Accelerated micro/macro Monte Carlo simulation of dilute polymer solutions

Transcription

1 Acceleraed micro/macro Mone Carlo simulaion of dilue polymer soluions Giovanni Samaey Scienific Compuing, Dep. of Compuer Science, K.U. Leuven Based on join work wih K. Debraban (U. Odense), T. Lelievre (ENPC, Paris), V. Lega (UCLouvain)

2 Micro-macro simulaion of dilue polymer soluions Macroscopic par : Navier-Sokes equaions for + u ru Coupling : non-newonian sress ensor (Kramers formula) p = hx F (X)i Id We =(1 ) u rp +div( p ) div(u) =0 Microscopic par : Sochasic differenial equaion (SDE) for he configuraion of an individual polymer dx = apple apple() X Laso, Öinger, J. Non-Newonian Fluid Mech. 47 (1993) F (X) d + p dw, 2We We 2

3 P (y, (X, ) ) Example 1 : Linear springs Sress ensor p hxf(x)i / X 2 Sign of X is irrelevan (lengh of spring), so Disribuion of X evolves owards a Gaussian Closed model for evoluion of he variance dx = apple apple() X 1 1 F (X) d + p dw, F(X) =X 2We We () h(y µ) 2 i µ = hxi!0 d d = 2 apple() =h(x We We µ) 2 i X

4 Example 2 : FENE springs Finiely exensible nonlinearly elasic (FENE) apple 1 1 dx = apple() X F (X) d + p dw, F(X) = 2We We Disribuion becomes non-gaussian (wih sharp peak) X 1 X 2 /b 0.4 Impossible o represen exacly wih a finie number of momens U [L] =(U l ) L l=1 U l = X 2l P ( X,) '( X ) =0.5 =1 = 0.50 = 1.00 Mone Carlo simulaion required (especially in higher dimensions!) X 4

5 Connecing he levels of descripion Microscopic level known model simulaion code available Macroscopic level only sae variables unknown evoluion equaions Simulaie Lifing Resricie * * + Δ Coarse ime-sepper is a wrapper around a microscopic simulaion Generic building block for compuaional muliscale algorihms 5

6 Coarse ime-sepper based bifurcaion analysis U() U( + ) = (U()) Time-sepper is a black box Direcly compue macroscopic seady saes and heir sabiliy Marix-vecor producs (Ū) U (U )=0 Use (marix-free) ieraive mehods (RPM, Newon-Krylov) -> equaion-free Ū Ū + v D (Ū + v) (Ū) v Kevrekidis e al., / Kevrekidis & S, Annual Review on Physical Chemisry 60: ,

7 Acceleraion of macroscopic simulaion Exploi a separaion in spaial and emporal scales Coarse projecive inegraion Exrapolae macroscopic sae forward in ime Pach dynamics Inerpolae beween microscopic simulaion in small subdomains x Gear, Kevrekidis, SISC. 24: , 2004 / Lafie, S, SISC, 2010, submied. S, Roose, Kevrekidis, SIAM MMS 4: , 2005 / S, Kevrekidis, Roose, JCP 213(1): ,

8 The heerogeneous muliscale mehods An alernaive formulaion Posulae a general form for he unknown macroscopic equaion Supplemen his equaion wih an esimaion of missing macroscopic quaniies from a microscopic simulaion - Iniializaion of he microscopic model from a given macroscopic sae - Esimaion of a macroscopic quaniy from microscopic daa This formulaion has advanages from a numerical analysis viewpoin E, Engquis, Vanden-Eijnden, e al.,

9 Quesions from a numerical analysis viewpoin During lifing, missing microscopic informaion is filled in based on he macroscopic sae. Wha are appropriae macroscopic sae variables? How accurae is he reconsrucion? Wha is he influence of lifing errors on macroscopic evoluion? During resricion, a macroscopic Simulaie sae is esimaed based on he microscopic sae. How big is he variance of he noise during resricion? Lifing Resricie How can his variance be reduced? How is variance affeced when exrapolaing in ime? * * + Δ How does exrapolaion affec sabiliy? Rousse, S, M3AS, 2011, submied. Frederix, S, Roose, ESIAM: M2AN, Gear, Kaper, Kevrekidis, Zagaris. SIAM J. Appl. Dyn. Sys. 4: , Debraban, S, ESIAM: M2AN, 2011, submied. Frederix, S, Vandekerckhove, Roose, Li, Nies. Discree Con Dyn-B 11: , Ghysels, S, Van Liedekerke, Tijskens, Ramon, Roose, In. J. Muliscale Comp. Engng. 8(4): , S, Lelievre, Lega, Compuers and Fluids,

10 Coarse ime-sepper for Mone Carlo simulaion Microscopic level dx = apple apple() X Simulaie 1 1 F (X ) d + p dw 2We We X k+1 = s X (X k,apple(), ) from momens o an ensemble Lifing Resricie from an ensemble o momens R : X 7! U L : U 7! X = {X j } J j=1 U l = 1 J P J j=1 f l(x j ) * * + Δ Macroscopic level U [L] =(U l ) L l=1 U l = X 2l du d = H(U,apple()) p = T (U) 10

11 Lifing operaor : consrained simulaion Simulae wih consrained macroscopic sae unil condiional equilibrium X,m+1 = s X (X,m,apple, )+ r X R(X,m+1 ), me 2 R L zodanig da R(X,m+1 )=U - Time inegraion, followed by projecion ono manifold defined by imposed macroscopic sae X,m+1 = arg min X,m+1 s X (X,m,, ) wih consrain R(X,m+1 )=U The resul of he lifing is hen given as (for M sufficienly large) X = L(U ):=X,M Consisen iniial condiion also by projecion of a nearby ensemble S, Lelievre, Lega, Compuers and Fluids 43: ,

12 Lifing induces a closure approximaion Experimen - Coarse ime-sepper wih very small ime sep - Macroscopic sae variables : U [L] =(U l ) L l=1, - (Much more expensive han full microscopic simulaion) U l = hx 2l i Lifing inroduces modeling error ha decreases for an increasing number of momens U 1 () p () L = 1 L = 2 L = 3 L = 4 FENE M1 20 p

13 Exrapolaion via coarse projecive inegraion Sar wih a given macroscopic sae U = U N Lif o he corresponding microscopic sae L : U = U N 7! X N = X N,M K K microscopic Simulae he ensemble over seps X N,k+1 = s X (X N,k,apple( N,k ), ), k =0,...K 1 Resric o macroscopic sae Exrapolae macroscopic sae U N,K = R(X N,K ) U N+1 = U N,K +( K ) UN,K U N K 13

14 Efficiency and accuracy of coarse projecive inegraion Coarse projecive inegraion is efficien if Number of consrained seps during lifing - The bigger he ime scale separaion ( ), he smaller M can be We! 0 (M + K) We! 0 Number of seps o esimae ime derivaive - Bu: in he limi when, he macroscopic model is known! - Real acceleraion is only possible for an inermediary regime During exrapolaion, esimaion noise is amplified wih a facor /K - An alernaive would be o use less paricles and no exrapolaion - For equal saisical error, coarse projecive inegraion requires as much compuaions as a full microscopic simulaion (assuming M=0!) 14

15 An alernaive exrapolaion sraegy Mulisep sae exrapolaion Projecive inegraion U N+1 = U N,K +( K ) UN,K U N K Mulisep sae exrapolaion - Exrapolae using he las poin of each sequence of microscopic simulaion U N+1 = U N,K +( K ) UN,K U N 1,K - Saisical error is unaffeced - Sysemaic error does ge amplified wih a facor /K - Bu we wan o exrapolae jus because we can olerae a larger sysemaic error! Sommeijer, Compu. Mah. Appl. 19 (6) (1990)

16 Maching: an alernaive for lifing Classical lifing : = N,K X N+1 X N+1 = P(U = L(U N+1 N+1 ; X N,K ) ) = N+1 - mach an ensemble on wih an exrapolaed macroscopic sae on - simulae wih macroscopic consrain unil condiional equilibrium (M seps) Alernaive : perform maching wihou consrained simulaion K - The ime gained during exrapolaion is no los during consrained simulaion - The projeced ensemble now also Debraban, S, ESIAM M2AN, 2010, submied. depends on he ensemble a he previous ime sep! 16

17 Accuracy of maching operaor rel. error in p Error Experimen - Macroscopic sae variables : U [L] =(U l ) L l=1, U l = hx 2l i - Simulae unil ime * X ( ) - Projec ono manifold defined by and compare wih Projecion inroduces a modeling error ha decreases wih - increasing number of momens - decreasing exrapolaion ime sep Error C L 10 2 L = 3 L = L = 5 O( ) 10 4 U [L] ( ) X ( ) 2-sample K-S es L p-value ,00E , ,25 7 0,84 17

18 Numerical illusraion Experimen - macroscopic sae variables - srongly ime dependen velociy gradien - adapive macroscopic ime sep U 1 = hx 2 i,u 2 = hxf(x)i apple() = 100 (1 ) exp( 4) Average gain of facor 4 in regime wihou srong scale separaion p () p() µ U 1 () µ U 1 ()

19 Conclusions Coarse projecive inegraion is a echnique o accelerae simulaion by inducing a numerical closure approximaion The numerical closure is imposed by he lifing and prohibis convergence o he macroscopic image of he microscopic dynamics Replacing he lifing by a projecion of he microscopic sae on he manifold defined by a cerain macroscopic sae allows for full convergence 19