55 research outputs found
A Function Space HMC Algorithm With Second Order Langevin Diffusion Limit
We describe a new MCMC method optimized for the sampling of probability
measures on Hilbert space which have a density with respect to a Gaussian; such
measures arise in the Bayesian approach to inverse problems, and in conditioned
diffusions. Our algorithm is based on two key design principles: (i) algorithms
which are well-defined in infinite dimensions result in methods which do not
suffer from the curse of dimensionality when they are applied to approximations
of the infinite dimensional target measure on \bbR^N; (ii) non-reversible
algorithms can have better mixing properties compared to their reversible
counterparts. The method we introduce is based on the hybrid Monte Carlo
algorithm, tailored to incorporate these two design principles. The main result
of this paper states that the new algorithm, appropriately rescaled, converges
weakly to a second order Langevin diffusion on Hilbert space; as a consequence
the algorithm explores the approximate target measures on \bbR^N in a number
of steps which is independent of . We also present the underlying theory for
the limiting non-reversible diffusion on Hilbert space, including
characterization of the invariant measure, and we describe numerical
simulations demonstrating that the proposed method has favourable mixing
properties as an MCMC algorithm.Comment: 41 pages, 2 figures. This is the final version, with more comments
and an extra appendix adde
On the study of slow-fast dynamics, when the fast process has multiple invariant measures
Motivated by applications to mathematical biology, we study the averaging
problem for slow-fast systems, {\em in the case in which the fast dynamics is a
stochastic process with multiple invariant measures}. We consider both the case
in which the fast process is decoupled from the slow process and the case in
which the two components are fully coupled. We work in the setting in which the
slow process evolves according to an Ordinary Differential Equation (ODE) and
the fast process is a continuous time Markov Process with finite state space
and show that, in this setting, the limiting (averaged) dynamics can be
described as a random ODE (that is, an ODE with random coefficients.)
Keywords. Multiscale methods, Processes with multiple equilibria, Averaging,
Collective Navigation, Interacting Piecewise Deterministic Markov Processes.Comment: 24 page
Non-mean-field Vicsek-type models for collective behavior
We consider interacting particle dynamics with Vicsek-type interactions, and their macroscopic Partial Differential Equation (PDE) limit, in the non-mean-field regime; that is, we consider the case in which each particle/agent in the system interacts only with a prescribed subset of the particles in the system (for example, those within a certain distance). In this non-mean-field regime the influence between agents (i.e. the interaction term) can be normalized either by the total number of agents in the system (global scaling) or by the number of agents with which the particle is effectively interacting (local scaling). We compare the behavior of the globally scaled and the locally scaled systems in many respects, considering for each scaling both the PDE and the corresponding particle model. In particular, we observe that both the locally and globally scaled particle system exhibit pattern formation (i.e. formation of traveling-wave-like solutions) within certain parameter regimes, and generally display similar dynamics. The same is not true of the corresponding PDE models. Indeed, while both PDE models have multiple stationary states, for the globally scaled PDE such (space-homogeneous) equilibria are unstable for certain parameter regimes, with the instability leading to traveling wave solutions, while they are always stable for the locally scaled one, which never produces traveling waves. This observation is based on a careful numerical study of the model, supported by further analysis
Asymptotic analysis for the generalized langevin equation
Various qualitative properties of solutions to the generalized Langevin
equation (GLE) in a periodic or a confining potential are studied in this
paper. We consider a class of quasi-Markovian GLEs, similar to the model that
was introduced in \cite{EPR99}. Geometric ergodicity, a homogenization theorem
(invariance principle), short time asymptotics and the white noise limit are
studied. Our proofs are based on a careful analysis of a hypoelliptic operator
which is the generator of an auxiliary Markov process. Systematic use of the
recently developed theory of hypocoercivity \cite{Vil04HPI} is made.Comment: 27 pages, no figures. Submitted to Nonlinearity
A Function Space HMC Algorithm With Second Order Langevin Diffusion Limit
We describe a new MCMC method optimized for the sampling of probability measures on Hilbert space which have a density with respect to a Gaussian; such measures arise in the Bayesian approach to inverse problems, and in conditioned diffusions. Our algorithm is based on two key design principles: (i) algorithms which are well defined in infinite dimensions result in methods which do not suffer from the curse of dimensionality when they are applied to approximations of the infinite dimensional target measure on R^N; (ii) nonreversible algorithms can have better mixing properties compared to their reversible counterparts. The method we introduce is based on the hybrid Monte Carlo algorithm, tailored to incorporate these two design principles. The main result of this paper states that the new algorithm, appropriately rescaled, converges weakly to a second order Langevin diffusion on Hilbert space; as a consequence the algorithm explores the approximate target measures on R^N in a number of steps which is independent of N. We also present the underlying theory for the limiting nonreversible diffusion on Hilbert space, including characterization of the invariant measure, and we describe numerical simulations demonstrating that the proposed method has favourable mixing properties as an MCMC algorithm
Ergodic properties of quasi-Markovian generalized Langevin equations with configuration dependent noise and non-conservative force
We discuss the ergodic properties of quasi-Markovian stochastic differential
equations, providing general conditions that ensure existence and uniqueness of
a smooth invariant distribution and exponential convergence of the evolution
operator in suitably weighted spaces, which implies the validity
of central limit theorem for the respective solution processes. The main new
result is an ergodicity condition for the generalized Langevin equation with
configuration-dependent noise and (non-)conservative force
Advanced glycation end product cross-link breaker attenuates diabetes-induced cardiac dysfunction by improving sarcoplasmic reticulum calcium handling
Diabetic heart disease is a distinct clinical entity that can progress to heart failure and sudden death. However, the mechanisms responsible for the alterations in excitation-contraction coupling leading to cardiac dysfunction during diabetes are not well known. Hyperglycemia, the landmark of diabetes, leads to the formation of advanced glycation end products (AGEs) on long-lived proteins, including sarcoplasmic reticulum (SR) Ca2+ regulatory proteins. However, their pathogenic role on SR Ca2+ handling in cardiac myocytes is unknown. Therefore, we investigated whether an AGE cross-link breaker could prevent the alterations in SR Ca2+ cycling that lead to in vivo cardiac dysfunction during diabetes. Streptozotocin-induced diabetic rats were treated with alagebrium chloride (ALT-711) for 8 weeks and compared to age-matched placebo-treated diabetic rats and healthy rats. Cardiac function was assessed by echocardiographic examination. Ventricular myocytes were isolated to assess SR Ca2+ cycling by confocal imaging and quantitative Western blots. Diabetes resulted in in vivo cardiac dysfunction and ALT-711 therapy partially alleviated diastolic dysfunction by decreasing isovolumetric relaxation time and myocardial performance index (MPI) (by 27 and 41% vs. untreated diabetic rats, respectively, P < 0.05). In cardiac myocytes, diabetes-induced prolongation of cytosolic Ca2+ transient clearance by 43% and decreased SR Ca2+ load by 25% (P < 0.05); these parameters were partially improved after ALT-711 therapy. SERCA2a and RyR2 protein expression was significantly decreased in the myocardium of untreated diabetic rats (by 64 and 36% vs. controls, respectively, P < 0.05), but preserved in the treated diabetic group compared to controls. Collectively, our results suggest that, in a model of type 1 diabetes, AGE accumulation primarily impairs SR Ca2+ reuptake in cardiac myocytes and that long-term treatment with an AGE cross-link breaker partially normalized SR Ca2+ handling and improved diabetic cardiomyopathy.Peer reviewedPhysiological Science
Using perturbed underdamped langevin dynamics to efficiently sample from probability distributions
In this paper we introduce and analyse Langevin samplers that consist of perturbations of the standard underdamped Langevin dynamics. The perturbed dynamics is such that its invariant measure is the same as that of the unperturbed dynamics. We show that appropriate choices of the perturbations can lead to samplers that have improved properties, at least in terms of reducing the asymptotic variance. We present a detailed analysis of the new Langevin sampler for Gaussian target distributions. Our theoretical results are supported by numerical experiments with non-Gaussian target measures
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