokes (Fischer, Hecht and Maday, 2005 [3]) - using fine andcoarse mesh in space




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Navier-Stokes (Fischer, Hecht and Maday, 2005 [3]) - using fine andcoarse mesh in space

Goal : Use the algorithm to solve more complicated problems


Simple Implementation


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S p e p e e tat o




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p p

Sample Problem : Find u =

u1 u2T

so that

u

t= Au, t (0, 100]

u = u0 = 1 0 T

, t = 0

where A is the 2 2 matrix given by

A =

0.02 0.20.2 0.02




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p p

t0 = 0 and T = 100

N = 100 and Tn = T = 1

Coarse scheme : Implicit Euler with T = T = 1

Fine scheme : Implicit Euler with t = 0.1Horizontal axis is u1 and the vertical axis is u2

fine scheme : 0.62 s

parareal scheme : 0.29496 s




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p pIteration 0




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Iteration 1




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Iteration 2




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Iteration 3




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Iteration 4


L. Baffico, S. Bernard, Y. Maday, G. Turinici, and G. Zrah.Parallel in time molecular dynamics simulations


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Parallel-in-time molecular-dynamics simulations.Phys. Rev. E, 66(5) :057701, Nov 2002.

Guillaume Bal and Yvon Maday.

A parareal time discretization for nonlinear PDEs with application tothe pricing of an american put.In L.F. Pavarino and A. Toselli, editors, Recent Developments inDomain Decomposition Methods, volume 23 of Lecture Notes inComputational Science and Engineering

, pages 189202.Springer-Verlag, Berlin, 2002.

Paul F. Fischer, Frdric Hecht, and Yvon Maday.A parareal in time semi-implicit approximation of the Navier-Stokesequations.

In Domain Decomposition Methods in Science and Engineering,volume 40 of Lecture Notes in Computational Science and Engineering,pages 433440. Springer, Berlin, 2005.

Jacques-Louis Lions, Yvon Maday, and Gabriel Turinici.


Rsolution dEDP par un schma en temps pararel.C. R. Acad. Sci. Paris Sr. I Math., 332(7) :661668, 2001.


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Y. Maday, E.M. R nquist, and G.A. Staff.The parareal-in-time algorithm : basics, stability and more.

2006.Yvon Maday and Gabriel Turinici.A parareal in time procedure for the control of partial differentialequations.C. R. Math. Acad. Sci. Paris, 335(4) :387392, 2002.

Yvon Maday and Gabriel Turinici.The parareal in time iterative solver : a further direction to parallelimplementation.In Domain decomposition methods in science and engineering,

volume 40 of Lecture Notes in Computational Science and Engineering,pages 441448. Springer, Berlin, 2005.

Gunnar Andreas Staff and Einar M. Rnquist.Stability of the parareal algorithm.


In Domain Decomposition Methods in Science and Engineering,volume 40 of Lecture Notes in Computational Science and Engineering,


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pages 449456. Springer, Berlin, 2005.


parareal algorithm

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