PARAREAL TIME DISCRETIZATION FOR PARABOLIC CONTROL PROBLEM

Transcription

1 PARAREAL TIME DISCRETIZATION FOR PARABOLIC CONTROL PROBLEM Daoud S Daoud Dept. of Mathematics Eastern Mediterranean University Famagusta-North Cyprus via Mersin 10-Turkey. e mail: daoud.daoud@emu.edu.tr 1/37 M5: Space-Time Parallel Methods for PDEs

2 OUTLINE 1. The Control Model Problem 2. The Parareal Algorithm 3. The Parreal-Inverse Algorithm 4. The Stability Analysis of Parreal- Control Algorithm 5. Discussion and Conclusions 2/37 M5: Space-Time Parallel Methods for PDEs

3 THE MODEL PROBLEM u t = u xx + p(t)u + φ(x, t), 0 x 1, 0 tu(x, 0) = f(x), 0 x 1 u(0, t) = g 0 (t), 0 < x T u(1, t) = g 1 (t), 0 < x T subject to the over specification condition at a point x 0 in the spatial domain (1) u(x 0, t) = E(t), (2) where f, g 0, g 1, E and φ are known functions while the functions u and p are unknown, and 1 < p(t) < 0 for t [0, T ]. 3/37 M5: Space-Time Parallel Methods for PDEs

4 CONTROL PROBLEM RELATED REFERENCES 1. J.R. Cannon, Yanping Lin, S.Wang, Determination of source parameter in parabolic equations. Meccanica, vol. 27:85-94, J.R. Cannon, Yanping Lin, S.Wang, Numerical procedures for the determination of an unknown coefficient in semi -linear parabolic equations. Inverse problems, vol. 10: , Yanping Lin, An inverse problem for a class of quasilinear parabolic equation, SIAM J. Math. Anal. Vol. 22: , /37 M5: Space-Time Parallel Methods for PDEs

5 The Engineering Applications of The Control Problem 1- Heat Conduction process 2- Thermoealsiticity 3-Chemical Diffusion and Control Theory 5/37 M5: Space-Time Parallel Methods for PDEs

6 THE CONTROL PROBLEM SOLUTION ALGORITHM Algorithm 1: The Solution of the Control Problem Algorithm Step 1: Consider the backward Euler scheme for the solution of (1) given by u n i t where p (t n+1 ) is the predicted value of p(t) at time T n+1 u n+1 i = h 2 (u n+1 i 1 2un+1 i + u n+1 i+1 ) + p (t n+1 )u n+1 i + φ(x 0, t n+1 ) (3) Step 2: Update the value of the function p (t n+1 ) using (2) u(x 0, t) = E(t), such that p(t n+1 ) l+1 = E (t n+1 ) h 2 (u n+1(l) x 0 h for l = 0, 1,... 2un+1(l) x 0 E(t n+1 ) + u n+1(l) x 0 +h ) φ(x 0, t n+1 ) The value of the pair (u n+1, p n+1(l+1) ) will be corrected using Step 1 and Step 2, recursively. Step 3: if u n+1(l+1) u n+1(l) < ε and p n+1(l+1) p n+1(l) < ε, then stop otherwise proceed with step 1 and step 2, ε is a pre specified accuracy. (4) 6/37 M5: Space-Time Parallel Methods for PDEs

7 RELATED REFERENCES TO THE PARAEAL ALGORITHM 1- J.-L. Lions, Y. Maday, and G. Turinici, Resolution d EDP par un scheme en temps parareal, C.R. Acad. Sci. Paris Ser. I math., 332: , G. Bal and Y. Maday, A parareal time discretization for non linear PDEs with application to the pricing of an american put, Proceeding of the Workshop on Domain Decomposition, Zurich, Switzerland, LNCSE series, Springer Verlag, G. Bal, Parallelization in time of(stochastic) ordinary differential equations, Math. Meth. Anal.Num (to appaer) C. Farhat and M. Chandesris, time decomposed parallel time-integrators;theory and feasibility studies for fluid, structure,and fluid-structire applications, Int.J.Numer.Meth. Engng. 58: , Y. Maday and G. Turininc, Parallel in time algorithms for quantum control: Parareal time discretization scheme, Int. J. Quant. Chem. 93: , G.A. Staff, Convergence and stability of the parareal algorithm: A numerical and theoretical investigation, preprint Numerics no. 2/2003, Norwegian University of Science and Technology, Norway. 7- G.A. Staff and E.M. Ronquist, Stability of the parareal algorithm, 7/37 M5: Space-Time Parallel Methods for PDEs

8 PARAREAL ALGORITHM Algorithm 2: Parareal Algorithm Step 1: For k = 0, and over Ω [t n, t n + n t] solve for u n+1 0 using coarse propagator u n+1 0 = G t (u n 0) n = 0, 1,..., N 1, whereu 0 0 = f(x). Step 2: Over each time interval [t n, t n + sδt], solve for v n+1 0, using fine propagator v n+1 0 = F sδt (v n 0 ) = i=s i=1 F iδt (v n 0 ) where v n 0 = u n 0 Step 3: Evaluate the jumps S n 0 = F sδt (u n 0) G t (u n 0), then u n+1 1 = G t (u n 0) + S n 0 8/37 M5: Space-Time Parallel Methods for PDEs

9 Step 4: Correction step; For k 1, and n = 1,..., N 1 u n+1 k = G t (u n k) + F sδt (u n k 1) G t (u n k 1) Step 5: If all jumps are not sufficiently small, the process can be iterated as follows u n+1 k = G t (u n k) + F sδt (u n k 1) G t (u n k 1) u 0 k = u 0 for all k 9/37 M5: Space-Time Parallel Methods for PDEs

10 The Solution of The Control Problem by Parareal Algorithm Algorithm 3: Parareal- Control Problem Algorithm Step 1: Over the domain Ω [t n, t n+1 ] and for k = 1, consider the coarse propagator i.e. u n+1 1 u n 1 t = (u xx ) n p(t n )u n+1 n = 0, 1,... N 1 The solution u n+1 1 denoted by G t (u n 1) p(t n+1 ) correction: Consider the correction of p(t) by the following relation p(t n+1 ) = E (t n+1 ) (u xx ) 1 (x0,t n+1 ) φ(x 0, t n+1 ) E(t n+1 ) Step 2: For k + 1 > 1 and over the domain Ω [t n, t n+1 ] consider the coarse propagator i.e. u n+1 k+1 un k+1 t = (u xx ) n+1 k+1 + p(t n)u n+1 k+1 n = 0, 1,... N 1 10/37 M5: Space-Time Parallel Methods for PDEs

11 p(t n+1 ) correction: Consider the correction of p(t) by the following relation p(t n+1 ) = E (t n+1 ) (u xx ) k+1 (x0,t n+1 ) f(x 0, t n+1 ) E(t n+1 ) The solution u n+1 k+1 is denoted by G t(u n k+1 ) Step 3: consider the fine propagator solution over Ω [t n, t n+l ], l = 1,..., s 1 Solve for u n+l k The solution u n+s k = u n+1 then for for l = 1,... s 1 where s = t δt The solution u n+1 k+1 u n+l 1 k δt = (u xx ) n+l 1 k k is denoted by F sδt (u n k ) + p(t n+l 1 )u n+l 1 k p(t n+l ) = E (t n+l ) u xx,k (x0,t n+l ) φ(x 0, t n+l ), E(t n+l ) is given by u n+1 k+1 = G t(u n k+1) + F sδt (u n k) G t (u n k) (5) 11/37 M5: Space-Time Parallel Methods for PDEs

12 Stability of the Parareal-Control Algorithm Let u(x, t) be the solution of such that u t = u xx + p(t)u(t), (6) u(0, t) = 0, u(1, t) = 0 and u(x, 0) = u 0. (7) The spatial derivative operator is approximated by the second order central difference approximation given by (u xx ) (xi,t) h 2 [u(x i+1, t) 2u(x i, t) + u(x i 1, t)] + O(h 2 ). (8) 12/37 M5: Space-Time Parallel Methods for PDEs

13 Consider the Fourier transform of the model problem, and over the Fourier domain the problem corresponding to (6) is given by û t = Q(ξ, t)û(ξ, t) (9) where Q(ξ, t) = q(ξ) + p(t), (p(t) = p(t)) such that Q(u) = q(ξ) = 2h 2 sin 2 (ξ/2), and h is the space mesh spacing. Then Q(ξ)û(ξ), Q(ξ, t) = q(ξ) + p(t) = 2h 2 sin 2 ( ξ ) + p(t). (10) 2 13/37 M5: Space-Time Parallel Methods for PDEs

14 For the numerical solution of the model problem we defined the coarse time interval t such that t n+1 = t n + t, and the fine solution time step size is give by δt, such that t δt = s, t = T N Over the time interval [t n, t n + t], and corresponding to (9), and (10) the amplification factor (Stability Function) of the considered schemes over the Fourier domain are given by, û n+1 (ξ, t n+1 ) = (1 Q(ξ, t n ) t) 1 û n (ξ, t n ) ( 1 = 1 + (2h 2 sin 2 ( ξ 2 n)) t) ) p(t ûn (ξ, t n ) (11) which is unconditional stable for p(t) < 0,. 14/37 M5: Space-Time Parallel Methods for PDEs

15 The amplification factor of the explicit scheme, over the time interval [t n, t n + sδt] is given by û n+1 (ξ, t n ) = s i=1 (1 + Q(ξ, t n+i 1)δt)û n (ξ, t n ) û n (ξ, t n ) = (12) s i=1 (1 + ( 2h 2 sin 2 ( ξ 2 ) + p(t n+i 1))δt)û n (ξ, t n ) and its conditional stable scheme according to the explicit scheme condition and for any p(t). 15/37 M5: Space-Time Parallel Methods for PDEs

16 û n k 1 û k n 1 û k 1 n 1 2 û k n 2 û k 1 n 2 û k 2 n 2 3 û k n 3 û k 1 n 3 û k 2 n 3 û k 3 n 3 4 û k n 4 û k 1 n 4 û k 2 n 4 û k 3 n 4 û k 4 n 4 The Parareal Iterative Scheme 16/37 M5: Space-Time Parallel Methods for PDEs

17 û n+1 k+1 = G t(û n k+1) + F sδt (û n k) G t (û n k) (13) Subtitling (11) and (12) in (13) then the solution û n+1 k+1 parareal algorithm, algorithm, is given by of the model problem by the û n+1 k+1 = (1 Q(ξ, t n) t) 1 û n k+1 + s (1 + Q(ξ, t n+i 1 )δt)û n k (1 Q(ξ, t n ) t) 1 û n k i=1 (14) At this point we will present the stability studies for the following cases case 1- δt = t (s = 1) for sequential algorithm case 2- δt = t s (s > 1) for Parareal algorithm 17/37 M5: Space-Time Parallel Methods for PDEs

18 Case I, δt = t For the first case and over the time interval [t n, t n + t] with time step δt = t, we obtain the following relation û n+1 k+1 = (1 Q(ξ, t n) t) 1 û n k+1 + (1 + Q(ξ, t n) t)û n k (1 Q(ξ, t n) t) 1 û n k Let (15) F (ξ, t n ) = (1 + Q(ξ, t n ) t) 1 (1 Q(ξ, t n ) t) 1 (16) Then (15) is re written as follows û n+1 k+1 = (1 + Q(ξ, t n) t)f (ξ, t n ) }{{} ûn k+1 + (1 + Q(ξ, t n ) t)(1 F (ξ, t n )) }{{} ûn k (17) where û 0 k+1 = û0, 18/37 M5: Space-Time Parallel Methods for PDEs

19 For the stability of the scheme given by (17) we will consider the approach of analysis by G. Staff Parareal algorithm-a survey of present work, 2003 Then the iterated solution û n+1 k+1 ) û n+1 k+1 = k i=0 ( n i is given by ((1 + Q(ξ, t n ) t)(1 F (ξ, t n ))) i ((1 + Q(ξ, t n ) t)f (ξ, t n )) n i û 0 and then the amplification factor (Stability Function) of the parreal algorithm (15), is given by T (ξ, t n ) = k i=0 ( n i ) ((1 + Q(ξ, t n ) t)(1 F (ξ, t n ))) i (1 + Q(ξ, t n ) t)f (ξ, t n )) n i (18) 19/37 M5: Space-Time Parallel Methods for PDEs

20 The algorithm is stable whenever Consider, sup 1 n N sup 1 k N T (ξ, t n ) 1 (19) T (ξ, t n ) < k i=0 ( n i ) ((1 + Q(ξ, t n ) t)(1 F (ξ, t n ))) i ((1 + Q(ξ, t n ) t)f (ξ, t n )) n i < n ( (1 + Q(ξ, t n ) t)(1 F (ξ, t n )) + (1 + Q(ξ, t n ) t)f (ξ, t n ) ) n i=0 T (ξ, t n ) < ( ((1 + Q(ξ, t n ) t)(1 F (ξ, t n ))) + ((1 + Q(ξ, t n ) t)f (ξ, t n )) ) n 20/37 M5: Space-Time Parallel Methods for PDEs

21 and hence for stability conditions is to show that the stability function, ρ(ξ, t n ), satisfying ρ(ξ, t n ) = (1 + Q(ξ, t n ) t)(1 F (ξ, t n )) (1 + Q(ξ, t n ) t)f (ξ, t n ) < 1 (20) and to investigate the necessary conditions for r and p(t) as well. Consider ρ(ξ, t n ) = (1 Q(ξ, t n ) t) 1 ( 1 + (Q(ξ, t n ) t) 2) < 1 Since Q(ξ, t n ) = 2h 2 sin 2 (ξ) + p(t n ), then 1+ ( 2h 2 sin 2 (ξ/2) + p(t n )) t ) 2 = ( 1 + 4r 2 sin 4 (ξ/2) 4r sin 2 (ξ/2)p(t n ) t + p 2 (t n ) t Therefore (1 + 4r 2 sin 4 (ξ/2) 4r sin 2 (ξ/2)p(t n ) t + p 2 (t n ) t 2) < 1 + 2r sin 2 (ξ/2) ( 2r sin 2 (ξ/2) 2p(t n ) t ) p(t n ) t for 1 < p(t n ) < 0 and for all n = 0, 1,..., N, r = t/h 2 21/37 M5: Space-Time Parallel Methods for PDEs

22 Then for r < 1/4 concluding that 1 + 2r sin 2 (ξ/2) ( 2r sin 2 (ξ/2) 2p(t n ) t ) p(t n ) t < 1 + 2r sin 2 (ξ/2) (2r 2p(t n ) t) p(t n ) t < 1 + 2r sin 2 (ξ/2) (1/2 2p(t n ) t) p(t n ) t < 1 + 2r sin 2 (ξ/2) (1/2 + 2 t) p(t n ) t < since r = t < 1/4, then h r sin 2 (ξ/2) (1/2 + 1/2) p(t n ) t = (1 Q(ξ, t n ) t) Therefore, for r = t h 2 < 1/4 and 1 < p(t) < 0 concluding that ρ(ξ, t n ) = (1 + Q(ξ, t n ) t)(1 F (ξ, t n )) (1 + Q(ξ, t n ) t)f (ξ, t n ) < 1 and then T (ξ, t n ) < 1 for all n 22/37 M5: Space-Time Parallel Methods for PDEs

23 The above analysis and discussions present the proof of the main result given by the following theorem. Theorem 1: For the given control model problem (1) solved by the parareal algorithm (Algorithm 3) given by, u n+1 k+1 = G t(û n k+1) + F δt (û n k) G t (û n k) (21) where G t and F δt are the coarse and fine solvers, respectively, and for s = 1 i.e. t = δt. Then ρ(ξ, t n ) < 1 for all δt = t < 1 h 2 h 2 4 and 1 < p(t) < 0, where ρ(ξ, t n ) is the stability function induced by (15), given by (20). 23/37 M5: Space-Time Parallel Methods for PDEs

24 Case 2: s = t/δt > 1 û n+1 k+1 = (1 Q(ξ, t n) t) 1 û n k+1 + s (1 + Q(ξ, t n+i 1 )δt)û n k (1 Q(ξ, t n ) t) 1 û n k In this case the stability function of the Parareal Scheme is given by i=1 ρ(ξ, t n ) = 2(1 Q(ξ, t n ) t) 1 (22) s (1 + Q(ξ, t n )δt (23) To investigate the necessary condition to satisfy ρ(ξ, t n ) < 1, Consider i=1 24/37 M5: Space-Time Parallel Methods for PDEs

25 For the first term (1 Q(ξ, t n ) t) 1 }{{} [ 2 (1 Q(ξ, t n ) t) i=1(1 + Q(ξ, t n+i 1 )δt) }{{} (1 Q(ξ, t n ) t) 1 < 1 for any r c = t/h 2 and for 1 < p(t) < 0, which in fact represent the stability of the coarse amplification factor ( Stability function). For the second term [ ] s 2 (1 Q(ξ, t n ) t) (1 + Q(ξ, t n+i 1 )δt) (25) i=1 ] (24) 25/37 M5: Space-Time Parallel Methods for PDEs

26 perform the multiplication in (25) concluding that s (1 + Q(ξ, t n+i 1 )δt) 1 + δt i=1 s Q(ξ, t n+i 1 ) + O(δt 2 ) (26) where Q(ξ, t n+i 1 ) = 2h 2 sin 2 ( ξ 2 ) + p(t n+i 1), and substitute (26) in (25) leading to { 2 (1 Q(ξ, tn ) t) [ 1 + δt s i=1 Q(ξ, t n+i 1) + O(δt 2 ) ]} = Q(ξ, t n ) t δt(1 Q(ξ, t n ) t) s i=1 Q(ξ, t n+i 1) 1 + Q(ξ, t n ) t s i=1 Q(ξ, t n+i 1)δt + O(δt 2 ) where Q(ξ, t n+i 1 ) = 2h 2 sin 2 (ξ/2) + p(t n+i 1 ), therefore i=1 26/37 M5: Space-Time Parallel Methods for PDEs

27 1 + Q(ξ, t n ) t δt s ( 2h 2 sin 2 (ξ/2) + p(t n+i 1 )) = i=1 1 + ( 2h 2 sin 2 (ξ/2) + p(t n ) ) t δt where r c = t Therefore 1 2r c sin 2 (ξ/2) + tp(t n ) + h 2, r f = δt h 2, and t δt = s. 1 2r c sin 2 (ξ/2) + tp(t n ) + }{{} <0 s ( 2h 2 sin 2 (ξ/2) + p(t n+i 1 ) ) i=1 s ( 2rf sin 2 (ξ/2) δtp(t n+i 1 ) ) i=1 s ( 2rf sin 2 (ξ/2) + δt ) < 1 + t i=1 27/37 M5: Space-Time Parallel Methods for PDEs

28 then [ ρ(ξ, t n ) = (1 Q(ξ, t n ) t) 1 2 (1 Q(ξ, t }{{} n ) t) ] + Q(ξ, t n+i 1 )δt) }{{ i=1(1 } for 1 < p(t n ) < 0 for all t n [0, T ]. < 1 28/37 M5: Space-Time Parallel Methods for PDEs

29 For the case when s = t/δt > 1 we will have the following theorem Theorem 2: For the given control model problem (1) solved by the parareal algorithm (Algorithm 3) given by, u n+1 k+1 = G t(û n k+1) + F sδt (û n k) G t (û n k) (27) where G t and F sδt are the coarse and fine solvers, respectively, and for s = t/δt > 1. Then ρ(ξ, t n ) < 1 for all r c = t/h 2 and r f = δt/h 2 satisfying the coarse and fine propagator stability constrained, respectively, and 1 < p(t) < 0, where ρ(ξ, t n ) is the stability function induced by (15), given by (23). 29/37 M5: Space-Time Parallel Methods for PDEs

30 Model Problem u t = u xx + p(t)u + φ(x, t) defined over the domain x Ω = (0, 1) and t [0, 1]. With exact solution given by u(x, t) = e t2 (cos(πx) + sin(πx)), p(t) is given by p(t) = 1 + t 2 and φ(x, t) = e t2 (π t 2 2t)(cos(πx) + sin(πx)). Subject to u(0, t) = e t2 u(1, t) = e t2 u(x, 0) = cos(πx) + sin(πx) 30/37 M5: Space-Time Parallel Methods for PDEs

31 For comparison purpose the above model problem has been solved using 1- Control Problem Algorithm (Algorithm 1) 2- Parareal-Control Problem Algorithm (Algorithm 2) 31/37 M5: Space-Time Parallel Methods for PDEs

32 Comments and Discussions 1. Due to the iterative process by the parareal algorithm the solution of the control problem does not require updating of the value of p(t), over each single coarse time step, as required by algorithm Due to inducing of the fine time step, in the parareal algorithm, the numerical results shows how accurate result we will achieve for u(x, t) and p(t) by algorithm 2 over the use of algorithm 1 (for u(x, t) see tables 1-4, and for p(t) see tables 5-8). 3. There is flexibility in this algorithm that we may use different spatial mesh spacing in the coarse solvers, due to its unconditional stable criteria. 4. From the sketch of the stability region by the polar graphs, as shown in figs 1-2, It shows how the stability region, amplification factor, effected for different values of (δt/h 2 ), and p(t) values as well. 5. The considered specific Case 1 is designated for the sequential type of algorithm to solve the Control problem, and by using Case 2 it shows that by Parareal algorithm the constrained on the ratio δt/h 2 is not required. 32/37 M5: Space-Time Parallel Methods for PDEs

33 Coarse r c = 0.2, Fine r f = 0.2 Coarse r c = 0.2, Fine r f = r f =0.2 r c = r f =0.2 r c =0.2 p(t)= /37 M5: Space-Time Parallel Methods for PDEs

34 Coarse r c = 0.25, Fine r f = 0.25 Coarse r c = 0.4, Fine r f = r f =0.25 r c = r f =0.4 r c = Coarse r c = 0.5, Fine r f = r f =0.5 r c = /37 M5: Space-Time Parallel Methods for PDEs

35 r c = 1.25 r c = 1 r c = 0.8 r c = 0.5 r c = 0.25 r c = 0.2 u(x, t) e e e e e e-4 p(t) e e-2 Table 1: The value of u(x, t) and p(t) using 5 corrections only for algorithm 1 k r c = 1.25 r c = 1 r c = 0.5 r c = 0.25 r f = 0.25 r f = 0.25 r f = 0.25 r f = e e e e e e e e e e e e e e e e e e e e-4 Table 2: The error for different s ratios, for r f = 0.25 k r c = 1.25 r c = 1 r c = 0.5 r c = r f = r f = r f = r f = e e e e e e e e e e e e e e e e e e e e-5 Table 3: The error for different s ratios, for r f = k r c = 1 r c = 0.8 r c = 0.4 r c = 0.2 r f = 0.2 r f = 0.2 r f = 0.2 r f = e e e e e e e e e e e e e e e e e e e e-5 Table 4: The error for different s ratios, for r f = /37 M5: Space-Time Parallel Methods for PDEs

36 r c = 1.25 r c = 1 r c = 0.8 r c = 0.5 r c = 0.25 r c = 0.2 u(x, t) e e e e e e-4 p(t) e e-2 Table 5: The value of u(x, t) and p(t) using 5 corrections only for algorithm 1 k r c = 1.25 r c = 1 r c = 0.5 r c = 0.25 r f = 0.25 r f = 0.25 r f = 0.25 r f = e e e e e e e-2 Table 6: The error of p(t) for different s ratios, for r f = 0.25 k r c = 1 r c = 0.8 r c = 0.4 r c = 0.2 r f = 0.2 r f = 0.2 r f = 0.2 r f = e e e e e e e-2 Table 7: The error of p(t) for different s ratios, for r f = 0.2 k r c = 1.25 r c = 1 r c = 0.5 r c = r f = r f = r f = r f = e e e e e e e e e-2 Table 8: The error p(t)for different s ratios, for r f = /37 M5: Space-Time Parallel Methods for PDEs

37 k r c = 1 r c = 1 r c = 1 r c = 0.5 r c = 0.5 r f = r f = 0.2 r f = 0.25 r f = 0.25 r f = e e e e e e e e e e e e e e e e e e e e e e e e e-4 Table 9: The solution for different ratios of Coarse and Fine solvers 37/37 M5: Space-Time Parallel Methods for PDEs