Academic literature on the topic 'Underdamped Langevin diffusion'

Abstract The diffusion behavior of particles moving in complex heterogeneous environment is a very topical issue. We characterize particle’s trajectory via an underdamped Langevin system driven by a Gaussian white noise with a time dependent diffusivity of velocity, together with a random relaxation timescale τ to parameterize the effect of complex medium. We mainly concern how the random parameter τ influences the diffusion behavior and ergodic property of this Langevin system. Besides, the comparison between the fixed and random initial velocity v 0 is conducted to show the effect of different initial ensembles. The heavy-tailed distribution of τ with finite mean is found to suppress the decay rate of the velocity correlation function and promote the diffusion behavior, playing a competition role to the time dependent diffusivity. More interestingly, a random v 0 with a specific distribution depending on random τ also enhances the diffusion. Both the random parameters τ and v 0 influence the dynamics of the Langevin system in a non-obvious way, which cannot be ignored even they have finite moments.

Langevin dynamics (LD) has been extensively studied theoretically and practically as a basic sampling technique. Recently, the incorporation of non-reversible dynamics into LD is attracting attention because it accelerates the mixing speed of LD. Popular choices for non-reversible dynamics include underdamped Langevin dynamics (ULD), which uses second-order dynamics and perturbations with skew-symmetric matrices. Although ULD has been widely used in practice, the application of skew acceleration is limited although it is expected to show superior performance theoretically. Current work lacks a theoretical understanding of issues that are important to practitioners, including the selection criteria for skew-symmetric matrices, quantitative evaluations of acceleration, and the large memory cost of storing skew matrices. In this study, we theoretically and numerically clarify these problems by analyzing acceleration focusing on how the skew-symmetric matrix perturbs the Hessian matrix of potential functions. We also present a practical algorithm that accelerates the standard LD and ULD, which uses novel memory-efficient skew-symmetric matrices under parallel-chain Monte Carlo settings.

The reduction of high-dimensional systems to effective models on a smaller set of variables is an essential task in many areas of science. For stochastic dynamics governed by diffusion processes, a general procedure to find effective equations is the conditioning approach. In this paper, we are interested in the spectrum of the generator of the resulting effective dynamics, and how it compares to the spectrum of the full generator. We prove a new relative error bound in terms of the eigenfunction approximation error for reversible systems. We also present numerical examples indicating that, if Kramers–Moyal (KM) type approximations are used to compute the spectrum of the reduced generator, it seems largely insensitive to the time window used for the KM estimators. We analyze the implications of these observations for systems driven by underdamped Langevin dynamics, and show how meaningful effective dynamics can be defined in this setting.

Cette thèse s'intéresse à l'analyse et la conception de méthodes de Monte Carlo par chaine de Markov (MCMC) utilisées dans l'échantillonnage en grande dimension. Elle est constituée de trois parties.La première introduit une nouvelle classe de chaînes de Markov et méthodes MCMC. Celles-ci permettent d'améliorer des méthodes MCMC à l'aide d'échantillons visant une restriction de la loi cible originale sur un domaine choisi par l'utilisateur. Cette procédure donne naissance à une nouvelle chaîne qui tire au mieux parti des propriétés de convergences des deux processus qui lui sont sous-jacents. En plus de montrer que cette chaîne vise toujours la mesure cible originale, nous établissons également des propriétés d'ergodicité sous des hypothèses faibles sur les noyaux de Markov mis en jeu.La seconde partie de ce document s'intéresse aux discrétisations de la diffusion de Langevin sous-amortie. Cette diffusion ne pouvant être calculée explicitement en général, il est classique de considérer des discrétisations. Cette thèse établie pour une large classe de discrétisations une condition de minoration uniforme en le pas de temps. Avec des hypothèses supplémentaires sur le potentiel, cela permet de montrer que ces discrétisations convergent géométriquement vers leur unique mesure de probabilité invariante en V-norme.La dernière partie étudie l'algorithme de Langevin non ajusté dans le cas où le gradient du potentiel est connu à une erreur uniformément bornée près. Cette partie fournie des bornes en V-norme et en distance de Wasserstein entre les itérations de l'algorithme avec le gradient exact et celle avec le gradient approché. Pour ce faire il est introduit une chaine de Markov auxiliaire qui borne la différence. Il est établi que cette chaîne auxiliaire converge en loi vers un processus dit collant déjà étudié dans la littérature pour la version continue de ce problème
This thesis focuses on the analysis and design of Markov chain Monte Carlo (MCMC) methods used in high-dimensional sampling. It consists of three parts.The first part introduces a new class of Markov chains and MCMC methods. These methods allow to improve MCMC methods by using samples targeting a restriction of the original target distribution on a domain chosen by the user. This procedure gives rise to a new chain that takes advantage of the convergence properties of the two underlying processes. In addition to showing that this chain always targets the original target measure, we also establish ergodicity properties under weak assumptions on the Markov kernels involved.The second part of this thesis focuses on discretizations of the underdamped Langevin diffusion. As this diffusion cannot be computed explicitly in general, it is classical to consider discretizations. This thesis establishes for a large class of discretizations a condition of uniform minimization in the time step. With additional assumptions on the potential, it shows that these discretizations converge geometrically to their unique V-invariant probability measure.The last part studies the unadjusted Langevin algorithm in the case where the gradient of the potential is known to within a uniformly bounded error. This part provides bounds in V-norm and in Wasserstein distance between the iterations of the algorithm with the exact gradient and the one with the approximated gradient. To do this, an auxiliary Markov chain is introduced that bounds the difference. It is established that this auxiliary chain converges in distribution to sticky process already studied in the literature for the continuous version of this problem