SDEs in large dimension and numerical methods Part 1: Sampling the - PowerPoint PPT Presentation

Introduction Free energy Thermodynamic integration Adaptive biasing techniques SDEs in large dimension and numerical methods Part 1: Sampling the canonical distribution T. Lelièvre CERMICS - Ecole des Ponts ParisTech & Matherials project-team - INRIA RICAM Winterschool, December 2016

Introduction Free energy Thermodynamic integration Adaptive biasing techniques Motivation The aim of molecular dynamics simulations is to understand the relationships between the macroscopic properties of a molecular system and its atomistic features. In particular, one would like to evaluate numerically macroscopic quantities from models at the microscopic scale. Many applications in various fields: biology, physics, chemistry, materials science. Various models: discrete state space (kinetic Monte Carlo, Markov State Model) or continuous state space (Langevin). The basic ingredient: a potential V which associates to a configuration ( x 1 , ..., x N ) = x ∈ R 3 N atom an energy V ( x 1 , ..., x N atom ) . The dimension d = 3 N atom is large (a few hundred thousand to millions).

Introduction Free energy Thermodynamic integration Adaptive biasing techniques Empirical force field Typically, V is a sum of potentials modelling interaction between two particles, three particles and four particles: � � � V = V 1 ( x i , x j ) + V 2 ( x i , x j , x k ) + V 3 ( x i , x j , x k , x l ) . i < j i < j < k i < j < k < l 5 ǫ = 1 , σ = 1 For example, 4 V 1 ( x i , x j ) = V LJ ( | x i − x j | ) 3 V LJ ( r ) where 2 � 12 − � σ �� σ � 6 � V LJ ( r ) = 4 ǫ is 1 r r the Lennard-Jones potential. 0 -1 0.6 0.8 1 1.2 1.4 1.6 1.8 2 r

Introduction Free energy Thermodynamic integration Adaptive biasing techniques Dynamics Newton equations of motion: � d X t = M − 1 P t dt d P t = −∇ V ( X t ) dt

Introduction Free energy Thermodynamic integration Adaptive biasing techniques Dynamics Newton equations of motion + thermostat: Langevin dynamics: � d X t = M − 1 P t dt d P t = −∇ V ( X t ) dt − γ M − 1 P t dt + � 2 γβ − 1 d W t where γ > 0. Langevin dynamics is ergodic wrt � � − β p t M − 1 p µ ( d x ) ⊗ Z − 1 exp d p with p 2 d µ = Z − 1 exp ( − β V ( x )) d x � where Z = exp ( − β V ( x )) d x is the partition function and β = ( k B T ) − 1 is proportional to the inverse of the temperature.

Introduction Free energy Thermodynamic integration Adaptive biasing techniques Dynamics Newton equations of motion + thermostat: Langevin dynamics: � d X t = M − 1 P t dt d P t = −∇ V ( X t ) dt − γ M − 1 P t dt + � 2 γβ − 1 d W t where γ > 0. Langevin dynamics is ergodic wrt � � − β p t M − 1 p µ ( d x ) ⊗ Z − 1 exp d p with p 2 d µ = Z − 1 exp ( − β V ( x )) d x � where Z = exp ( − β V ( x )) d x is the partition function and β = ( k B T ) − 1 is proportional to the inverse of the temperature. In the following, we focus on the overdamped Langevin (or gradient) dynamics � 2 β − 1 d W t , d X t = −∇ V ( X t ) dt + which is also ergodic wrt µ .

Introduction Free energy Thermodynamic integration Adaptive biasing techniques Macroscopic quantities of interest These dynamics are used to compute macroscopic quantities: (i) Thermodynamic quantities (averages wrt µ of some observables): stress, heat capacity, free energy,... � T � R d ϕ ( x ) µ ( d x ) ≃ 1 E µ ( ϕ ( X )) = ϕ ( X t ) dt . T 0 (ii) Dynamical quantities (averages over trajectories): diffusion coefficients, viscosity, transition rates,... M E ( F (( X t ) t ≥ 0 )) ≃ 1 � F (( X m t ) t ≥ 0 ) . M m = 1 Difficulties: (i) high-dimensional problem ( N ≫ 1); (ii) X t is a metastable process and µ is a multimodal measure.

Introduction Free energy Thermodynamic integration Adaptive biasing techniques Metastability: energetic and entropic barriers A two-dimensional schematic picture 2 2.0 1.5 1.5 1 1.0 0.5 0.5 0 0.0 x y -0.5 -0.5 -1 -1.0 -1.5 -1.5 -2 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 0 2000 4000 6000 8000 10000 x Iterations 3 3 2 2 1 1 0 0 x y -1 -1 -2 -2 -3 -3 -6 -4 -2 0 2 4 6 0 2000 4000 6000 8000 10000 x Iterations − → • Slow convergence of trajectorial averages • Transitions between metastable states are rare events

Introduction Free energy Thermodynamic integration Adaptive biasing techniques A toy model for solvation Influence of the solvation on a dimer conformation [Dellago, Geissler] . �� Compact state. Stretched state. The particles interact through a pair potential: truncated LJ for all particles except the two monomers (black particles) which interact through a double-well potential. A slow variable is the distance between the two monomers.

Introduction Free energy Thermodynamic integration Adaptive biasing techniques A toy example in material sciences The 7 atoms Lennard Jones cluster in 2D. (a) C 0 , V = − 12 . 53 (b) C 1 , V = − 11 . 50 (c) C 2 , V = − 11 . 48 (d) C 3 , V = − 11 . 40 Figure: Low energy conformations of the Lennard-Jones cluster. − → simulation

Introduction Free energy Thermodynamic integration Adaptive biasing techniques Simulations of biological systems Unbinding of a ligand from a protein (Diaminopyridine-HSP90, Courtesy of SANOFI) Elementary time-step for the molecular dynamics = 10 − 15 s Dissociation time = 0 . 5 s Challenge: bridge the gap between timescales

SDEs in large dimension and numerical methods Part 1: Sampling the - PowerPoint PPT Presentation

Introduction Free energy Thermodynamic integration Adaptive biasing techniques SDEs in large dimension and numerical methods Part 1: Sampling the canonical distribution T. Lelivre CERMICS - Ecole des Ponts ParisTech & Matherials

SDEs in large dimension and numerical methods Part 2: Sampling metastable dynamics T. Lelivre

Scientific Computing I Part I Module 4: Numerical Methods for ODE Basic Numerical Methods

Dimension reduction numerical methods for Bermudan options Scott Sues Probability, Numerics, and

Numerical Methods for Solving Large Scale Eigenvalue Problems Lecture 2, February 28, 2018:

Scientific Computing I Methods Module 4: Numerical Methods for ODE Michael Bader Lehrstuhl

Scientific Computing I Methods Module 4: Numerical Methods for ODE Michael Bader Lehrstuhl

Large deviation principle for SDEs with discontinuous coefficients D. Sobolieva 1 1 Department

A second order discretization and efficient simulation for Backward SDEs Konstantinos Manolarakis

Computing the quasipotential for nongradient SDEs in 3D Samuel F. Potter Joint work with Maria

Bayesian Probabilistic Numerical Methods (Part I) Chris. J. Oates Newcastle University Alan

Numerical methods for dynamical systems Alexandre Chapoutot ENSTA Paris master CPS IP Paris

Numerical methods for dynamical systems Alexandre Chapoutot ENSTA Paris master CPS IP Paris

Numerical Methods for Large-Scale Ill-Posed Inverse Problems Julianne Chung University of

Numerical Integration and Monte Carlo Integration Elementary schemes for 1D integrals with no

Numerical Methods for Linear Discrete Ill-Posed Problems Lothar Reichel Como, May 2018 Part 1:

Numerical methods for dynamical systems Alexandre Chapoutot ENSTA Paris master CPS IP Paris

Bridge numbers of links and minimal numbers of meridian generators of link groups Yeonhee JANG

The dual space of precompact groups Salvador Hern andez Universitat Jaume I The dual space of

Van Kampen diagrams are bicolimits Pawel Sobocinski, University of Southampton (joint work with

Homotopy groups, braids and links Jie Wu National University of Singapore Novosibirsk workshop

The Phoibos Analyzer Series Electron Spectrometers for Hard x-ray Photoemission Spectroscopy

Higher Structures in Algebraic Quantum Field Theory Alexander Schenkel School of Mathematical

Programming and Proving with Higher Inductive Types Dan Licata Wesleyan University Department

Graph rewriting with cloning Dominique Duval based on work with Rachid Echahed and Fr ed