## Publications

57 results found

Abdulle A, Pavliotis GA, Vaes U, Specral methods for multiscale stochastic differential equations, *SIAM/ASA Journal on Uncertainty Quantification*, ISSN: 2166-2525

This paper presents a new method for the solution of multiscale stochastic differential equations atthe diffusive time scale. In contrast to averaging-based methods, e.g., the heterogeneous multiscalemethod (HMM) or the equation-free method, which rely on Monte Carlo simulations, in this paperwe introduce a new numerical methodology that is based on a spectral method. In particular, we usean expansion in Hermite functions to approximate the solution of an appropriate Poisson equation,which is used in order to calculate the coefficients of the homogenized equation. Spectral convergenceis proved under suitable assumptions. Numerical experiments corroborate the theory and illustratethe performance of the method. A comparison with the HMM and an application to singularlyperturbed stochastic PDEs are also presented.

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

© 2017 The Authors We present a novel control methodology to control the roughening processes of semilinear parabolic stochastic partial differential equations in one dimension, which we exemplify with the stochastic Kuramoto-Sivashinsky equation. The original equation is split into a linear stochastic and a nonlinear deterministic equation so that we can apply linear feedback control methods. Our control strategy is then based on two steps: first, stabilize the zero solution of the deterministic part and, second, control the roughness of the stochastic linear equation. We consider both periodic controls and point actuated ones, observing in all cases that the second moment of the solution evolves in time according to a power-law until it saturates at the desired controlled value.

Tomlin RJ, Papageorgiou DT, Pavliotis GA,
et al., 2017, Three-dimensional wave evolution on electrified falling films, *JOURNAL OF FLUID MECHANICS*, Vol: 822, Pages: 54-79, ISSN: 0022-1120

We consider the full three-dimensional dynamics of a thin falling liquid film on a flat plate inclined at some non-zero angle to the horizontal. In addition to gravitational effects, the flow is driven by an electric field which is normal to the substrate far from the flow. This extends the work of Tseluiko & Papageorgiou (J. Fluid Mech., vol. 556, 2006b, pp. 361–386) by including transverse dynamics. We study both the cases of overlying and hanging films, where the liquid lies above or below the substrate, respectively. Starting with the Navier–Stokes equations coupled with electrostatics, a fully nonlinear two-dimensional Benney equation for the interfacial dynamics is derived, valid for waves that are long compared to the film thickness. The weakly nonlinear evolution is governed by a Kuramoto–Sivashinsky equation with a non-local term due to the electric field effect. The electric field term is linearly destabilising and produces growth rates proportional to $|\unicode[STIX]{x1D743}|^{3}$ , where $\unicode[STIX]{x1D743}$ is the wavenumber vector of the perturbations. It is found that transverse gravitational instabilities are always present for hanging films, and this leads to unboundedness of nonlinear solutions even in the absence of electric fields – this is due to the anisotropy of the nonlinearity. For overlying films and a restriction on the strength of the electric field, the equation is well-posed in the sense that it possesses bounded solutions. This two-dimensional equation is studied numerically for the case of periodic boundary conditions in order to assess the effects of inertia, electric field strength and the size of the periodic domain. Rich dynamical behaviours are observed and reported. For subcritical Reynolds number flows, a sufficiently strong electric field can promote non-trivial dynamics for some choices of domain size, leading to fully two-dimensional evolutions of the interface. We also observe two-dimensiona

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

In this paper we study the diffusion approximation of a swarming model given by a systemof interacting Langevin equations with nonlinear friction. The diffusion approximationrequires the calculation of the drift and diffusion coefficients that are given as averages ofsolutions to appropriate Poisson equations. We present a new numerical method for computingthese coefficients that is based on the calculation of the eigenvalues and eigenfunctionsof a Schr¨odinger operator. These theoretical results are supported by numerical simulationsshowcasing the efficiency of the method.

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Duncan AB, Lelievre T, Pavliotis GA,
et al., 2016, Variance Reduction Using Nonreversible Langevin Samplers, *JOURNAL OF STATISTICAL PHYSICS*, Vol: 163, Pages: 457-491, ISSN: 0022-4715

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 nonreversiblecomponent 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.

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- Citations: 5

Thompson AB, Gomes SN, Pavliotis GA,
et al., 2016, Stabilising falling liquid film flows using feedback control, *PHYSICS OF FLUIDS*, Vol: 28, Pages: 012107-012107, ISSN: 1070-6631

© 2016 AIP Publishing LLC. Falling liquid films become unstable due to inertial effects when the fluid layer is sufficiently thick or the slope sufficiently steep. This free surface flow of a single fluid layer has industrial applications including coating and heat transfer, which benefit from smooth and wavy interfaces, respectively. Here, we discuss how the dynamics of the system are altered by feedback controls based on observations of the interface height, and supplied to the system via the perpendicular injection and suction of fluid through the wall. In this study, we model the system using both Benney and weighted-residual models that account for the fluid injection through the wall. We find that feedback using injection and suction is a remarkably effective control mechanism: the controls can be used to drive the system towards arbitrary steady states and travelling waves, and the qualitative effects are independent of the details of the flow modelling. Furthermore, we show that the system can still be successfully controlled when the feedback is applied via a set of localised actuators and only a small number of system observations are available, and that this is possible using both static (where the controls are based on only the most recent set of observations) and dynamic (where the controls are based on an approximation of the system which evolves over time) control schemes. This study thus provides a solid theoretical foundation for future experimental realisations of the active feedback control of falling liquid films.

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- Citations: 3

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

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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

Kalliadasis S, Krumscheid S, Pavliotis GA,
et al., 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

© 2015 Elsevier Inc.. In many applications it is desirable to infer coarse-grained models from observational data. The observed process often corresponds only to a few selected degrees of freedom of a high-dimensional dynamical system with multiple time scales. In this work we consider the inference problem of identifying an appropriate coarse-grained model from a single time series of a multiscale system. It is known that estimators such as the maximum likelihood estimator or the quadratic variation of the path estimator can be strongly biased in this setting. Here we present a novel parametric inference methodology for problems with linear parameter dependency that does not suffer from this drawback. Furthermore, we demonstrate through a wide spectrum of examples that our methodology can be used to derive appropriate coarse-grained models from time series of partial observations of a multiscale system in an effective and systematic fashion.

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- Citations: 3

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

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

© 2013 © The authors 2013. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. 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 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 one 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.

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- Citations: 3

Latorre JC, Kramer PR, Pavliotis GA, 2014, Numerical methods for computing effective transport properties of flashing Brownian motors, *Journal of Computational Physics*, Vol: 257, Pages: 57-82, ISSN: 0021-9991

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,
et al., 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

We study phase field equations in perforated domains for arbitrary free energies. These equations have found numerous applications in a wide spectrum of both science and engineering problems with homogeneous environments. Here, we focus on strongly heterogeneous materials with perforations such as porous media. To the best of our knowledge, we provide the first derivation of upscaled equations for general free energy densities. In view of the versatile applications of phase field equations, we expect that our study will lead to new modelling and computational perspectives for interfacial transport and phase transformations in strongly heterogeneous environments. © 2014 Elsevier Ltd. All rights reserved.

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- Citations: 5

Krumscheid S, Pavliotis GA, Kalliadasis S,
et al., 2013, SEMIPARAMETRIC DRIFT AND DIFFUSION ESTIMATION FOR MULTISCALE DIFFUSIONS, *MULTISCALE MODELING & SIMULATION*, Vol: 11, Pages: 442-473, ISSN: 1540-3459

We consider the problem of statistical inference for the effective dynamics of multiscale diffusion processes with (at least) two widely separated characteristic time scales. More precisely, we seek to determine parameters in the effective equation describing the dynamics on the longer diffusive time scale, i.e., in a homogenization framework. We examine the case where both the drift and the diffusion coefficients in the effective dynamics are space dependent and depend on multiple unknown parameters. It is known that classical estimators, such as maximum likelihood and quadratic variation of the path estimators, fail to obtain reasonable estimates for parameters in the effective dynamics when based on observations of the underlying multiscale diffusion. We propose a novel algorithm for estimating both the drift and the diffusion coefficients in the effective dynamics based on a semiparametric framework. We demonstrate by means of extensive numerical simulations of a number of selected examples that the algorithm performs well when applied to data from a multiscale diffusion. These examples also illustrate that the algorithm can be used effectively to obtain accurate and unbiased estimates. © 2013 Society for Industrial and Applied Mathematics.

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- Citations: 4

Latorre JC, Pavliotis GA, Kramer PR, 2013, Corrections to Einstein’s Relation for Brownian Motion in a Tilted Periodic Potential, *Journal of Statistical Physics*, Vol: 150, Pages: 776-803, ISSN: 0022-4715

Lelièvre T, Nier F, Pavliotis GA,
et al., 2013, Optimal Non-reversible Linear Drift for the Convergence to Equilibrium of a Diffusion, *Journal of Statistical Physics*, Vol: 152, Pages: 237-274, ISSN: 0022-4715

Papaefthymiou ES, Papageorgiou DT, Pavliotis GA,
et al., 2013, Nonlinear interfacial dynamics in stratified multilayer channel flows, *JOURNAL OF FLUID MECHANICS*, Vol: 734, Pages: 114-143, ISSN: 0022-1120

Abstract The dynamics of viscous immiscible pressure-driven multilayer flows in channels are investigated using a combination of modelling, analysis and numerical computations. More specifically, the particular system of three stratified layers with two internal fluid-fluid interfaces is considered in detail in order to identify the nonlinear mechanisms involved due to multiple fluid surface interactions. The approach adopted is analytical/asymptotic and is valid for interfacial waves that are long compared with the channel height or individual undisturbed liquid layer thicknesses. This leads to a coupled system of fully nonlinear partial differential equations of Benney type that contain a small slenderness parameter that cannot be scaled out of the problem. This system is in turn used to develop a consistent coupled system of weakly nonlinear evolution equations, and it is shown that this is possible only if the underlying base-flow and fluid parameters satisfy certain conditions that enable a synchronous Galilean transformation to be performed at leading order. Two distinct canonical cases (all terms in the equations are of the same order) are identified in the absence and presence of inertia, respectively. The resulting systems incorporate all of the active physical mechanisms at Reynolds numbers that are not large, namely, nonlinearities, inertia-induced instabilities (at non-zero Reynolds number) and surface tension stabilization of sufficiently short waves. The coupled system supports several instabilities that are not found in single long-wave equations including, transitional instabilities due to a change of type of the flux nonlinearity from hyperbolic to elliptic, kinematic instabilities due to the presence of complex eigenvalues in the linearized advection matrix leading to a resonance between the interfaces, and the possibility of long-wave instabilities induced by an interaction between the flux function of the system and the surface tension terms. All

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Schmuck M, Pradas M, Kalliadasis S,
et al., 2013, New Stochastic Mode Reduction Strategy for Dissipative Systems, *PHYSICAL REVIEW LETTERS*, Vol: 110, ISSN: 0031-9007

We present a new methodology for studying non-Hamiltonian nonlinear systems based on an information theoretical 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 defined 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 support our method and reveal that the long-time underlying stochastic process of the fast (unresolved) modes obeys a universal distribution that does not depend on the initial conditions and which we rigorously derive by the maximum entropy principle.

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- Citations: 9

Abdulle A, Pavliotis GA, Abdulle A,
et al., 2012, Numerical methods for stochastic partial differential equations with multiple scales, *JOURNAL OF COMPUTATIONAL PHYSICS*, Vol: 231, Pages: 2482-2497, ISSN: 0021-9991

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- Citations: 3

Goddard BD, Nold A, Savva N,
et al., 2012, General Dynamical Density Functional Theory for Classical Fluids, *PHYSICAL REVIEW LETTERS*, Vol: 109, ISSN: 0031-9007

We study the dynamics of a colloidal fluid including inertia and hydrodynamic interactions, two effects which strongly influence the nonequilibrium properties of the system. We derive a general dynamical density functional theory which shows very good agreement with full Langevin dynamics. In suitable limits, we recover existing dynamical density functional theories and a Navier-Stokes-like equation with additional nonlocal terms.

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- Citations: 27

Goddard BD, Pavliotis GA, Kalliadasis S,
et al., 2012, THE OVERDAMPED LIMIT OF DYNAMIC DENSITY FUNCTIONAL THEORY: RIGOROUS RESULTS, *MULTISCALE MODELING & SIMULATION*, Vol: 10, Pages: 633-663, ISSN: 1540-3459

Consider the overdamped limit for a system of interacting particles in the presence of hydrodynamic interactions. For two-body hydrodynamic interactions and one- and two-body potentials, a Smoluchowski-type evolution equation is rigorously derived for the one-particle distribution function. This new equation includes a novel definition of the diffusion tensor. A comparison with existing formulations of dynamic density functional theory is also made. © 2012 Society for Industrial and Applied Mathematics.

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- Citations: 11

Nolen J, Pavliotis GA, Stuart AM, 2012, Multiscale Modelling and Inverse Problems, Multiscale Modelling and Inverse Problems (wNumerical Analysis of Multiscale Problems

Ottobre M, Pavliotis GA, Pravda-Starov K,
et al., 2012, Exponential return to equilibrium for hypoelliptic quadratic systems, *JOURNAL OF FUNCTIONAL ANALYSIS*, Vol: 262, Pages: 4000-4039, ISSN: 0022-1236

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- Citations: 10

Pavliotis GA, Pokern Y, Stuart AM, 2012, Parameter estimation for multiscale diffusions: an overview, Statistical methods for stochastic differential equations, Boca Raton, FL, Publisher: CRC Press, Pages: 429-472

Pradas M, Pavliotis GA, Kalliadasis S,
et al., 2012, Additive noise effects in active nonlinear spatially extended systems, *EUROPEAN JOURNAL OF APPLIED MATHEMATICS*, Vol: 23, Pages: 563-591, ISSN: 0956-7925

We examine the effects of pure additive noise on spatially extended systems with quadratic nonlinearities. We develop a general multi-scale 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. © 2012 Cambridge University Press.

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- Citations: 12

Schmuck M, Pradas M, Pavliotis GA,
et al., 2012, Upscaled phase-field models for interfacial dynamics in strongly heterogeneous domains, *PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES*, Vol: 468, Pages: 3705-3724, ISSN: 1364-5021

We derive a new, effective macroscopic Cahn-Hilliard equation whose homogeneous free energy is represented by fourth-order polynomials, which form the frequently applied double-well potential. This upscaling is done for perforated/strongly heterogeneous domains. To the best knowledge of the authors, this seems to be the first attempt of upscaling the Cahn-Hilliard equation in such domains. The new homogenized equation should have a broad range of applicability owing to the well-known versatility of phase-field models. The additionally introduced feature of systematically and reliably accounting for confined geometries by homogenization allows for new modelling and numerical perspectives in both science and engineering. Our results are applied to wetting dynamics in porous media and to a single channel with strongly heterogeneous walls. © 2012 The Royal Society.

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Ottobre M, Pavliotis GA, Ottobre M,
et al., 2011, Asymptotic analysis for the generalized Langevin equation, *NONLINEARITY*, Vol: 24, Pages: 1629-1653, ISSN: 0951-7715

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Pradas M, Tseluiko D, Kalliadasis S,
et al., 2011, Noise Induced State Transitions, Intermittency, and Universality in the Noisy Kuramoto-Sivashinksy Equation, *PHYSICAL REVIEW LETTERS*, Vol: 106, ISSN: 0031-9007

Consider the effect of pure additive noise on the long-time dynamics of the noisy Kuramoto-Sivashinsky (KS) equation close to the instability onset. When the noise acts only on the first stable mode (highly degenerate), the KS solution undergoes several state transitions, including critical on-off intermittency and stabilized states, as the noise strength increases. Similar results are obtained with the Burgers equation. Such noise-induced transitions are completely characterized through critical exponents, obtaining the same universality class for both equations, and rigorously explained using multiscale techniques.

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- Citations: 27

Savva N, Pavliotis GA, Kalliadasis S,
et al., 2011, Contact lines over random topographical substrates. Part 1. Statics, *JOURNAL OF FLUID MECHANICS*, Vol: 672, Pages: 358-383, ISSN: 0022-1120

We investigate theoretically the statistics of the equilibria of two-dimensional droplets over random topographical substrates. The substrates are appropriately represented as families of certain stationary random functions parametrized by a characteristic amplitude and wavenumber. In the limit of shallow topographies and small contact angles, a linearization about the flat-substrate equilibrium reveals that the droplet footprint is adequately approximated by a zero-mean, normally distributed random variable. The theoretical analysis of the statistics of droplet shift along the substrate is highly non-trivial. However, for weakly asymmetric substrates it can be shown analytically that the droplet shift approaches a Cauchy random variable; for fully asymmetric substrates its probability density is obtained via Padé approximants. Generalization to arbitrary stationary random functions does not change qualitatively the behaviour of the statistics with respect to the characteristic amplitude and wavenumber of the substrate. Our theoretical results are verified by numerical experiments, which also suggest that on average a random substrate neither enhances nor reduces droplet wetting. To address the question of the influence of substrate roughness on wetting, a stability analysis of the equilibria must be performed so that we can distinguish between stable and unstable equilibria, which in turn requires modelling the dynamics. This is the subject of Part 2 of this study.

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- Citations: 20

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