81 results found
Nüsken N, Pavliotis GA, 2019, Constructing sampling schemes via coupling: Markov semigroups and optimal transport, SIAM/ASA Journal on Uncertainty Quantification, Vol: 7, Pages: 324-382, ISSN: 2166-2525
In this paper we develop a general framework for constructing and analyzing coupled Markov chain Monte Carlo samplers, allowing for both (possibly degenerate) diffusion and piecewise deterministic Markov processes. For many performance criteria of interest, including the asymptotic variance, the task of finding efficient couplings can be phrased in terms of problems related to optimal transport theory. We investigate general structural properties, proving a singularity theorem that has both geometric and probabilistic interpretations. Moreover, we show that those problems can often be solved approximately and support our findings with numerical experiments. For the particular objective of estimating the variance of a Bayesian posterior, our analysis suggests using novel techniques in the spirit of antithetic variates. Addressing the convergence to equilibrium of coupled processes we furthermore derive a modified Poincaré inequality.
Gomes SN, Kalliadasis S, Pavliotis GA, et al., 2019, Dynamics of the Desai-Zwanzig model in multiwell and random energy landscapes, Physical Review E, Vol: 99, ISSN: 2470-0045
We analyze a variant of the Desai-Zwanzig model [J. Stat. Phys. 19, 1 (1978)]. In particular, we study stationary states of the mean field limit for a system of weakly interacting diffusions moving in a multiwell potential energy landscape, coupled via a Curie-Weiss type (quadratic) interaction potential. The location and depth of the local minima of the potential are either deterministic or random. We characterize the structure and nature of bifurcations and phase transitions for this system, by means of extensive numerical simulations and of analytical calculations for an explicitly solvable model. Our numerical experiments are based on Monte Carlo simulations, the numerical solution of the time-dependent nonlinear Fokker-Planck (McKean-Vlasov) equation, the minimization of the free-energy functional, and a continuation algorithm for the stationary solutions.
Durán-Olivencia MA, Gvalani RS, Kalliadasis S, et al., 2019, Instability, rupture and fluctuations in thin liquid films: Theory and computations, Journal of Statistical Physics, Vol: 174, Pages: 579-604, ISSN: 0022-4715
Thin liquid films are ubiquitous in natural phenomena and technological applications. They have been extensively studied via deterministic hydrodynamic equations, but thermal fluctuations often play a crucial role that needs to be understood. An example of this is dewetting, which involves the rupture of a thin liquid film and the formation of droplets. Such a process is thermally activated and requires fluctuations to be taken into account self-consistently. In this work we present an analytical and numerical study of a stochastic thin-film equation derived from first principles. Following a brief review of the derivation, we scrutinise the behaviour of the equation in the limit of perfectly correlated noise along the wall-normal direction, as opposed to the perfectly uncorrelated limit studied by Grün et al. (J Stat Phys 122(6):1261–1291, 2006). We also present a numerical scheme based on a spectral collocation method, which is then utilised to simulate the stochastic thin-film equation. This scheme seems to be very convenient for numerical studies of the stochastic thin-film equation, since it makes it easier to select the frequency modes of the noise (following the spirit of the long-wave approximation). With our numerical scheme we explore the fluctuating dynamics of the thin film and the behaviour of its free energy in the vicinity of rupture. Finally, we study the effect of the noise intensity on the rupture time, using a large number of sample paths as compared to previous studies.
Tomlin RJ, Gomes SN, Pavliotis GA, et al., 2019, Optimal Control of Thin Liquid Films and Transverse Mode Effects, SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS, Vol: 18, Pages: 117-149, ISSN: 1536-0040
Abdulle A, Pavliotis GA, Vilmart G, 2019, Accelerated convergence to equilibrium and reduced asymptotic variance for Langevin dynamics using Stratonovich perturbations, Comptes Rendus Mathematique, ISSN: 1631-073X
© 2019 In this paper, we propose a new approach for sampling from probability measures in, possibly, high-dimensional spaces. By perturbing the standard overdamped Langevin dynamics by a suitable Stratonovich perturbation that preserves the invariant measure of the original system, we show that accelerated convergence to equilibrium and reduced asymptotic variance can be achieved, leading, thus, to a computationally advantageous sampling algorithm. The new perturbed Langevin dynamics is reversible with respect to the target probability measure and, consequently, does not suffer from the drawbacks of the nonreversible Langevin samplers that were introduced in C.-R. Hwang et al. (1993)  and studied in, e.g., T. Lelièvre et al. (2013)  and A.B. Duncan et al. (2016) , while retaining all of their advantages in terms of accelerated convergence and reduced asymptotic variance. In particular, the reversibility of the dynamics ensures that there is no oscillatory transient behaviour. The improved performance of the proposed methodology, in comparison to the standard overdamped Langevin dynamics and its nonreversible perturbation, is illustrated on an example of sampling from a two-dimensional warped Gaussian target distribution.
Schmuck M, Pavliotis GA, Kalliadasis S, 2019, Recent advances in the evolution of interfaces: Thermodynamics, upscaling, and universality, Computational Materials Science, Vol: 156, Pages: 441-451, ISSN: 0927-0256
We consider the evolution of interfaces in binary mixtures permeating strongly heterogeneous systems such as porous media. To this end, we first review available thermodynamic formulations for binary mixtures based on general reversible-irreversible couplings and the associated mathematical attempts to formulate a non-equilibrium variational principle in which these non-equilibrium couplings can be identified as minimizers. Based on this, we investigate two microscopic binary mixture formulations fully resolving heterogeneous/perforated domains: (a) a flux-driven immiscible fluid formulation without fluid flow; (b) a momentum-driven formulation for quasi-static and incompressible velocity fields. In both cases we state two novel, reliably upscaled equations for binary mixtures/multiphase fluids in strongly heterogeneous systems by systematically taking thermodynamic features such as free energies into account as well as the system's heterogeneity defined on the microscale such as geometry and materials (e.g. wetting properties). In the context of (a), we unravel a universality with respect to the coarsening rate due to its independence of the system's heterogeneity, i.e. the well-known O(t1/3)-behaviour for homogeneous systems holds also for perforated domains. Finally, the versatility of phase field equations and their thermodynamic foundation relying on free energies, make the collected recent developments here highly promising for scientific, engineering and industrial applications for which we provide an example for lithium batteries.
Duncan A, Zygalakis K, Pavliotis G, 2018, Nonreversible Langevin Samplers: Splitting Schemes, Analysis and Implementation
For a given target density, there exist an infinite number of diffusion processes which are ergodic with respect to this density. As observed in a number of papers, samplers based on nonreversible diffusion processes can significantly outperform their reversible counterparts both in terms of asymptotic variance and rate of convergence to equilibrium. In this paper, we take advantage of this in order to construct efficient sampling algorithms based on the Lie-Trotter decomposition of a nonreversible diffusion process into reversible and nonreversible components. We show that samplers based on this scheme can significantly outperform standard MCMC methods, at the cost of introducing some controlled bias. In particular, we prove that numerical integrators constructed according to this decomposition are geometrically ergodic and characterise fully their asymptotic bias and variance, showing that the sampler inherits the good mixing properties of the underlying nonreversible diffusion. This is illustrated further with a number of numerical examples ranging from highly correlated low dimensional distributions, to logistic regression problems in high dimensions as well as inference for spatial models with many latent variables.
Gomes SN, Pavliotis GA, 2018, Mean Field Limits for Interacting Diffusions in a Two-Scale Potential, JOURNAL OF NONLINEAR SCIENCE, Vol: 28, Pages: 905-941, ISSN: 0938-8974
Duncan AB, Nusken N, Pavliotis GA, 2017, Using Perturbed Underdamped Langevin Dynamics to Efficiently Sample from Probability Distributions, JOURNAL OF STATISTICAL PHYSICS, Vol: 169, Pages: 1098-1131, ISSN: 0022-4715
Tomlin RJ, Papageorgiou DT, Pavliotis GA, 2017, Three-dimensional wave evolution on electrified falling films, JOURNAL OF FLUID MECHANICS, Vol: 822, Pages: 54-79, ISSN: 0022-1120
Gomes SN, Kalliadasis S, Papageorgiou DT, et al., 2017, Controlling roughening processes in the stochastic Kuramoto-Sivashinsky equation, PHYSICA D-NONLINEAR PHENOMENA, Vol: 348, Pages: 33-43, ISSN: 0167-2789
Gomes SN, Papageorgiou DT, Pavliotis GA, 2017, Stabilizing non-trivial solutions of the generalized Kuramoto-Sivashinsky equation using feedback and optimal control, IMA JOURNAL OF APPLIED MATHEMATICS, Vol: 82, Pages: 158-194, ISSN: 0272-4960
Abdulle A, Pavliotis GA, Vaes U, 2017, Spectral Methods for Multiscale Stochastic Differential Equations, SIAM-ASA JOURNAL ON UNCERTAINTY QUANTIFICATION, Vol: 5, Pages: 720-761, ISSN: 2166-2525
Craster R, Guenneau S, Hutridurga H, et al., 2017, Regularized transformation optics for transient heat transfer, 2017 11th International Congress on Engineered Material Platforms for Novel Wave Phenomena (METAMATERIALS), Publisher: IEEE, Pages: 127-129
Bonnaillie-Noel V, Carrillo JA, Goudon T, et al., 2016, Efficient numerical calculation of drift and diffusion coefficients in the diffusion approximation of kinetic equations, IMA JOURNAL OF NUMERICAL ANALYSIS, Vol: 36, Pages: 1536-1569, ISSN: 0272-4979
Duncan AB, Kalliadasis S, Pavliotis GA, et al., 2016, Noise-induced transitions in rugged energy landscapes, PHYSICAL REVIEW E, Vol: 94, ISSN: 2470-0045
Duncan AB, Lelievre T, Pavliotis GA, 2016, Variance Reduction Using Nonreversible Langevin Samplers, JOURNAL OF STATISTICAL PHYSICS, Vol: 163, Pages: 457-491, ISSN: 0022-4715
Thompson AB, Gomes SN, Pavliotis GA, et al., 2016, Stabilising falling liquid film flows using feedback control, PHYSICS OF FLUIDS, Vol: 28, ISSN: 1070-6631
Krumscheid S, Pradas M, Pavliotis GA, et al., 2015, Data-driven coarse graining in action: Modeling and prediction of complex systems, PHYSICAL REVIEW E, Vol: 92, ISSN: 2470-0045
Ottobre M, Pavliotis GA, Pravda-Starov K, 2015, Some remarks on degenerate hypoelliptic Ornstein-Uhlenbeck operators, JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, Vol: 429, Pages: 676-712, ISSN: 0022-247X
Kalliadasis S, Krumscheid S, Pavliotis GA, 2015, A new framework for extracting coarse-grained models from time series with multiscale structure, JOURNAL OF COMPUTATIONAL PHYSICS, Vol: 296, Pages: 314-328, ISSN: 0021-9991
Gomes SN, Pradas M, Kalliadasis S, et al., 2015, Controlling spatiotemporal chaos in active dissipative-dispersive nonlinear systems, PHYSICAL REVIEW E, Vol: 92, ISSN: 2470-0045
Schmuck M, Pradas M, Pavliotis GA, et al., 2015, A new mode reduction strategy for the generalized Kuramoto-Sivashinsky equation, IMA JOURNAL OF APPLIED MATHEMATICS, Vol: 80, Pages: 273-301, ISSN: 0272-4960
Duncan AB, Elliott CM, Pavliotis GA, et al., 2015, A Multiscale Analysis of Diffusions on Rapidly Varying Surfaces, JOURNAL OF NONLINEAR SCIENCE, Vol: 25, Pages: 389-449, ISSN: 0938-8974
Joubaud R, Pavliotis GA, Stoltz G, 2015, Langevin Dynamics with Space-Time Periodic Nonequilibrium Forcing, JOURNAL OF STATISTICAL PHYSICS, Vol: 158, Pages: 1-36, ISSN: 0022-4715
Hartmann C, Latorre JC, Zhang W, et al., 2014, Optimal control of multiscale systems using reduced-order models, Journal of Computational Dynamics, Vol: 1, Pages: 279-306
© American Institute of Mathematical Sciences. We study optimal control of diffusions with slow and fast variables and address a question raised by practitioners: is it possible to first eliminate the fast variables before solving the optimal control problem and then use the optimal control computed from the reduced-order model to control the original, high-dimensional system? The strategy "first reduce, then optimize"-rather than "first optimize, then reduce"-is motivated by the fact that solving optimal control problems for high-dimensional multiscale systems is numerically challenging and often computationally prohibitive. We state suffcient and necessary conditions, under which the "first reduce, then control" strategy can be employed and discuss when it should be avoided. We further give numerical examples that illustrate the "first reduce, then optmize" approach and discuss possible pitfalls.
Pavliotis GA, 2014, Stochastic Processes and Applications Diffusion Processes, the Fokker-Planck and Langevin Equations, Publisher: Springer, ISBN: 9781493913220
This book presents various results and techniques from the theory of stochastic processes that are useful in the study of stochastic problems in the natural sciences.
Schmuck M, Pavliotis GA, Kalliadasis S, 2014, Effective macroscopic interfacial transport equations in strongly heterogeneous environments for general homogeneous free energies, APPLIED MATHEMATICS LETTERS, Vol: 35, Pages: 12-17, ISSN: 0893-9659
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