Mathematical analysis of an Adaptive Biasing Potential method for - PowerPoint PPT Presentation

Mathematical analysis of an Adaptive Biasing Potential method for diffusions Charles-Edouard Bréhier Joint work with Michel Benaïm (Neuchâtel, Switzerland) CNRS & Université Lyon 1, Institut Camille Jordan (France) C-E Bréhier (CNRS-Lyon) Convergence of ABP 1 / 31

Plan of the talk Motivations 1 The Adaptive Biasing Potential method 2 Consistency and elements of proof 3 Self-interacting diffusions Analysis of the ABP method Extensions 4 Examples Abstract framework C-E Bréhier (CNRS-Lyon) Convergence of ABP 2 / 31

Motivations: setting Goal: estimating averages � 1 � T d φ ( x ) e − β V ( x ) dx , φ d µ β = Z ( β ) where V : T d → R . From now on β = 1. The probability distribution µ is ergodic for the overdamped Langevin dynamics: √ dX 0 t = −∇ V ( X 0 t ) dt + 2 dW t . Difficulty: slow convergence of temporal averages (metastability) � t 1 � φ ( X s ) ds → φ d µ. t t →∞ 0 C-E Bréhier (CNRS-Lyon) Convergence of ABP 3 / 31

Context Use of an importance sampling strategy: change of the drift coefficient in the dynamics. A nice review B. M. Dickson. Survey of adaptive biasing potentials: comparisons and outlook. Current Opinion in Structural Biology , 2017. Related to the following techniques: umbrella sampling S. Marsili, A. Barducci, R. Chelli, P. Procacci, and V. Schettino. Self-healing umbrella sampling: a non-equilibrium approach for quantitative free energy calculations. The Journal of Physical Chemistry B , 2006. G. Fort, B. Jourdain, T. Lelièvre, and G. Stoltz. Self-healing umbrella sampling: convergence and efficiency. Stat. Comput. , 2017. C-E Bréhier (CNRS-Lyon) Convergence of ABP 4 / 31

Context Use of an importance sampling strategy: change of the drift coefficient in the dynamics. A nice review B. M. Dickson. Survey of adaptive biasing potentials: comparisons and outlook. Current Opinion in Structural Biology , 2017. Related to the following techniques: metadynamics A. Laio and M. Parrinello. Escaping free-energy minima. Proceedings of the National Academy of Sciences , 2002. A. Barducci, G. Bussi, and M. Parrinello. Well-tempered metadynamics: a smoothly converging and tunable free-energy method. Physical review letters , 2008. C-E Bréhier (CNRS-Lyon) Convergence of ABP 4 / 31

Context Use of an importance sampling strategy: change of the drift coefficient in the dynamics. A nice review B. M. Dickson. Survey of adaptive biasing potentials: comparisons and outlook. Current Opinion in Structural Biology , 2017. Related to the following techniques: Wang-Landau F. Wang and D. Landau. Determining the density of states for classical statistical models: A random walk algorithm to produce a flat histogram. Physical Review E , 2001. F. Wang and D. Landau. Efficient, multiple-range random walk algorithm to calculate the density of states. Physical review letters , 2001. G. Fort, B. Jourdain, E. Kuhn, T. Lelièvre, and G. Stoltz. Convergence of the Wang-Landau algorithm. Math. Comp. , 2015. C-E Bréhier (CNRS-Lyon) Convergence of ABP 4 / 31

Biasing methods for diffusions Adaptive Biasing Force (ABF) √ � −∇ V ( X t )+ F t ( X t ) � dX t = dt + 2 dW t . Adaptive Biasing Potential (ABP) √ � −∇ V ( X t )+ ∇V t ( X t ) � dX t = dt + 2 dW t . References on ABF J. Comer, J. C. Gumbart, J. Hénin, T. Lelièvre, A. Pohorille, and C. Chipot. The adaptive biasing force method: Everything you always wanted to know but were afraid to ask. The Journal of Physical Chemistry B , 2014. T. Lelièvre, M. Rousset, and G. Stoltz. Long-time convergence of an adaptive biasing force method. Nonlinearity , 2008. The bias F t depends on the law L ( X t ) . C-E Bréhier (CNRS-Lyon) Convergence of ABP 5 / 31

Biasing methods for diffusions Adaptive Biasing Force (ABF) √ � −∇ V ( X t )+ F t ( X t ) � dX t = dt + 2 dW t . Adaptive Biasing Potential (ABP) √ � −∇ V ( X t )+ ∇V t ( X t ) � dX t = dt + 2 dW t . We study a continuous time version of the ABP method proposed in B. Dickson, F. Legoll, T. Lelièvre, G. Stoltz, and P. Fleurat-Lessard. Free energy calculations: An efficient adaptive biasing potential method. J. Phys. Chem. B , 2010. The bias V t depends on the past trajectory of the process, X r for 0 ≤ r ≤ t . C-E Bréhier (CNRS-Lyon) Convergence of ABP 5 / 31

Plan of the talk Motivations 1 The Adaptive Biasing Potential method 2 Consistency and elements of proof 3 Self-interacting diffusions Analysis of the ABP method Extensions 4 Examples Abstract framework C-E Bréhier (CNRS-Lyon) Convergence of ABP 6 / 31

Biasing the potential Reaction coordinate: ξ : x ∈ T d �→ x 1 ∈ T . Biasing: for A : T → R , √ dX A ( X A t = −∇ � V − A ◦ ξ � t ) dt + 2 dW t . Ergodic invariant law is modified ∝ e − V ( x )+ A ( ξ ( x )) . Why it is useful: � weighted averages converge to φ d µ (consistency) convergence is faster when A is well-chosen: free energy can be made adaptive C-E Bréhier (CNRS-Lyon) Convergence of ABP 7 / 31

The free energy A ⋆ : T → R is given by � e − A ⋆ ( z ) = T d − 1 Z ( 1 ) − 1 e − V ( z , x 2 ,..., x d ) dx 2 . . . dx d . Acceleration of the sampling = removing free energy barriers. Choosing A = A ⋆ : flat histogram for ξ ( X A ⋆ t ) � t 1 ) ds → δ ξ ( X A ⋆ t →∞ dz . t s 0 Adaptive algorithm: A t → t →∞ A ∞ ≈ A ⋆ . C-E Bréhier (CNRS-Lyon) Convergence of ABP 8 / 31

The Adaptive Biasing Potential method Two unknowns: X t and A t . Dynamics for X t : √ � � dX t = −∇ V − A t ◦ ξ ( X t ) dt + 2 dW ( t ) . Computation of the bias A t : convolution of a kernel function K � e − A t ( z ) = � � T d K z , ξ ( x ) µ t ( dx ) , with the weighted empirical averages � t 0 e − A r ◦ ξ ( X r ) δ X r dr µ t = µ 0 + . � t 1 + 0 e − A r ◦ ξ ( X r ) dr C-E Bréhier (CNRS-Lyon) Convergence of ABP 9 / 31

Convergence results √ � , e − A t ( z ) = � � � � dX t = −∇ V − A t ◦ ξ ( X t ) dt + 2 dW ( t ) K z , ξ ( x ) µ t ( dx ) � t 0 e − A r ◦ ξ ( X r ) δ X r dr µ t = µ 0 + . � t 0 e − A r ◦ ξ ( X r ) dr 1 + Theorem (Benaïm-B.) Almost surely, µ t → t →∞ µ , A t → t →∞ A ∞ , with µ ( dx ) = e − V ( x ) dx and e − A ∞ ( z ) = � T d K ( z , ξ ( · )) d µ . C-E Bréhier (CNRS-Lyon) Convergence of ABP 10 / 31

Convergence results √ � , e − A t ( z ) = � � � � dX t = −∇ V − A t ◦ ξ ( X t ) dt + 2 dW ( t ) K z , ξ ( x ) µ t ( dx ) � t 0 e − A r ◦ ξ ( X r ) δ X r dr µ t = µ 0 + . � t 0 e − A r ◦ ξ ( X r ) dr 1 + Theorem (Benaïm-B.) Almost surely, µ t → t →∞ µ , A t → t →∞ A ∞ , with µ ( dx ) = e − V ( x ) dx and e − A ∞ ( z ) = � T d K ( z , ξ ( · )) d µ . consistent with non-adaptive versions ( A t = A ∀ t ≥ 0) A ∞ � = A ⋆ due to the kernel function. C-E Bréhier (CNRS-Lyon) Convergence of ABP 10 / 31

Comments on the efficiency Convergence of histograms: almost surely � t 1 t →∞ e A ∞ ( z ) − A ⋆ ( z ) dz . δ ξ ( X s ) ds → t 0 Asymptotic variance: same as in non-adaptive version with A = A ∞ . The choice of the kernel function K is important. Assumption K : T × T → ( 0 , ∞ ) is of class C ∞ and positive. � K ( z , · ) dz = 1 . Examples: K ( z , ζ ) ∝ e − | z − ζ | 2 vs. K ( z , ζ ) = e − A ( z ) . 2 δ C-E Bréhier (CNRS-Lyon) Convergence of ABP 11 / 31

Plan of the talk Motivations 1 The Adaptive Biasing Potential method 2 Consistency and elements of proof 3 Self-interacting diffusions Analysis of the ABP method Extensions 4 Examples Abstract framework C-E Bréhier (CNRS-Lyon) Convergence of ABP 12 / 31

Role of the weighted empirical measures µ t √ dX t = −∇ � V − A t ◦ ξ � ( X t ) dt + 2 dW ( t ) , � t 0 e − A r ◦ ξ ( X r ) δ X r dr µ t = µ 0 + , � t 0 e − A r ◦ ξ ( X r ) dr 1 + � � � � � exp − A t ( z ) = K z , ξ ( x ) µ t ( dx ) . Considering A t = A ( µ t ) , may be interpreted as a self-interacting diffusion, with unknowns X t and µ t √ � � dX t = −∇ V − A ( µ t ) ◦ ξ ( X t ) dt + 2 dW ( t ) , � t 0 e − A ( µ r ) ◦ ξ ( X r ) δ X r dr µ t = µ 0 + � t 0 e − A ( µ r ) ◦ ξ ( X r ) dr 1 + C-E Bréhier (CNRS-Lyon) Convergence of ABP 13 / 31

Self-interacting diffusions Classical formulation: √ dY t = −∇ V ( Y t , ν t ) dt + 2 dW t with � t 1 � � � V ( y , · ) d ν V ( y , ν ) = , ν t = ν 0 + δ Y r dr . 1 + t 0 M. Benaïm, M. Ledoux, and O. Raimond. Self-interacting diffusions. Probab. Theory Related Fields , 2002. Main differences: type of the coupling µ t in ABP is a weighted empirical distribution. C-E Bréhier (CNRS-Lyon) Convergence of ABP 14 / 31

Stochastic approximation: the ODE method � t √ 1 � � dY t = −∇ V ( Y t , ν t ) dt + 2 dW t , ν t = ν 0 + δ Y r dr . 1 + t 0 The empirical distribution solves the random ODE 1 d ν t � � δ Y t − ν t dt = . 1 + t Asymptotic time-scale separation: slow-fast system ( t → ∞ ). A. Benveniste, M. Métivier, and P. Priouret. Adaptive algorithms and stochastic approximations . Springer-Verlag, 1990. H. J. Kushner and G. G. Yin. Stochastic approximation and recursive algorithms and applications . Springer-Verlag, 2003. C-E Bréhier (CNRS-Lyon) Convergence of ABP 15 / 31