RATIO BASED PARALLEL TIME INTEGRATION (RAPTI) FOR INITIAL VALUE PROBLEMS

Transcription

1 RATIO BASED PARALLEL TIME INTEGRATION (RAPTI) FOR INITIAL VALUE PROBLEMS Nabil R. Nassif Noha Makhoul Karam Yeran Soukiassian Mathematics Department, American University of Beirut IRISA, Campus Beaulieu, Université de Rennes I, Rennes, France Computer Science Department, American University of Beirut, Beirut Lebanon RAPTI - Nassif Karam Soukiassian August 2006 p.1/28

2 Outline Introduction and review Rescaling Methodology RAPTI for diffusion reaction problems RAPTI for Two Species Logistic Lotka Volterra RAPTI for Three Species Logistic Lotka Volterra Numerical Results Conclusion and Future Work RAPTI - Nassif Karam Soukiassian August 2006 p.2/28

3 Introduction A course grid consists of time slices. On each slice we intend to integrate independently" an IVP, starting with its own initial value. This induces gaps between the end value of one slice and the starting value of the next slice. 2 A course grid consisting of four time slices Slices Gaps y(t) Fine Grid (within a slice) RAPTI - Nassif Karam Soukiassian August 2006 p.3/

4 Introduction: General steps of a predictor-corrector scheme for parallel time integration 1. Choice of a coarse grid that defines large time-slices. 2. Prediction of the initial values of the solution on each time-slice, using for example a Euler-implicit scheme. 3. Iteration until convergence of the following process: Solving independent evolution problems on each time-slice, using any method on a fine grid, leading to discontinuity jump between the end value of one slice and the predicted initial value on the next one. Correction of the slices initial values by propagation of the jumps, RAPTI - Nassif Karam Soukiassian August 2006 p.4/28

5 Introduction: Review of Previous Work The first implementation has been suggested by Nievergelt and led to multiple shooting methods (1964). Variants of this method were then developed by several authors, in particular by Chartier and Philippe (1993). Lions, Maday and Turinici proposed a new version the parareal algorithm in 2001, updated by Maday and Bal in 2002, Farhat-Chandesris in 2003 and Tromeur-Dervout-Guibert in Farhat et al in 2005 proposed a change to improve the performance of the parareal algorithm for second-order hyperbolic systems. RAPTI - Nassif Karam Soukiassian August 2006 p.5/28

6 Rescaling Methodology... Find y : [0, T b ) R, 0 < T b, such that: (1) 8 < : dy dt = f(y), 0 < t < T b y(0) = y 0 > 0. (2) (3) t = T n 1 + β n s, y(t) = y n 1 + α n z(s). 8 dz >< ds = g n(z) := β n f(y α n 1 + α n z(s)), 0 < s < s n n z(0) = 0. >: z(s n ) = S. (4) (5) y(t n ) = y n 1 + y n 1 S = y n 1 (1 + S) y n = (1 + S) n ; T n = T n 1 + β n s n. y n = y(t n) y n 1 y(t n 1 ) = 1 + S. RAPTI - Nassif Karam Soukiassian August 2006 p.6/28

7 Rescaling Methodology Two main advantages: 1. Extremely accurate numerical approach, since each IVP (3) can be solved up to any precision, with starting homogeneous condition z(0) = 0 and a fixed ending condition z(s n ) = S (met with an adaptive procedure) 2. The sequence of IVP (3) are defined independently and can be implemented either sequentally or in parallel. RAPTI - Nassif Karam Soukiassian August 2006 p.7/28

8 RAPTI for Diffusion-Reaction Problems... Consider the Cauchy problem where one seeks a vector function U : [0, ) R k, that verifies: (6) 8 < du dt = AU + F (U) : U(0) = U 0 R k (7) 8 u >< t u = aup, p 1, a > 0 for x Ω =] 1; 1[ and 0 < t < u(x, t) = 0 for x Ω and 0 < t < >: u(x, 0) = p(x) > 0 for x Ω. RAPTI - Nassif Karam Soukiassian August 2006 p.8/28

9 RAPTI for Diffusion-Reaction Problems... (8) 8 < U(t) = U n 1 + diag(α n )Z(s) : t = T n 1 + β n s α n R k 0 s s n, T n 1 t T n (9) 8 < β n = : α n = U n 1 1 U n 1 p 1 (10) 8 dz >< ds = G n(z) = F (U n 1 + diag(α n )Z(s)) Z(0) = 0 >: Z(s n ) = S 0 s < s n (S being the cut-off value) U n = U(T n ) = U n 1 +diag(u n 1 )Z(s n ) = diag(u n 1 )(e+z(s n )); T n = T n 1 +β n s n. (11) (12) r n = (e + Z(s n )). (13) U n = diag(r n )U n 1. RAPTI - Nassif Karam Soukiassian August 2006 p.9/28

10 RAPTI for Diffusion-Reaction Problems Time versus the solution U n for the diffusion reaction problem, where each solution is a vector consisting of 8 components The componenets of the solution U(T5,4) U(T5,3) U(T5,2) U(T8,4) U(T8,3) U(T8,2) 2 U(T5,1) U(T8,1) 1 T1 T2 T3 T4 T5 T6 T7 T8 T9 T Tn RAPTI - Nassif Karam Soukiassian August 2006 p.10/28

11 RAPTI for diffusion-reaction problem: Algorithm 1. Sequential run on n s slices, until transition vector r n stabilizes. 2. For following slices, n > n s, estimate ratios r p n, based on r e n (n n s) and get predicted initial values for each slice n > n s. 3. Execute parallel computations on each n th slice, n > n s, with starting value U p n 1 leading to an end value U c n. Compute the sequence G n of gaps at the end of each n th slice, as G n = U c n U p n. 4. Test convergence: n s < n n conv, G n ɛ g tol. 5. Update n s by n conv and repeat steps 3 to 5 until a maximum time T is reached. RAPTI - Nassif Karam Soukiassian August 2006 p.11/28

12 RAPTI for Two Species Logistic Lotka-Volterra... (14) dx = ax bxy dt ex2, dy dt = cy + dxy fy2, x(0) = x 0, y(0) = y 0. where a, b, c, d, e, f, x 0, y 0 are given positive parameters. The stable equilibrium point of the above system is given by: M E = (x E, y E ) = ( af+bc ef+bd, ad ec ef+bd ) RAPTI - Nassif Karam Soukiassian August 2006 p.12/28

13 RAPTI for Two Species Logistic Lotka-Volterra 3.5 Logistic Lotka Volterra with 2 Species M(T1)=(x1,y1) (xe,ye) (x2,y2) M0=(x0,y0) RAPTI - Nassif Karam Soukiassian August 2006 p.13/28

14 RAPTI for 2-Species Logistic Lotka-Volterra: Rescaling (15) 8 < du dt = AU + F (U) : U(0) = U 0 with t 0 = T n β n s, 0 U(t) = 1 U n diag(α n )Z(s) 1 (α n R 2 0 ) U x A, U 0 x 0 A, A a 0 A, F (x, y) y y 0 0 c bxy ex2 dxy fy 2 1 A RAPTI - Nassif Karam Soukiassian August 2006 p.14/28

15 RAPTI for 2 Species Logistic Lotka-Volterra: Rescaling By selecting α n = U n 1, β n = 1, then Z(s) verifies: (16) 8 >< dz = diag( 1 )A(U ds α n 1 + diag(α n )Z(s)) + diag( 1 )F (U n α n 1 + n diag(α n )Z(s)), 0 < s < s n Z(0) = 0 >: Arg( M E M(T n )) = Arg( M E M(T n 1 )) + 2π Stopping criterion is based on finding a period T such that: (17) Arg( M E M(nT )) = Arg( M E M(0)) + 2π n 1 RAPTI - Nassif Karam Soukiassian August 2006 p.15/28

16 RAPTI for 2-species Logistic Lotka-Volterra: Defining the ratios (18) r n = M E M(T n ), n = 1, 2, M E M(T n 1 ) Choice based on the fact that in the classic Lotka-Volterra model, the ratios r n = M E M(T n ) M E M(T 0 ) = 1 n 4 Classic Lotka Volterra with 2 Species M(Tn)=M0=(x0,y0) ME=(xE,yE) RAPTI - Nassif Karam Soukiassian August 2006 p.16/28

17 RAPTI for 2-Species Logistic Lotka-Volterra: Stopping criterion The trajectories of the Logistic Lotka-Volterra with 2-species in the xy-plane circulate counterclockwise and spiral inwards towards the equilbrium point (x E, y E ). The ratios stabilize r n = M E M(T n ) M E M(T n 1 ) the beginning of each slice. C for n n s allowing us to predict the initial values at 3.5 Logistic Lotka Volterra with 2 Species M(T1)=(x1,y1) (xe,ye) (x2,y2) M0=(x0,y0) RAPTI - Nassif Karam Soukiassian August 2006 p.17/28

18 RAPTI for 2-Species Logistic Lotka-Volterra: Algorithm 1. Sequential run on n s slices, until the ratios r n stabilize 2. Determine the total number of slices n t given by: 6 n t = n s + 4 log ɛ s tol 7 rns 5 log C 3. For following slices, n > n s, estimate ratios rn, p based on rn e (n n s) and get predicted initial values for each slice n > n s given by: Un p =M E + C n n s U ns M E sign(u ns M E ) using the following conditions: n M E M(T (n) has the same direction of M E M(T (0) ME M(T (n) p = M E M(T (n s ) (C) n n s n > n s 4. Execute parallel computations on each n th slice, n > n s, with starting value U p n 1 leading to an end value U c n. Compute the sequence G n of gaps at the end of each n th slice, as G n = U c n U p n. 5. Test convergence: n s < n n conv, G n ɛ g tol. 6. Update n s by n conv and repeat steps 2 to 6 until a maximum time T is reached. RAPTI - Nassif Karam Soukiassian August 2006 p.18/28

19 RAPTI for 3-Species Logistic Lotka-Volterra... (19) 8 >< >: dx dt dy dt dz dt = ax bxy hx2 = cy + dxy eyz iy2 = fx + gyz jz2 x(0) = x 0, y(0) = y 0, z(0) = z 0. where a, b, c, d, e, f, g, h, i, j, x 0, y 0, z 0 are given positive parameters. A study of the equilibrium points of the system lead to 8 of them, of which there exists a unique stable point given by: N E = (x E, y E, z E ) = ( bjc bfe+age+aij ghe+hij+dbj, jhc+jda+fhe ghe+hij+dbj, hif ghc+gda dbf ) ghe+hij+dbj RAPTI - Nassif Karam Soukiassian August 2006 p.19/28

20 RAPTI for 3-Species Logistic Lotka-Volterra Logistic Lotka Volterra with 3 Species First Slice Second Slice 5 4 Z axis Third Species (x0,y0,z0) (xe,ye,ze) (xe,ye) 3 Y axis Second Species X axis First Species The upper part represents the trajectories of the Logistic Lotka-Volterra with 3-species in the xyz-plane. The lower part represents the projection of the solution in the xy-plane which has a similar behaviour to that of the 2-species logistic Lotka-Volterra system. The ratios r n = M E M(T n ) M E M(T n 1 ) C, ρ n = N E N(T n ) N E N(T n 1 ) D for n n s RAPTI - Nassif Karam Soukiassian August 2006 p.20/28

21 RAPTI for 3-species Lotka-Volterra: Algorithm Sequential run on ns slices, until the ratios r n and ρ n stabilize Determine the total 0 number of slices nt given 1 by 6 n t = n s + 4 log ɛs tol 7 6 rns 5, 4 log ɛs tol 7 ρns 5A log C log D For following slices, n > ns, estimate ratios r p n and ρ p n, based on r e n and re n (n n s ) and get predicted initial values for each slice n > n s given by: U p n =M E + C n n s U ns M E sign(u ns M E ) using the following conditions: n M E M(T (n) has the same direction of M E M(T (0) ME M(T (n) p = M E M(T (n s ) (C) n n s n > n s NE N(T (n) p = N E N(T (n s ) (D) n n s n > n s lead to predict the starting value at every slice n, n > n s (Note that for n = n s + 1, U p n 1 = U e n 1 ) Execute parallel computations on each n th slice, n > n s, with starting value U p n 1 leading to an end value U c n. Compute the sequence G n of gaps at the end of each n th slice, as G n = U c n U p n. Test convergence: ns < n n conv, G n ɛ g tol. Update ns by n conv and repeat steps 2 to 6 until arapti maximum - Nassif time Karam T Soukiassian is reached. August 2006 p.21/28

22 Numerical Results List of tolerances used in the RaPTI algorithm. Tolerance ɛ r tol ɛ g tol ɛ l tol ɛ s tol Functionality Determine n s Determine number of iterations Determine stopping criteria within a slice Determine number of slices RAPTI - Nassif Karam Soukiassian August 2006 p.22/28

23 Numerical Results Diffusion Reaction-1 The results of running the diffusion reaction problem: (20) u 2 u = au p, p 1, a > 0, x ] 1; 1[, 0 < t < t x 2 u(x, t) = 0 for x Ω and 0 < t < u(x, 0) = p(x) > 0 for x Ω. p(x) = 2 x + 2 for x 0.5 p(x) = 2 3 x for x 0.5 p = 0.8, h = 1 8, S = 2, a = 3 RAPTI - Nassif Karam Soukiassian August 2006 p.23/28

24 Numerical Results Diffusion Reaction-2 n s ɛ r tol ɛ g tol ɛ l tol Number of Slices Iterations RAPTI - Nassif Karam Soukiassian August 2006 p.24/28

25 Numerical Results 2-species Logistic Lotka Volterra(1) a = 1, b = 1, c = 1, d = 1, e = 1 2, f = 1 5, x 0 = 10, y 0 = 5, τ = n s ɛ r tol ɛ g tol ɛ l tol ɛ s tol Nb of Slices Iter a = 1, b = 1, c = 1, d = 1, e = 1 2, f = 1 5, x 0 = 1, y 0 = 5, τ = n s ɛ r tol ɛ g tol ɛ l tol ɛ s tol Nb of Slices Iter RAPTI - Nassif Karam Soukiassian August 2006 p.25/28

26 Numerical Results 2-species Logistic Lotka Volterra(2) a = 2, b = 1, c = 1, d = 1, e = 1 2, f = 1 5, x 0 = 1, y 0 = 5, τ = n s ɛ r tol ɛ g tol ɛ l tol ɛ s tol Nb of Slices Iter a = 2, b = 1, c = 1, d = 2, e = 1 2, f = 1 5, x 0 = 1, y 0 = 5, τ = n s ɛ r tol ɛ g tol ɛ l tol ɛ s tol Nb of Slices Iter RAPTI - Nassif Karam Soukiassian August 2006 p.26/28

27 Numerical Results 3-species Logistic Lotka Volterra(1) a = 3, b = 1, c = 1, d = 1, e = 1, f = 1, g = 1, h = 1 2, i = 1 5, j = 1 2, x 0 = 1, y 0 = 1, z 0 = 2, τ = n s ɛ r tol ɛ g tol ɛ l tol ɛ s tol Nb of Slices Iter a = 3, b = 1, c = 1, d = 2, e = 1, f = 1, g = 1, h = 1 2, i = 1 5, j = 1 2, x 0 = 1, y 0 = 1, z 0 = 2, τ = n s ɛ r tol ɛ g tol ɛ l tol ɛ s tol Nb of Slices Iter RAPTI - Nassif Karam Soukiassian August 2006 p.27/28

28 Numerical Results 3-species Logistic Lotka Volterra(2) a = 3, b = 1, c = 1, d = 1, e = 1, f = 1, g = 1, h = 1 2, i = 1 5, j = 1 2, x 0 = 1, y 0 = 2, z 0 = 5, τ = n s ɛ r tol ɛ g tol ɛ l tol ɛ s tol Nb of Slices Iter RAPTI - Nassif Karam Soukiassian August 2006 p.28/28