33 research outputs found
Long time asymptotics of a Brownian particle coupled with a random environment with non-diffusive feedback force
We study the long time behavior of a Brownian particle moving in an
anomalously diffusing field, the evolution of which depends on the particle
position. We prove that the process describing the asymptotic behaviour of the
Brownian particle has bounded (in time) variance when the particle interacts
with a subdiffusive field; when the interaction is with a superdiffusive field
the variance of the limiting process grows in time as t^{2{\gamma}-1}, 1/2 <
{\gamma} < 1. Two different kinds of superdiffusing (random) environments are
considered: one is described through the use of the fractional Laplacian; the
other via the Riemann-Liouville fractional integral. The subdiffusive field is
modeled through the Riemann-Liouville fractional derivative.Comment: 45 page
Asymptotic Analysis for Markovian models in non-equilibrium Statistical Mechanics
This thesis is mainly concerned with the problem of exponential convergence to equilibrium
for open classical systems. We consider a model of a small Hamiltonian system coupled to
a heat reservoir, which is described by the Generalized Langevin Equation (GLE) and we
focus on a class of Markovian approximations to the GLE. The generator of these Markovian
dynamics is an hypoelliptic non-selfadjoint operator. We look at the problem of exponential
convergence to equilibrium by using and comparing three different approaches: classic ergodic
theory, hypocoercivity theory and semiclassical analysis (singular space theory). In particular,
we describe a technique to easily determine the spectrum of quadratic hypoelliptic operators
(which are in general non-selfadjoint) and hence obtain the exact rate of convergence to
equilibrium
Bounding stationary averages of polynomial diffusions via semidefinite programming
We introduce an algorithm based on semidefinite programming that yields
increasing (resp. decreasing) sequences of lower (resp. upper) bounds on
polynomial stationary averages of diffusions with polynomial drift vector and
diffusion coefficients. The bounds are obtained by optimising an objective,
determined by the stationary average of interest, over the set of real vectors
defined by certain linear equalities and semidefinite inequalities which are
satisfied by the moments of any stationary measure of the diffusion. We
exemplify the use of the approach through several applications: a Bayesian
inference problem; the computation of Lyapunov exponents of linear ordinary
differential equations perturbed by multiplicative white noise; and a
reliability problem from structural mechanics. Additionally, we prove that the
bounds converge to the infimum and supremum of the set of stationary averages
for certain SDEs associated with the computation of the Lyapunov exponents, and
we provide numerical evidence of convergence in more general settings
Uniform in time convergence of numerical schemes for stochastic differential equations via Strong Exponential stability: Euler methods, Split-Step and Tamed Schemes
We prove a general criterion providing sufficient conditions under which a
time-discretiziation of a given Stochastic Differential Equation (SDE) is a
uniform in time approximation of the SDE. The criterion is also, to a certain
extent, discussed in the paper, necessary. Using such a criterion we then
analyse the convergence properties of numerical methods for solutions of SDEs;
we consider Explicit and Implicit Euler, split-step and (truncated) tamed Euler
methods. In particular, we show that, under mild conditions on the coefficients
of the SDE (locally Lipschitz and strictly monotonic), these methods produce
approximations of the law of the solution of the SDE that converge uniformly in
time. The theoretical results are verified by numerical examples.Comment: 50 pages, 2 figure
Non-stationary phase of the MALA algorithm
The Metropolis-Adjusted Langevin Algorithm (MALA) is a Markov Chain Monte
Carlo method which creates a Markov chain reversible with respect to a given
target distribution, pi^N, with Lebesgue density on R^N; it can hence be used
to approximately sample the target distribution. When the dimension N is large
a key question is to determine the computational cost of the algorithm as a
function of N. One approach to this question, which we adopt here, is to derive
diffusion limits for the algorithm. The family of target measures that we
consider in this paper are, in general, in non-product form and are of interest
in applied problems as they arise in Bayesian nonparametric statistics and in
the study of conditioned diffusions. Furthermore, we study the situation, which
arises in practice, where the algorithm is started out of stationarity. We
thereby significantly extend previous works which consider either only measures
of product form, when the Markov chain is started out of stationarity, or
measures defined via a density with respect to a Gaussian, when the Markov
chain is started in stationarity. We prove that, in the non-stationary regime,
the computational cost of the algorithm is of the order N^(1/2) with dimension,
as opposed to what is known to happen in the stationary regime, where the cost
is of the order N^(1/3).Comment: 37 pages. arXiv admin note: text overlap with arXiv:1405.489
Optimal scaling of the MALA algorithm with irreversible proposals for Gaussian targets
It is well known in many settings that reversible Langevin diffusions in
confining potentials converge to equilibrium exponentially fast. Adding
irreversible perturbations to the drift of a Langevin diffusion that maintain
the same invariant measure accelerates its convergence to stationarity. Many
existing works thus advocate the use of such non-reversible dynamics for
sampling. When implementing Markov Chain Monte Carlo algorithms (MCMC) using
time discretisations of such Stochastic Differential Equations (SDEs), one can
append the discretization with the usual Metropolis-Hastings accept-reject step
and this is often done in practice because the accept--reject step eliminates
bias. On the other hand, such a step makes the resulting chain reversible. It
is not known whether adding the accept-reject step preserves the faster mixing
properties of the non-reversible dynamics. In this paper, we address this gap
between theory and practice by analyzing the optimal scaling of MCMC algorithms
constructed from proposal moves that are time-step Euler discretisations of an
irreversible SDE, for high dimensional Gaussian target measures. We call the
resulting algorithm the \imala, in comparison to the classical MALA algorithm
(here {\em ip} is for irreversible proposal). In order to quantify how the cost
of the algorithm scales with the dimension , we prove invariance principles
for the appropriately rescaled chain. In contrast to the usual MALA algorithm,
we show that there could be two regimes asymptotically: (i) a diffusive regime,
as in the MALA algorithm and (ii) a ``fluid" regime where the limit is an
ordinary differential equation. We provide concrete examples where the limit is
a diffusion, as in the standard MALA, but with provably higher limiting
acceptance probabilities. Numerical results are also given corroborating the
theory
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
Diffusion Limit for the Random Walk Metropolis Algorithm out of stationarity
The Random Walk Metropolis (RWM) algorithm is a Metropolis–Hastings Markov Chain Monte Carlo algorithm designed to sample from a given target distribution π^N with Lebesgue density on R^N. Like any other Metropolis–Hastings algorithm, RWM constructs a Markov chain by randomly proposing a new position (the “proposal move”), which is then accepted or rejected according to a rule which makes the chain reversible with respect to π^N. When the dimension N is large, a key question is to determine the optimal scaling with N of the proposal variance: if the proposal variance is too large, the algorithm will reject the proposed moves too often; if it is too small, the algorithm will explore the state space too slowly. Determining the optimal scaling of the proposal variance gives a measure of the cost of the algorithm as well. One approach to tackle this issue, which we adopt here, is to derive diffusion limits for the algorithm. Such an approach has been proposed in the seminal papers (Ann. Appl. Probab. 7 (1) (1997) 110–120; J. R. Stat. Soc. Ser. B. Stat. Methodol. 60 (1) (1998) 255–268). In particular, in (Ann. Appl. Probab. 7 (1) (1997) 110–120) the authors derive a diffusion limit for the RWM algorithm under the two following assumptions: (i) the algorithm is started in stationarity; (ii) the target measure π^N is in product form. The present paper considers the situation of practical interest in which both assumptions (i) and (ii) are removed. That is (a) we study the case (which occurs in practice) in which the algorithm is started out of stationarity and (b) we consider target measures which are in non-product form. Roughly speaking, we consider target measures that admit a density with respect to Gaussian; such measures arise in Bayesian nonparametric statistics and in the study of conditioned diffusions. We prove that, out of stationarity, the optimal scaling for the proposal variance is O(N^(−1)), as it is in stationarity. In this optimal scaling, a diffusion limit is obtained and the cost of reaching and exploring the invariant measure scales as O(N). Notice that the optimal scaling in and out of stationatity need not be the same in general, and indeed they differ e.g. in the case of the MALA algorithm (Stoch. Partial Differ. Equ. Anal Comput. 6 (3) (2018) 446–499). More importantly, our diffusion limit is given by a stochastic PDE, coupled to a scalar ordinary differential equation; such an ODE gives a measure of how far from stationarity the process is and can therefore be taken as an indicator of convergence. In this sense, this paper contributes understanding to the old-standing problem of monitoring convergence of MCMC algorithms