Time Parallel Time Integration

Transcription

1 Time Inegraion Universiy of Geneva SSSTC Symposium, Geneva, Augus 17

2 Solving Evoluion Problems in Parallel? The Lorenz equaions: d d dy d dz d = σ +σy = z +r y = y bz Simpler: du d = f(u), u( ) = u du Euler: d u( n+1) u( n) u n+1 = u n + f(u n ) u u

3 Solving Evoluion Problems in Parallel? The Lorenz equaions: d d dy d dz d = σ +σy = z +r y = y bz Simpler: du d = f(u), u( ) = u du Euler: d u( n+1) u( n) u 1 = u + f(u ) u u u

4 Solving Evoluion Problems in Parallel? The Lorenz equaions: d d dy d dz d = σ +σy = z +r y = y bz Simpler: du d = f(u), u( ) = u du Euler: d u( n+1) u( n) u u 2 = u 1 + f(u 1 ) u u 1 u

5 Solving Evoluion Problems in Parallel? The Lorenz equaions: d d dy d dz d = σ +σy = z +r y = y bz Simpler: du d = f(u), u( ) = u du Euler: d u( n+1) u( n) u u 3 = u 2 + f(u 2 ) u u 1 u 2 u

6 Solving Evoluion Problems in Parallel? The Lorenz equaions: d d dy d dz d = σ +σy = z +r y = y bz Simpler: du d = f(u), u( ) = u du Euler: d u( n+1) u( n) u u 4 = u 3 + f(u 3 ) u u 1 u 2 u 3 u

7 Solving Evoluion Problems in Parallel? The Lorenz equaions: d d dy d dz d = σ +σy = z +r y = y bz Simpler: du d = f(u), u( ) = u du Euler: d u( n+1) u( n) u u 5 = u 4 + f(u 4 ) u u 1 u 2 u 3 u 4 u

8 Solving Evoluion Problems in Parallel? The Lorenz equaions: d d dy d dz d = σ +σy = z +r y = y bz Simpler: du d = f(u), u( ) = u du Euler: d u( n+1) u( n) u u n+1 = u n + f(u n ) u u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u9 u1 u11 u

9 Solving Evoluion Problems in Parallel? The Lorenz equaions: d d dy d dz d = σ +σy = z +r y = y bz Simpler: du d = f(u), u( ) = u du Euler: d u( n+1) u( n) Top5 Supercompuer Lis - June 17: 1. Sunway TaihuLigh (China, cores) 2. Tianhe-2 Milky-Way-2 (China, 3 1 cores) 3. Piz Dain (Swizerland, cores) How can one use so many cores o solve such a problem faser? 3 4 5

10 Over he Course of Time small scale ITERATIVE Picard Lindeloef 1893/4 large scale small scale DIRECT 1964 Miranker Liniger 1967 Shampine Was 1969 Lelarasmee Ruehli Sangiovanni Vincenelli 1982 Hackbusch 1984 Aelson Verwer 1985 Lubich Osermann 1987 Gear 1988 Jackson Norse 1986 Womble 199 Bellen Zennaro 1989 Worley 1991 Charier Philippe 1993 Hairer Norse Wanner 1992 Horon Vandewalle 1995 Burrage 1995 Saha Sadel Tremaine 1996 Gander 1996 Gander Halpern Naaf 1999 Sheen Sloan Thomee 1999 Lions Maday Turinici 1 1 Emme Minion 1/12 Gander Kwok Mandal 13 Gander Neumueller 14 Maday Ronquis 8 Chrislieb Macdonald Ong 1 Gander Gueel 13 5 Years of ime parallel ime inegraion (G, 15)

11 Jörg : Parallel for Inegraing Ordinary Differenial Equaions. Comm. of he ACM, For he las years, one has ried o speed up numerical compuaion mainly by providing ever faser compuers. Today, as i appears ha one is geing closer o he maimal speed of elecronic componens, emphasis is pu on allowing operaions o be performed in parallel. In he near fuure, much of numerical analysis will have o be recas in a more parallel form.

12 Jörg : Parallel for Inegraing Ordinary Differenial Equaions. Comm. of he ACM, For he las years, one has ried o speed up numerical compuaion mainly by providing ever faser compuers. Today, as i appears ha one is geing closer o he maimal speed of elecronic componens, emphasis is pu on allowing operaions o be performed in parallel. In he near fuure, much of numerical analysis will have o be recas in a more parallel form.

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14 Muliple du d = f(u), u() = u, [,1] Spli [,1] ino [, 1 3 ], [1 3, 2 3 ], [2 3,1] and solve du du 1 du 2 = f(u ), = f(u 1 ), = f(u 2 ), d d d u () = U, u 1 ( 1 3 ) = U 1, u 2 ( 2 3 ) = U 2, ogeher wih he maching condiions U = u, U 1 = u ( 1 3,U ), U 2 = u 1 ( 2 3,U 1) Solving his sysem wih Newon gives he recurrence U k+1 n+1 = u n( n+1,u k n )+ u n U n ( n+1,u k n )(Uk+1 n U k n ). (see Philippe Charier and Bernard Philippe 1993)

15 The Algorihm Résoluion d EDP par un schéma en emps pararéel. Lions, Maday, Turinici, C. R. Acad. Sci. Paris, 1 Elle a pour principale moivaion les problèmes en emps réel, d où la erminologie proposée de pararéel. Algorihm for u = f(u): one needs 1. Coarse solver G( 2, 1,u 1 ) 2. Fine solver F( 2, 1,u 1 ) U n+1 = G( n+1, n,u n) U k+1 n+1 = F( n+1, n,u k n )+G( n+1, n,u k+1 n ) G( n+1, n,u k n )

16 The Algorihm Résoluion d EDP par un schéma en emps pararéel. Lions, Maday, Turinici, C. R. Acad. Sci. Paris, 1 Elle a pour principale moivaion les problèmes en emps réel, d où la erminologie proposée de pararéel. Algorihm for u = f(u): one needs 1. Coarse solver G( 2, 1,u 1 ) 2. Fine solver F( 2, 1,u 1 ) U n+1 = G( n+1, n,u n) U k+1 n+1 = F( n+1, n,u k n )+G( n+1, n,u k+1 n ) G( n+1, n,u k n ) G, Vandewalle 7: 1. is muliple shooing Un+1 k+1 = u n( n+1,un k )+ u n ( n+1,un k U )(Uk+1 n Un k ) n wih he Jacobian approimaed by differences on a coarser grid. 2. is also a ime muligrid mehod wih aggressive coarsening.

17 Precise Convergence Esimae for Theorem (G, Hairer 7) Le F( n+1, n,un k) denoe he eac soluion a n+1 and G( n+1, n,un k ) be a one sep mehod wih local runcaion error bounded by C 1 T p+1. If G( + T,,) G( + T,,y) (1+C 2 T) y, hen k ma 1 n N u(n) Uk n C1 Tk(p+1) (1+C 2 T) N 1 k (N j) ma k! 1 n N u(n) U n j=1 (C1T)k e C2(T (k+1) T) T pk ma k! 1 n N u(n) U n. G and Hairer: Nonlinear Convergence Analysis for he Algorihm, Domain Decomposiion in Science and Engineering XVII, Springer-Verlag, 7.

18 Resuls for he Lorenz Equaions Suggesed by Jean-Pierre Eckmann (4) ẋ = σ +σy ẏ = z +r y ż = y bz Parameers: σ = 1, r = 28 and b = 8 3 = chaoic regime. Iniial condiions: (,y,z)() = (,5, 5) Simulaion ime: [,T = 1] Discreizaion: Fourh order Runge Kua, T = T = T ,

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31 for Parabolic Problem Discreize u A 1 B 2 A 2 B 3 A B n A n = u +f in space and ime u 1 u 2 u 3. u n = f 1 f 2 f 3. f n Soluion imes in seconds for sequenial and mulilevel mehod on one processor (G, Neumüller 16) dof forward subsiuion mulilevel

32 3D Hea Equaion Parallelizaion Resuls Scaling resuls on he Vienna Scienific Cluser VSC-2 Weak Scaling Srong Scaling 1 1 cores T dof ier ime T dof ier ime (all simulaions performed by M. Neumüller)

33 Time parallelizaion is currenly a very acive area of research Muliple leading o he parareal algorihm and mulilevel mehods are jus one of many approaches Review is available a gander