We present a unified framework to analyze the global convergence of Langevin dynamics based algorithms for nonconvex finite-sum optimization with n component functions. At the core of our analysis is a direct analysis of the ergodicity of the numerical approximations to Langevin dynamics, which leads to faster convergence rates. Specifically, we show that gradient Langevin dynamics (GLD) and st...

Brownian Dynamics (BD), also known as Langevin Dynamics, and Dissipative Particle Dynamics (DPD) are implicit solvent methods commonly used in models of soft matter and biomolecular systems. The interaction of the numerous solvent particles with larger particles is coarse-grained as a Langevin thermostat is applied to individual particles or to particle pairs. The Langevin thermostat requires a...

Langevin equation for a nano-particle suspended in a laminar fluid flow was analytically studied. The Brownian motion generated from molecular bombardment was taken as a Wiener stochastic process and approximated by a Gaussian white noise. Euler-Maruyama method was used to solve the Langevin equation numerically. The accuracy of Brownian simulation was checked by performing a series of simulati...

This work introduces and studies a new family of velocity jump Markov processes directly amenable to exact simulation with the following two properties: (i) trajectories converge in law, when time-step parameter vanishes, towards given Langevin or Hamiltonian dynamics; (ii) stationary distribution process is always exactly by product Gaussian (for velocities) any target log-density. The itself,...

We develop a new powerful method to reproduce in silico single-molecule manipulation experiments. We demonstrate that flexible polymers such as DNA can be simulated using rigid body dynamics thanks to an original implementation of Langevin dynamics in an open source library called Open Dynamics Engine. We moreover implement a global thermostat which accelerates the simulation sampling by two or...

This chapter reviews numerical simulations of quantum field theories based on stochastic quantization and the Langevin equation. The topics discussed include renormalization of finite stepsize algorithms, Fourier acceleration, and the relation of the Langevin equation to hybrid stochastic algorithms and hybrid Monte Carlo. Invited chapter to appear in the special supplement “Stochastic Quantiza...

Deterministic dynamical systems of Langevin type recently introduced by Beck & Roepsdorf Physica 145 A (1987) 1] generate Langevin dynamics in a suitable limit. We discuss the transition from chaos to randomness in these systems under the point of view of the attractor dimension and dynamical en-tropy and show that the limit of a stochastic process is reached via a shift of the deterministic pr...

We generalize the Langevin Dynamics through the mirror descent framework for first-order sampling. The näıve approach of incorporating Brownian motion into the mirror descent dynamics, which we refer to as Symmetric Mirrored Langevin Dynamics (S-MLD), is shown to connected to the theory of Weighted Hessian Manifolds. The S-MLD, unfortunately, contains the hard instance of Cox–Ingersoll–Ross pro...

Gradient-based Monte Carlo sampling algorithms, like Langevin dynamics and Hamiltonian Monte Carlo, are important methods for Bayesian inference. In large-scale settings, full-gradients are not affordable and thus stochastic gradients evaluated on mini-batches are used as a replacement. In order to reduce the high variance of noisy stochastic gradients, [Dubey et al., 2016] applied the standard...

: We present molecular dynamics simulation of phonon thermal conductivity of semiconductor nanoribbons with an account for phonon quantum statistics. In our semiquantum molecular dynamics simulation, dynamics of the system is described with the use of classical Newtonian equations of motion where the effect of phonon quantum statistics is introduced through random Langevin-like forces with a sp...