On the discretization of Feynman-Kac semi-groups. Application to - PowerPoint PPT Presentation

On the discretization of Feynman-Kac semi-groups. Application to rare events sampling and Di ff usion Monte Carlo. PhD Seminar Grégoire Ferré – Gabriel Stoltz CERMICS - ENPC www.enpc.fr Wednesday, May 10 th , 2017

Outline 1. Introduction: Two apparently unrelated problems 2. Our problem: Time step bias 3. Error on the invariant measure 4. Discretizing Feynman-Kac semi-groups Grégoire Ferré , Gabriel Stoltz CERMICS - ENPC On the discretization of Feynman-Kac semi-groups. Application to rare events sampling and Di ff usion Monte Carlo. 2 / 20

1. Introduction: Two apparently unrelated problems

Schrödinger ground state Schrödinger operator, describes the energy of a sytem: H = − ∆ + V , where V is a potential. Typical example: � N � V ( x 1 ,..., x N ) = V 1 ( x i ) + V 2 ( x i − x j ) . i = 1 i � j Grégoire Ferré , Gabriel Stoltz CERMICS - ENPC On the discretization of Feynman-Kac semi-groups. Application to rare events sampling and Di ff usion Monte Carlo. 4 / 20

Schrödinger ground state Schrödinger operator, describes the energy of a sytem: H = − ∆ + V , where V is a potential. Typical example: � N � V ( x 1 ,..., x N ) = V 1 ( x i ) + V 2 ( x i − x j ) . i = 1 i � j Goal (for electronic structure calculation, ground state energy, properties of materials, etc), compute the ground state energy E 0 , lowest eigenvalue H ψ = E 0 ψ. Grégoire Ferré , Gabriel Stoltz CERMICS - ENPC On the discretization of Feynman-Kac semi-groups. Application to rare events sampling and Di ff usion Monte Carlo. 4 / 20

Averages and fluctuations Example. Consider the dynamics � 2 β − 1 dB t . dX t = −∇ V ( X t ) dt + Goal: estimate a long time average � t � ϕ t = 1 µ ( dx ) = Z − 1 e − β V dx . ϕ ( X s ) ds − → ϕ d µ, t t → + ∞ 0 D Grégoire Ferré , Gabriel Stoltz CERMICS - ENPC On the discretization of Feynman-Kac semi-groups. Application to rare events sampling and Di ff usion Monte Carlo. 5 / 20

Averages and fluctuations Example. Consider the dynamics � 2 β − 1 dB t . dX t = −∇ V ( X t ) dt + Goal: estimate a long time average � t � ϕ t = 1 µ ( dx ) = Z − 1 e − β V dx . ϕ ( X s ) ds − → ϕ d µ, t t → + ∞ 0 D Problem: for a finite time t , ϕ t � µ ( ϕ ) New goal: estimate probabilities of fluctuations around the mean. Grégoire Ferré , Gabriel Stoltz CERMICS - ENPC On the discretization of Feynman-Kac semi-groups. Application to rare events sampling and Di ff usion Monte Carlo. 5 / 20

Large deviations in time Idea of large deviations: � 1 � � t ≍ e − tI ( a ) , ϕ ( X s ) ds = a P t 0 where I is the rate function . This suggests the formula 1 I ( a ) = − lim t log P ϕ t ( a ) . t →∞ Grégoire Ferré , Gabriel Stoltz CERMICS - ENPC On the discretization of Feynman-Kac semi-groups. Application to rare events sampling and Di ff usion Monte Carlo. 6 / 20

Large deviations in time Idea of large deviations: � 1 � � t ≍ e − tI ( a ) , ϕ ( X s ) ds = a P t 0 where I is the rate function . This suggests the formula 1 I ( a ) = − lim t log P ϕ t ( a ) . t →∞ Fact : I ( µ ( ϕ )) = 0. Grégoire Ferré , Gabriel Stoltz CERMICS - ENPC On the discretization of Feynman-Kac semi-groups. Application to rare events sampling and Di ff usion Monte Carlo. 6 / 20

Large deviations in time Idea of large deviations: � 1 � � t ≍ e − tI ( a ) , ϕ ( X s ) ds = a P t 0 where I is the rate function . This suggests the formula 1 I ( a ) = − lim t log P ϕ t ( a ) . t →∞ Fact : I ( µ ( ϕ )) = 0. Donsker-Varadhan [1975]: if one sets λ ( k ) := sup { ka − I ( a ) } , a ∈ R Then λ ( k ) is the largest eigenvalue of L + k ϕ where L is the generator of ( X t ) . We are back to a problem of ground state estimation. Grégoire Ferré , Gabriel Stoltz CERMICS - ENPC On the discretization of Feynman-Kac semi-groups. Application to rare events sampling and Di ff usion Monte Carlo. 6 / 20

Probabilistic representation of ground state energy Consider the generator L of a process ( X t ) and ( λ, h W ) the principal eigenvalue and eigenfunction of L + W . Feynman-Kac formula gives � 0 W ( X s ) ds � � t ∼ e λ t , E e so that λ has the representation � 0 W ( X s ) ds � � t 1 λ = lim t log E . e t →∞ Grégoire Ferré , Gabriel Stoltz CERMICS - ENPC On the discretization of Feynman-Kac semi-groups. Application to rare events sampling and Di ff usion Monte Carlo. 7 / 20

Probabilistic representation of ground state energy Consider the generator L of a process ( X t ) and ( λ, h W ) the principal eigenvalue and eigenfunction of L + W . Feynman-Kac formula gives � 0 W ( X s ) ds � � t ∼ e λ t , E e so that λ has the representation � 0 W ( X s ) ds � � t 1 λ = lim t log E . e t →∞ We then turn to the following more general quantity, for an observable ϕ , � 0 W ( X s ) ds � � t � E µ ϕ ( X t ) e Φ t ( µ )( ϕ ) = � 0 W ( X s ) ds � − → ϕ h W d ν, � t t →∞ D E µ e where ( L + W ) h W = λ h W . Grégoire Ferré , Gabriel Stoltz CERMICS - ENPC On the discretization of Feynman-Kac semi-groups. Application to rare events sampling and Di ff usion Monte Carlo. 7 / 20

2. Our problem: Time step bias

Statistical approximation Path average over a set of replicas ( X m t ) M m = 1 with initial distribution, � M � t 1 0 W ( X m s ) ds ϕ ( X m t ) e M m = 1 Φ t ( µ )( ϕ ) ≈ � M � t 1 0 W ( X m s ) ds e M m = 1 Grégoire Ferré , Gabriel Stoltz CERMICS - ENPC On the discretization of Feynman-Kac semi-groups. Application to rare events sampling and Di ff usion Monte Carlo. 9 / 20

Statistical approximation Path average over a set of replicas ( X m t ) M m = 1 with initial distribution, � M � t 1 0 W ( X m s ) ds ϕ ( X m t ) e M m = 1 Φ t ( µ )( ϕ ) ≈ � M � t 1 0 W ( X m s ) ds e M m = 1 Various sampling techniques: interacting particle systems, populations dynamics... Problem : huge variance of the exponential weights. Grégoire Ferré , Gabriel Stoltz CERMICS - ENPC On the discretization of Feynman-Kac semi-groups. Application to rare events sampling and Di ff usion Monte Carlo. 9 / 20

Discretization error Idea • Fact : it is impossible to run a continuous simulation on a computer, • Solution : discretize with a time step ∆ t , • Problem : this induces a bias in the result. Grégoire Ferré , Gabriel Stoltz CERMICS - ENPC On the discretization of Feynman-Kac semi-groups. Application to rare events sampling and Di ff usion Monte Carlo. 10 / 20

Discretization error Idea • Fact : it is impossible to run a continuous simulation on a computer, • Solution : discretize with a time step ∆ t , • Problem : this induces a bias in the result. We discretize the process ( X t ) into a Markov chain ( x n ) with evolution operator Q ∆ t and we study quantities of the form: � � ϕ ( x n ) e ∆ t � n − 1 i = 0 W ( x i ) � E µ Φ n , ∆ t ( µ )( ϕ ) = − → � � ϕ d ν W , ∆ t . e ∆ t � n − 1 n →∞ i = 0 W ( x i ) D E µ Grégoire Ferré , Gabriel Stoltz CERMICS - ENPC On the discretization of Feynman-Kac semi-groups. Application to rare events sampling and Di ff usion Monte Carlo. 10 / 20

Discretization error Idea • Fact : it is impossible to run a continuous simulation on a computer, • Solution : discretize with a time step ∆ t , • Problem : this induces a bias in the result. We discretize the process ( X t ) into a Markov chain ( x n ) with evolution operator Q ∆ t and we study quantities of the form: � � ϕ ( x n ) e ∆ t � n − 1 i = 0 W ( x i ) � E µ Φ n , ∆ t ( µ )( ϕ ) = − → � � ϕ d ν W , ∆ t . e ∆ t � n − 1 n →∞ i = 0 W ( x i ) D E µ Natural question: is ν W , ∆ t close to ν W := h W ν . Grégoire Ferré , Gabriel Stoltz CERMICS - ENPC On the discretization of Feynman-Kac semi-groups. Application to rare events sampling and Di ff usion Monte Carlo. 10 / 20

3. Error on the invariant measure

Discretization of a Markov process Let’s go back to the linear case: � E x [ ϕ ( X t )] − → ϕ d ν. t →∞ D The Markov process ( X t ) is discretized into a Markov chain ( x n ) with evolution operator Q ∆ t ( ϕ )( x ) = E [ ϕ ( x n + 1 ) | x n = x ] Grégoire Ferré , Gabriel Stoltz CERMICS - ENPC On the discretization of Feynman-Kac semi-groups. Application to rare events sampling and Di ff usion Monte Carlo. 12 / 20

Discretization of a Markov process Let’s go back to the linear case: � E x [ ϕ ( X t )] − → ϕ d ν. t →∞ D The Markov process ( X t ) is discretized into a Markov chain ( x n ) with evolution operator Q ∆ t ( ϕ )( x ) = E [ ϕ ( x n + 1 ) | x n = x ] We then have for a long time average of the Markov chain � E x [ ϕ ( x n )] − → ϕ d ν ∆ t . t →∞ D Grégoire Ferré , Gabriel Stoltz CERMICS - ENPC On the discretization of Feynman-Kac semi-groups. Application to rare events sampling and Di ff usion Monte Carlo. 12 / 20