Algorithms for Ensemble Control. B Leimkuhler University of Edinburgh

Transcription

1 Algorithms for Ensemble Control B Leimkuhler University of Edinburgh Model Reduction Across Disciplines Leicester 2014

2 (Stochastic) Ensemble Control extended ODE/SDE system design target ensemble SDE invariant measure

3 Thermostat extended Hamiltonian system (heat bath) temperature T Thermostat = ODE/SDE with prescribed unique invariant density (typically Boltzmann-Gibbs)

4 Thermostats Define: A = a(z)ρ can (z)dz stationary average C(τ) = ϕ τ (z) T Bzρ can (z)dz autocorrelation function A thermostat generates trajectories z(t) such that t  := lim t 1 t 0 t Ĉ(τ) := lim t 1 t 0 a(z(s))ds = A z(s + τ) T Bz(s)ds C(τ) (T.L.)

5 Control of Model Error Fourier modes of semi-discrete Burgers Equation log energy undissipated finite model with artificial dissipation slope = -2 Can we correct the energy decay relation using a thermostat -like device? log frequency k

6 Some Questions Many choices for reduced system with same invariant measure - how to design/choose? Ergodicity? How to promote rapid mixing, convergence? How does the SDE approach equilibrium? Role of dynamics? Relation to natural timescales. What is the effect of numerical discretization? (Invariant measures of numerical methods) How does the SDE discretization approach equilibrium? Can we correct model error using ensemble controls? (i.e. retroactively repair damaged models)

7 Ex: Brownian dynamics dx = U(X)dt + 2dW invariant measure: ρ eq = e U under certain conditions unique steady state of the Fokker-Planck equation: ρ t = L BDρ L BDρ = [ρ U]+Δρ

8 Ex: Langevin Dynamics dq = M 1 pdt dp =[ U(q) γp]dt + 2γk B TM 1/2 dw Fokker-Planck Operator: γ = friction parameter L LDη = (M 1 p) q η + U p η + γ p (pη)+γk B T Δη Preserves Gibbs distribution: L LDρ β =0 mass weighted partial Laplacian

9 Properties of L LD Under suitable conditions Discrete Spectrum, Spectral Gap Hypocoercive (but degenerate in the limit of small friction) Ergodic lim f,ρ(,t) = f,ρ β t Exponential convergence in an appropriate norm e tl Ke λ γt λ γ > 0

10 Hypoellipticity A 2nd order differential operator with C coefficients is hypoelliptic if its zeros are C Let U be a compact, connected, invariant subset for an SDE. dx = X 0 (x)dt + r j=1 X j (x)dw j If the corresponding Kolmogorov operator is hypoelliptic on U, then the flow is ergodic on U. Acknowledgement: Hairer s Lecture Notes Hörmander Villani Hairer

11 Hörmander condition The vector fields X 0 (x),...,x r (x) satisfy a Hörmander condition if Span{X 0 (x),...,x r (x), [X i,x j ](x), [X i, [X j,x k ]](x)...} = R N Theorem 1. Let U R N be open. If X 0,X 1 : U R N are two vectorfields that satisfy Hörmander s condition at every z U, then the operator L which is defined by L ρ := N i=1 z i (ρ(z)x 0,i (z)) N i,j=1 2 z i z j (ρ(z)x 1,i (z)x 1,j (z)) is hypoelliptic.

12 Langevin dynamics [Stuart, Mattingley, Higham 02] dx = pdt H = p 2 /2+U(x) f(x) = U (x) dp = f(x)dt pdt + 2dW HC: invariant measure: Lyapunov function b 0 =(p, f(x) p); b 1 =(0, 1) [ ] [ [b 0, b 1 ]= f b (x) 1 1 = 1 ρ = e H positive measure on open sets Therefore, Langevin dynamics is ergodic ]

13 Highly Degenerate Diffusions

14 Nose-Hoover H = p2 2 + U(q) Newtonian dynamics q = p ṗ = U (q) ξp μ ξ = p 2 θ preserves but not ergodic

15 Nose-Hoover H = p2 2 + U(q) Newtonian dynamics q = p ṗ = U (q) ξp μ ξ = p 2 θ preserves but not ergodic

16 Nose-Hoover H = p2 2 + U(q) Newtonian dynamics q = p ṗ = U (q) ξp μ ξ = p 2 θ governor preserves but not ergodic

17 Nose-Hoover H = p2 2 + U(q) Newtonian dynamics q = p ṗ = U (q) ξp μ ξ = p 2 θ Gibbs Governor governor preserves but not ergodic

18 Need for Stochastics z z Chains: z ξ q = p 4 ṗ = q ξp ξ 1 = μ 1 1 (p2 kt) ξ 1 ξ 2 ξ 2 = μ 1 2 (μ 1ξ1 2 kt) μ 1 =0.2, μ 2 =1 1 p q 2

19 Designer Diffusions L. Noorizadeh, Theil JSP 2009, L., Phys Rev E, 2010 L., Noorizadeh, Penrose JSP 2011 design to preserve extended Gibbs distribution OU ρ = ρ (X)e Ξ2 /2 weak coupling to stochastic perturbation

20 Nose-Hoover-Langevin dq = pdt dp = V ξp dξ = μ 1 [p T p nkt]dt γξdt + 2kTγ/μdW Unification of Nosé-Hoover and Langevin thermostats Generalizes NH thermostat Includes kinetic energy regulator Single scalar stochastic variable

21 Prop: Let the given system preserve Suppose the system is defined on where is a smooth compact submanifold Further suppose that the Lie algebra spanned by f,g spans at every point of Then the given system is ergodic on

22 Ergodicity of NHL dq = pdt dp = V ξp dξ = μ 1 [p T p nkt]dt γξdt + 2kTγ/μdW Let the potential have the form V = q T Bq then, under a mild non-resonance assumption, the NHL equations are ergodic on a large set. Proof: just check the Hörmander condition!

23 Ex: Nose-Hoover Langevin on a harmonic system f = [ p Bq ], g = [ 0 p ] S{f,g} = Lie algebra (ideal) generated by f,g Prop: [ B C k = k 1 p B k q [ B D k = k p B k q ] ] S{f,g} S{f,g}

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25 parameter dependence Autocorrelation Functions [L., Noorizadeh and Penrose, J. Stat. Phys. 2011] quantify efficiency of different thermostats accumulation of error in dynamics vs convergence rate

26 Vortex Method

27 Onsager, 1949 Statistical Hydrodynamics Oliver Bühler, 2002: a numerical study Point Vortices [Dubinkina, Frank and L., SIAM MMS 2010] A point vortex model for N vortices in a cylinder + boundary terms R Γ i ẋ i = J xi H

28 Onsager s Prediction Positive temperatures: Strong vortices of opposite sign tend to approach each other Negative temperatures:... vortices of the same sign will tend to cluster---preferably the strongest ones---so as to use up excess energy at the least possible cost in terms of degrees of freedom... the weaker vortices, free to roam practically at random will yield rather erratic and disorganized contributions to the flow. Strong vortices of the same sign will cluster

29 Buhler (2002) Simulation 4 strong 96 weak vortices sign indefinite, 0 net circulation in each group fixed ang. mom. Simulation results supported Onsager s predictions Use *finite* bath - not the Gibbsian model

30 Modified Canonical Statistics Assume the subsystem and reservoir variables decoupled in the Hamiltonian H(X A,X B )=H A (X A )+H B (X B ) Notation: Ω(E) =vol{x H(X) [E,E + de)} S(E) =lnω(e) Then: Prob{X A H = E} Ω B (E H A (X A )) = exp(s B (E H A )) = exp(s B (E) S B(E)H A + S B(E)H A 2 + ) exp( βh A + γha 2 + ) assume finite bath energy Gibbs statistics assume finite bath energy variance Generalized Bath Model

31 Modified Gibbs ρ finite e βh γh2 Modified stochastic control law: Gibbs: +OU(ζ) modified Gibbs: +OU(ζ) Allows direct comparison with Bühler s results GBK thermostat gives a model reduction

32 Experimental parameters β { 0.006, , 0.01} α =0.5, σ = 0.4 γ = β 2E 0, E 0 {628, 221, 197} t [1500, 12000] ρ(ζ) ζ t [0, 1000] β =

33 Distance between like signed vortices x i x j ++ β>0 β 0 β<0 Buhler 02 Gibbs modified Gibbs

34 Vortex clustering, N=12

35 Vortex clustering, N=12

36 Burgers/KdV

37 Thermostat controls in Burgers-KdV [Bajars, Frank and L., Nonlinearity, 2013] Rationale Discretized PDE models, e.g. Euler fluid equations, have a multiscale structure Energy flows from low to high modes: turbulent cascade Under discretization, the cascade is destabilized leading either to an artificial increase in energy at fine scales, or, if dissipation is used, artificial decrease First steps: try to preserve a target equilibrium ensemble

38 Can we use a molecular thermostat to control the ensemble in a semi-discrete Burgers/KdV model? Hamiltonian system energy

39 Truncated, discrete model Two other first integrals total momentum M, total kinetic energy E

40 Proposed mixed distribution: Now - design a highly degenerate thermostat Notes: The Hörmander condition is too hard for us to show we couple to the high wave numbers and demonstrate ergodicity using numerics E and Mattingley - prove HC for coupling to slow modes (opposite of what we want)

41 Burgers H Dist Kinetic Energy weak perturbation GBK(n*=15) GBK(n*=m): results using a thermostat applied only to modes m...n

42 Burgers Convergence of expected value of Hamiltonian convergence of averages is observed in all cases, but is very slow for GBK(n*=15)

43 c k : autocorrellation function for kth mode c 1 c 1 c3 c 5

44 2D Incompressible Navier Stokes - 5 slides omitted.

45 Conclusions 1. SDE-based thermostats are versatile tools to approximate averages with respect to given density 2. Degenerate thermostats allow for efficient recovery, i.e., with small perturbation of dynamics 3. They can be applied beyond MD, e.g. in fluid dynamics (and more broadly) 4. Potentially valuable for model correction, data assimilation, etc., i.e. to restore properties of the equilibrium ideal system to a corrupted set of equations.