arXiv subjects

G. A. Pavliotis

Publications and source records attributed to G. A. Pavliotis.

At least 37 records · Page 2Linked to original sources

arXiv2016

Noise-induced transitions in rugged energy landscapes

We consider the problem of an overdamped Brownian particle moving in multiscale potential with N + 1 characteristic length scales: the macroscale and N separated microscales. We show that the coarse-grained dynamics is given by an overdamped Langevin equation with respect to the free energy and with a space dependent diffusion tensor, the calculation of which requires the solution of N fully coupled Poisson equations. We study in detail the structure of the bifurcation diagram for one-dimensional problems and we show that the multiscale structure in the potential leads to hysteresis effects and to noise-induced transitions. Furthermore, we obtain an explicit formula for the effective diffusion coefficient for a self-similar separable potential and we investigate the limit of infinitely many small scales.

cond-mat.stat-mech↗

arXiv2015

Variance Reduction using Nonreversible Langevin Samplers

A standard approach to computing expectations with respect to a given target measure is to introduce an overdamped Langevin equation which is reversible with respect to the target distribution, and to approximate the expectation by a time-averaging estimator. As has been noted in recent papers, introducing an appropriately chosen nonreversible component to the dynamics is beneficial, both in terms of reducing the asymptotic variance and of speeding up convergence to the target distribution. In this paper we present a detailed study of the dependence of the asymptotic variance on the deviation from reversibility. Our theoretical findings are supported by numerical simulations.

math.NA↗

arXiv2015

Efficient numerical calculation of drift and diffusion coefficients in the diffusion approximation of kinetic equations

In this paper we study the diffusion approximation of a swarming model given by a system of interacting Langevin equations with nonlinear friction. The diffusion approximation requires the calculation of the drift and diffusion coefficients that are given as averages of solutions to appropriate Poisson equations. We present a new numerical method for computing these coefficients that is based on the calculation of the eigenvalues and eigenfunctions of a Schrödinger operator. These theoretical results are supported by numerical simulations showcasing the efficiency of the method.

math.NA↗

arXiv2013

A multiscale analysis of diffusions on rapidly varying surfaces

Lateral diffusion of molecules on surfaces plays a very important role in various biological processes, including lipid transport across the cell membrane, synaptic transmission and other phenomena such as exo- and endocytosis, signal transduction, chemotaxis and cell growth. In many cases, the surfaces can possess spatial inhomogeneities and/or be rapidly changing shape. Using a generalisation of the model for a thermally excited Helfrich elastic membrane, we consider the problem of lateral diffusion on quasi-planar surfaces, possessing both spatial and temporal fluctuations. Using results from homogenisation theory, we show that, under the assumption of scale separation between the characteristic length and time scales of the membrane fluctuations and the characteristic scale of the diffusing particle, the lateral diffusion process can be well approximated by a Brownian motion on the plane with constant diffusion tensor $D$ which depends in a highly nonlinear way on the detailed properties of the surface. The effective diffusion tensor will depend on the relative scales of the spatial and temporal fluctuations and, for different scaling regimes, we prove the existence of a macroscopic limit in each case.

math.AP↗

arXiv2013

A New Mode Reduction Strategy for the Generalized Kuramoto-Sivashinsky Equation

Consider the generalized Kuramoto-Sivashinsky (gKS) equation. It is a model prototype for a wide variety of physical systems, from flame-front propagation, and more general front propagation in reaction-diffusion systems, to interface motion of viscous film flows. Our aim is to develop a systematic and rigorous low-dimensional representation of the gKS equation. For this purpose, we approximate it by a renormalization group (RG) equation which is qualitatively characterized by rigorous error bounds. This formulation allows for a new stochastic mode reduction guaranteeing optimality in the sense of maximal information entropy. Herewith, noise is systematically added to the reduced gKS equation and gives a rigorous and analytical explanation for its origin. These new results would allow to reliably perform low-dimensional numerical computations by accounting for the neglected degrees of freedom in a systematic way. Moreover, the presented reduction strategy might also be useful in other applications where classical mode reduction approaches fail or are too complicated to be implemented.

math-ph↗

arXiv2013

A new stochastic mode reduction strategy for dissipative systems

We present a new methodology for studying non-Hamiltonian nonlinear systems based on an information theoretic extension of a renormalization group technique using a modified maximum entropy principle. We obtain a rigorous dimensionally reduced description for such systems. The neglected degrees of freedom by this reduction are replaced by a systematically deefined stochastic process under a constraint on the second moment. This then forms the basis of a computationally efficient method. Numerical computations for the generalized Kuramoto-Sivashinsky equation sup- port our method and reveal that the long-time underlying stochastic process of the fast (unresolved) modes obeys a universal distribution which does not depend on the initial conditions and which we rigorously derive by the maximum entropy principle.

physics.comp-ph↗

arXiv2012

Exponential return to equilibrium for hypoelliptic quadratic systems

We study the problem of convergence to equilibrium for evolution equations associated to general quadratic operators. Quadratic operators are non-selfadjoint differential operators with complex-valued quadratic symbols. Under appropriate assumptions, a complete description of the spectrum of such operators is given and the exponential return to equilibrium with sharp estimates on the rate of convergence is proven. Some applications to the study of chains of oscillators and the generalized Langevin equation are given.

math-ph↗

arXiv2012

Additive noise effects in active nonlinear spatially extended systems

We examine the effects of pure additive noise on spatially extended systems with quadratic nonlinearities. We develop a general multiscale theory for such systems and apply it to the Kuramoto-Sivashinsky equation as a case study. We first focus on a regime close to the instability onset (primary bifurcation), where the system can be described by a single dominant mode. We show analytically that the resulting noise in the equation describing the amplitude of the dominant mode largely depends on the nature of the stochastic forcing. For a highly degenerate noise, in the sense that it is acting on the first stable mode only, the amplitude equation is dominated by a pure multiplicative noise, which in turn induces the dominant mode to undergo several critical state transitions and complex phenomena, including intermittency and stabilisation, as the noise strength is increased. The intermittent behaviour is characterised by a power-law probability density and the corresponding critical exponent is calculated rigorously by making use of the first-passage properties of the amplitude equation. On the other hand, when the noise is acting on the whole subspace of stable modes, the multiplicative noise is corrected by an additive-like term, with the eventual loss of any stabilised state. We also show that the stochastic forcing has no effect on the dominant mode dynamics when it is acting on the second stable mode. Finally, in a regime which is relatively far from the instability onset, so that there are two unstable modes, we observe numerically that when the noise is acting on the first stable mode, both dominant modes show noise-induced complex phenomena similar to the single-mode case.

cond-mat.stat-mech↗

arXiv2012

Noise induced state transitions, intermittency and universality in the noisy Kuramoto-Sivashinsky equation

We analyze the effect of pure additive noise on the long-time dynamics of the noisy Kuramoto-Sivashinsky (KS) equation in a regime close to the instability onset. We show that when the noise is highly degenerate, in the sense that it acts only on the first stable mode, the solution of the KS equation undergoes several transitions between different states, including a critical on-off intermittent state that is eventually stabilized as the noise strength is increased. Such noise-induced transitions can be completely characterized through critical exponents, obtaining that both the KS and the noisy Burgers equation belong to the same universality class. The results of our numerical investigations are explained rigorously using multiscale techniques.

cond-mat.stat-mech↗

arXiv2012

Corrections to Einstein's relation for Brownian motion in a tilted periodic potential

In this paper we revisit the problem of Brownian motion in a tilted periodic potential. We use homogenization theory to derive general formulas for the effective velocity and the effective diffusion tensor that are valid for arbitrary tilts. Furthermore, we obtain power series expansions for the velocity and the diffusion coefficient as functions of the external forcing. Thus, we provide systematic corrections to Einstein's formula and to linear response theory. Our theoretical results are supported by extensive numerical simulations. For our numerical experiments we use a novel spectral numerical method that leads to a very efficient and accurate calculation of the effective velocity and the effective diffusion tensor.

math-ph↗

arXiv2011

Numerical methods for stochastic partial differential equations with multiples scales

A new method for solving numerically stochastic partial differential equations (SPDEs) with multiple scales is presented. The method combines a spectral method with the heterogeneous multiscale method (HMM) presented in [W. E, D. Liu, and E. Vanden-Eijnden, Comm. Pure Appl. Math., 58(11):1544--1585, 2005]. The class of problems that we consider are SPDEs with quadratic nonlinearities that were studied in [D. Blomker, M. Hairer, and G.A. Pavliotis, Nonlinearity, 20(7):1721--1744, 2007.] For such SPDEs an amplitude equation which describes the effective dynamics at long time scales can be rigorously derived for both advective and diffusive time scales. Our method, based on micro and macro solvers, allows to capture numerically the amplitude equation accurately at a cost independent of the small scales in the problem. Numerical experiments illustrate the behavior of the proposed method.

math.NA↗

arXiv2010

Multiscale Modelling and Inverse Problems

The need to blend observational data and mathematical models arises in many applications and leads naturally to inverse problems. Parameters appearing in the model, such as constitutive tensors, initial conditions, boundary conditions, and forcing can be estimated on the basis of observed data. The resulting inverse problems are often ill-posed and some form of regularization is required. These notes discuss parameter estimation in situations where the unknown parameters vary across multiple scales. We illustrate the main ideas using a simple model for groundwater flow. We will highlight various approaches to regularization for inverse problems, including Tikhonov and Bayesian methods. We illustrate three ideas that arise when considering inverse problems in the multiscale context. The first idea is that the choice of space or set in which to seek the solution to the inverse problem is intimately related to whether a homogenized or full multiscale solution is required. This is a choice of regularization. The second idea is that, if a homogenized solution to the inverse problem is what is desired, then this can be recovered from carefully designed observations of the full multiscale system. The third idea is that the theory of homogenization can be used to improve the estimation of homogenized coefficients from multiscale data.

math.ST↗

arXiv2010

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.

math-ph↗

arXiv2010

Asymptotic Analysis of the Green-Kubo Formula

A detailed study of various distinguished limits of the Green-Kubo formula for the self-diffusion coefficient is presented in this paper. First, an alternative representation of the Green-Kubo formula in terms of the solution of a Poisson equation is derived when the microscopic dynamics is Markovian. Then, the techniques developed in \cite{golden2, AvelMajda91} are used to obtain a Stieltjes integral representation formula for the symmetric and antisymmetric parts of the diffusion tensor. The effect of irreversible microscopic dynamics on the diffusion coefficient is analyzed and various asymptotic limits of physical interest are studied. Several examples are presented that confirm the findings of our theory.

math-ph↗

arXiv2009

Estimating eddy diffusivities from noisy Lagrangian observations

The problem of estimating the eddy diffusivity from Lagrangian observations in the presence of measurement error is studied in this paper. We consider a class of incompressible velocity fields for which is can be rigorously proved that the small scale dynamics can be parameterised in terms of an eddy diffusivity tensor. We show, by means of analysis and numerical experiments, that subsampling of the data is necessary for the accurate estimation of the eddy diffusivity. The optimal sampling rate depends on the detailed properties of the velocity field. Furthermore, we show that averaging over the data only marginally reduces the bias of the estimator due to the multiscale structure of the problem, but that it does significantly reduce the effect of observation error.

math-ph↗

arXiv2008

Calculating Effective Diffusivities in the Limit of Vanishing Molecular Diffusion

In this paper we study the problem of the numerical calculation (by Monte Carlo Methods) of the effective diffusivity for a particle moving in a periodic divergent-free velocity filed, in the limit of vanishing molecular diffusion. In this limit traditional numerical methods typically fail, since they do not represent accurately the geometry of the underlying deterministic dynamics. We propose a stochastic splitting method that takes into account the volume preserving property of the equations motion in the absence of noise, and when inertial effects can be neglected. An extension of the method is then proposed for the cases where the noise has a non trivial time-correlation structure and when inertial effects cannot be neglected. Modified equations are used to perform backward error analysis. The new stochastic geometric integrators are shown to outperform standard Euler-based integrators. Various asymptotic limits of physical interest are investigated by means of numerical experiments, using the new integrators.

math.NA↗

arXiv2008

Maximum Likelihood Drift Estimation for Multiscale Diffusions

We study the problem of parameter estimation using maximum likelihood for fast/slow systems of stochastic differential equations. Our aim is to shed light on the problem of model/data mismatch at small scales. We consider two classes of fast/slow problems for which a closed coarse-grained equation for the slow variables can be rigorously derived, which we refer to as averaging and homogenization problems. We ask whether, given data from the slow variable in the fast/slow system, we can correctly estimate parameters in the drift of the coarse-grained equation for the slow variable, using maximum likelihood. We show that, whereas the maximum likelihood estimator is asymptotically unbiased for the averaging problem, for the homogenization problem maximum likelihood fails unless we subsample the data at an appropriate rate. An explicit formula for the asymptotic error in the log likelihood function is presented. Our theory is applied to two simple examples from molecular dynamics.

math.ST↗

arXiv2008

Diffusive Transport in Periodic Potentials: Underdamped Dynamics

In this paper we present a systematic and rigorous method for calculating the diffusion tensor for a Brownian particle moving in a periodic potential which is valid in arbitrary dimensions and for all values of the dissipation. We use this method to obtain an explicit formula for the diffusion coefficient in one dimension which is valid in the underdamped limit, and we also obtain higher order corrections to the Lifson-Jackson formula for the diffusion coefficient in the overdamped limit. A numerical method for calculating the diffusion coefficient is also developed and is shown to perform extremely well for all values of the dissipation.

cond-mat.stat-mech↗