Accelerated micro/macro Monte Carlo simulation of dilute polymer solutions Giovanni Samaey Scientific Computing, Dept. of Computer Science, K.U. Leuven Based on joint work with K. Debrabant (U. Odense), T. Lelievre (ENPC, Paris), V. Legat (UCLouvain)
Micro-macro simulation of dilute polymer solutions Macroscopic part : Navier-Stokes equations for solvent ✓ @ u ◆ Re @ t + u · r u = (1 � ✏ ) ∆ u � r p + div( ⌧ p ) div( u ) = 0 Coupling : non-Newtonian stress tensor (Kramers’ formula) ✏ ⌧ p = We h X ⌦ F ( X ) i � Id Microscopic part : Stochastic differential equation (SDE) for the configuration of an individual polymer 1 � 1 dX = κ ( t ) X − 2We F ( X ) dt + dW t , √ We 2 Laso, Öttinger, J. Non-Newtonian Fluid Mech. 47 (1993) 1-20.
Example 1 : Linear springs 1 � 1 dX = κ ( t ) X − 2We F ( X ) dt + F ( X ) = X dW t , √ We ⌦ X 2 ↵ • Stress tensor τ p ⇠ h XF ( X ) i / µ = h X i ! 0 • Sign of X is irrelevant (length of spring), so • Distribution of X evolves towards a Gaussian ✓ 1 ◆ Σ + 2 d Σ • Closed model for evolution of the variance dt = − 2 κ ( t ) − 2We We 0 . 8 0 . 5 Σ = h ( X � µ ) 2 i P ( X, t ) 0 . 6 0 . 45 Σ ( t ) h ( y � µ ) 2 i ρ ( y, t ) 0 . 4 0 . 4 0 . 2 0 . 35 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 3 − 2 − 1 0 1 2 3 4 t X t
Example 2 : FENE springs • Finitely extensible nonlinearly elastic (FENE) 1 � 1 X dX = κ ( t ) X − 2We F ( X ) dt + dW t , F ( X ) = √ 1 − X 2 /b We • Distribution becomes non-Gaussian (with sharp peak) 0 . 4 t ∗ = 0 . 5 δ = 0 . 50 δ = 1 . 00 • Impossible to represent exactly 0 . 3 with a finite number of moments P ( | X | , t ) t ∗ = 1 ϕ ( | X | ) 0 . 2 U [ L ] = ( U l ) L l =1 ⌦ X 2 l ↵ 0 . 1 U l = • Monte Carlo simulation required 0 (especially in higher dimensions !) 0 1 2 3 4 5 6 7 | X | | X | 4
Connecting the levels of description Simulatie Microscopic level • known model • simulation code available Lifting Restrictie Macroscopic level • only state variables • unknown evolution equations t* + Δ t t* • Coarse time-stepper is a wrapper around a microscopic simulation • Generic building block for computational multiscale algorithms 5
Coarse time-stepper based bifurcation analysis U ( t ) U ( t + τ ) = Φ τ ( U ( t )) • Time-stepper is a black box Matrix-vector products • Directly compute macroscopic steady Φ τ ( ¯ U ) states and their stability U ∗ − Φ τ ( U ∗ ) = 0 D Φ τ ( ¯ U ) · v ¯ U • Use (matrix-free) iterative methods (RPM, Φ τ ( ¯ U + � · v ) Newton-Krylov) -> equation-free ¯ U + � · v Kevrekidis et al., 2000 - ... / Kevrekidis & S, Annual Review on Physical Chemistry 60:321-344, 2009 6
Acceleration of macroscopic simulation Exploit a separation in spatial and temporal scales Coarse projective integration Extrapolate macroscopic state forward in time ∆ t � δ t δ t Patch dynamics Interpolate between microscopic simulation in small subdomains x Gear, Kevrekidis, SISC. 24:1091-1106, 2004 / Lafitte, S, SISC, 2010, submitted. S, Roose, Kevrekidis, SIAM MMS 4:278-306, 2005 / S, Kevrekidis, Roose, JCP 213(1):264-287, 2006. 7
The heterogeneous multiscale methods An alternative formulation • Postulate a general form for the unknown macroscopic equation • Supplement this equation with an estimation of missing macroscopic quantities from a microscopic simulation - Initialization of the microscopic model from a given macroscopic state - Estimation of a macroscopic quantity from microscopic data • This formulation has advantages from a numerical analysis viewpoint E, Engquist, Vanden-Eijnden, et al., 2003 - ... 8
Questions from a numerical analysis viewpoint During restriction , a During lifting , missing Simulatie macroscopic state is estimated microscopic information is based on the microscopic state. filled in based on the macroscopic state . • How big is the variance of the noise during restriction ? • What are appropriate Lifting Restrictie macroscopic state • How can this variance be reduced ? variables ? • How is variance affected • How accurate is the when extrapolating in time ? reconstruction ? t* + Δ t t* • How does extrapolation affect • What is the influence of lifting stability ? errors on macroscopic evolution ? Rousset, S, M3AS, 2011, submitted. Frederix, S, Roose, ESIAM: M2AN, 2010. Gear, Kaper, Kevrekidis, Zagaris. SIAM J. Appl. Dyn. Syst. 4:711-732, 2005. Debrabant, S, ESIAM: M2AN, 2011, submitted. Frederix, S, Vandekerckhove, Roose, Li, Nies. Discrete Cont Dyn-B 11: 855-874, 2009. Ghysels, S, Van Liedekerke, Tijskens, Ramon, Roose, Int. J. Multiscale Comp. Engng. 8(4):411-422, 2010. S, Lelievre, Legat, Computers and Fluids, 2010. 9
Coarse time-stepper for Monte Carlo simulation 1 � 1 d X = κ ( t ) X − 2We F ( X ) dt + d W t Microscopic level √ We X k +1 = s X ( X k , κ ( t ) , δ t ) Simulatie from moments to an from an ensemble to ensemble moments Lifting Restrictie R : X 7! U L : U 7! X = { X j } J P J U l = 1 j =1 f l ( X j ) j =1 J t* + Δ t t* U [ L ] = ( U l ) L d U dt = H ( U , κ ( t )) Macroscopic level l =1 ⌦ X 2 l ↵ U l = τ p = T ( U ) 10
Lifting operator : constrained simulation • Simulate with constrained macroscopic state until conditional equilibrium X ∗ ,m +1 = s X ( X ∗ ,m , κ ∗ , δ t ) + Λ r X R ( X ∗ ,m +1 ) , met Λ 2 R L zodanig dat R ( X ∗ ,m +1 ) = U ∗ - Time integration, followed by projection onto manifold defined by imposed macroscopic state X ∗ ,m +1 = arg min � � X ∗ ,m +1 − s X ( X ∗ ,m , κ ∗ , δ t ) � � with constraint R ( X ∗ ,m +1 ) = U ∗ • The result of the lifting is then given as (for M sufficiently large) X ∗ = L ( U ∗ ) := X ∗ ,M • Consistent initial condition also by projection of a nearby ensemble S, Lelievre, Legat, Computers and Fluids 43:119-133, 2011. 11
Lifting induces a closure approximation • Experiment - Coarse time-stepper with very small time step U [ L ] = ( U l ) L U l = h X 2 l i - Macroscopic state variables : l =1 , - (Much more expensive than full microscopic simulation) • Lifting introduces modeling error that decreases for an increasing number of moments 40 150 U 1 ( t ) τ p ( t ) L = 1 L = 2 125 L = 3 L = 4 30 FENE 100 M 1 20 τ p 75 50 10 25 0 0 12 t t 0 1 2 3 4 0 1 2 3 4 t t
Extrapolation via coarse projective integration • Start with a given macroscopic state U = U N • Lift to the corresponding microscopic state ∆ t � δ t L : U = U N 7! X N = X N,M K δ t • Simulate the ensemble over microscopic K steps X N,k +1 = s X ( X N,k , κ ( t N,k ) , δ t ) , k = 0 , . . . K − 1 • Restrict to macroscopic state U N,K = R ( X N,K ) • Extrapolate macroscopic state U N +1 = U N,K + ( ∆ t − K δ t ) U N,K − U N K δ t 13
Efficiency and accuracy of coarse projective integration • Coarse projective integration is efficient if ∆ t � ( M + K ) δ t Number of steps to Number of constrained estimate time derivative steps during lifting - The bigger the time scale separation ( ), the smaller M can be We → 0 - But: in the limit when , the macroscopic model is known ! We → 0 - Real acceleration is only possible for an intermediary regime • During extrapolation, estimation noise is amplified with a factor ∆ t/K δ t - An alternative would be to use less particles and no extrapolation - For equal statistical error, coarse projective integration requires as much computations as a full microscopic simulation (assuming M=0 !) 14
An alternative extrapolation strategy Multistep state extrapolation U N +1 = U N,K + ( ∆ t − K δ t ) U N,K − U N • Projective integration K δ t • Multistep state extrapolation - Extrapolate using the last point of each sequence of microscopic simulation U N +1 = U N,K + ( ∆ t − K δ t ) U N,K − U N − 1 ,K ∆ t - Statistical error is unaffected ∆ t/K δ t - Systematic error does get amplified with a factor - But we want to extrapolate just because we can tolerate a larger systematic error ! 15 Sommeijer, Comput. Math. Appl. 19 (6) (1990) 37–49.
Matching: an alternative for lifting • “Classical” lifting : t = t N,K - match an ensemble on X N +1 = L ( U N +1 ) X N +1 = P ( U N +1 ; X N,K ) with an extrapolated macroscopic t = t N +1 state on - simulate with macroscopic constraint until conditional equilibrium (M steps) • Alternative : perform matching without ∆ t � δ t constrained simulation K δ t - The time gained during extrapolation is not lost during constrained simulation - The projected ensemble now also Debrabant, S , ESIAM M2AN, 2010, submitted. depends on the ensemble at the previous time step ! 16
Accuracy of matching operator • Experiment U [ L ] = ( U l ) L U l = h X 2 l i - Macroscopic state variables : l =1 , - Simulate until time t* U [ L ] ( t ∗ ) X ( t ∗ ) - Project onto manifold defined by and compare with X ( t ∗ − ∆ t ) • Projection introduces a modeling error that decreases with - increasing number of moments Error ∼ C L ∆ t - decreasing extrapolation time step 2-sample K-S test 10 − 2 L p-value L = 3 L = 4 3 0 rel. error in τ p Error L = 5 10 − 3 O ( ∆ t ) 4 7,00E-06 10 − 4 5 0,28 6 0,25 10 − 5 7 0,84 0 . 001 0 . 01 ∆ t 17 ∆ t
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