14 results found
Dutta R, Gomes SN, Kalise D, et al., 2021, Using mobility data in the design of optimal lockdown strategies for the COVID-19 pandemic, PLOS COMPUTATIONAL BIOLOGY, Vol: 17, ISSN: 1553-734X
Cimpeanu R, Gomes SN, Papageorgiou DT, 2021, Active control of liquid film flows: beyond reduced-order models, NONLINEAR DYNAMICS, Vol: 104, Pages: 267-287, ISSN: 0924-090X
Gomes SN, Pavliotis GA, Vaes U, 2020, Mean field limits for interacting diffusions with colored noise: phase transitions and spectral numerical methods, SIAM: Multiscale Modeling and Simulation, Vol: 18, Pages: 1343-1370, ISSN: 1540-3459
In this paper we consider systems of weakly interacting particles driven by colored noise in a bistable potential, and we study the effect of the correlation time of the noise on the bifurcation diagram for the equilibrium states. We accomplish this by solving the corresponding McKean--Vlasov equation using a Hermite spectral method, and we verify our findings using Monte Carlo simulations of the particle system. We consider both Gaussian and non-Gaussian noise processes, and for each model of the noise we also study the behavior of the system in the small correlation time regime using perturbation theory. The spectral method that we develop in this paper can be used for solving linear and nonlinear, local and nonlocal (mean field) Fokker--Planck equations, without requiring that they have a gradient structure.
Tomlin R, Noronha Moreira Antunes Gomes S, 2019, Point-actuated feedback control of multidimensional interfaces, IMA Journal of Applied Mathematics, Vol: 84, Pages: 1112-1142, ISSN: 0272-4960
We consider the application of feedback control strategies with point actuators to multidimensional evolv-ing interfaces in order to stabilise desired states. We take a Kuramoto–Sivashinsky equation as a test case;this equation arises in the study of thin liquid films, exhibiting a wide range of dynamics in differentparameter regimes, including unbounded growth and full spatiotemporal chaos. The controls correspondphysically to mass-flux actuators located in the substrate on which the liquid film lies. In the case of par-tial state observability, we utilise a proportional control strategy where forcing at a point depends only onthe local observation. We find that point-actuated controls may inhibit unbounded growth of a solution,if the actuators are sufficient in number and in strength, and can exponentially stabilise the desired state.We investigate actuator arrangements, and find that the equidistant case is the most favourable for con-trol performance, with a large drop in effectiveness for poorly arranged actuators. Proportional controlsare also used to synchronise two chaotic solutions. When the interface is fully observable, we constructmodel-based controls using the linearisation of the governing equation. These improve on proportionalcontrols, and are applied to stabilise non-trivial steady and travelling wave solutions.
Thompson AB, Gomes SN, Denner F, et al., 2019, Robust low-dimensional modelling of falling liquid films subject to variable wall heating, Journal of Fluid Mechanics, Vol: 877, Pages: 844-881, ISSN: 0022-1120
Accurate low-dimensional models for the dynamics of falling liquid films subject to localized or time-varying heating are essential for applications that involve patterning or control. However, existing modelling methodologies either fail to respect fundamental thermodynamic properties or else do not accurately capture the effects of advection and diffusion on the temperature profile. We argue that the best-performing long-wave models are those that give the surface temperature implicitly as the solution of an evolution equation in which the wall temperature alone (and none of its derivatives) appears as a source term. We show that, for both flat and non-uniform films, such a model can be rationally derived by expanding the temperature field about its free-surface values. We test this model in linear and nonlinear regimes, and show that its predictions are in remarkable quantitative agreement with full Navier–Stokes calculations regarding the surface temperature, the internal temperature field and the surface displacement that would result from temperature-induced Marangoni stresses.
Gomes SN, Stuart AM, Wolfram M-T, 2019, Parameter estimation for macroscopic pedestrian dynamics models from microscopic data, SIAM Journal on Applied Mathematics, Vol: 79, Pages: 1475-1500, ISSN: 0036-1399
In this paper we develop a framework for parameter estimation in macroscopicpedestrian models using individual trajectories---microscopic data. We consider a unidirectional flowof pedestrians in a corridor and assume that the velocity decreases with the average density accordingto the fundamental diagram. Our model is formed from a coupling between a density dependentstochastic differential equation and a nonlinear partial differential equation for the density, andis hence of McKean--Vlasov type. We discuss identifiability of the parameters appearing in thefundamental diagram from trajectories of individuals, and we introduce optimization and Bayesianmethods to perform the identification. We analyze the performance of the developed methodologies invarious situations, such as for different in- and outflow conditions, for varying numbers of individualtrajectories, and for differing channel geometries.
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.
Tomlin R, Gomes SN, Pavliotis G, 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
We consider the control of a three-dimensional thin liquid film on a flat substrate, inclined at a nonzero angle to the horizontal. Controls are applied via same-fluid blowing and suction through the substrate surface. The film may be either overlying or hanging, where the liquid lies above or below the substrate, respectively. We study the weakly nonlinear evolution of the fluid interface, which is governed by a forced Kuramoto--Sivashinsky equation in two space dimensions. The uncontrolled problem exhibits three ranges of dynamics depending on the incline of the substrate: stable flat film solution, bounded chaotic dynamics, or unbounded exponential growth of unstable transverse modes. We proceed with the assumption that we may actuate at every location on the substrate. The main focus is the optimal control problem, which we first study in the special case that the forcing may only vary in the spanwise direction. The structure of the Kuramoto--Sivashinsky equation allows the explicit construction of optimal controls in this case using the classical theory of linear quadratic regulators. Such controls are employed to prevent the exponential growth of transverse waves in the case of a hanging film, revealing complex dynamics for the streamwise and mixed modes. Next, we consider the optimal control problem in full generality and prove the existence of an optimal control. For numerical simulations, an iterative gradient descent algorithm is employed. Finally, we consider the effects of transverse mode forcing on the chaotic dynamics present in the streamwise and mixed modes for the case of a vertical film flow. Coupling through nonlinearity allows us to reduce the average energy in solutions without directly forcing the dominant linearly unstable modes.
Noronha Moreira Antunes Gomes ST, 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
In this paper we study the combined mean field and homogenization limits for a system of weakly interacting diffusions moving in a two-scale, locally periodic confining potential, of the form considered in . We show that, although the mean field and homogenization limits commute for finite times, they do not, in general, commute in the long time limit. In particular, the bifurcation diagrams for the stationary states can be different depending on the order with which we take the two limits. Furthermore, we construct the bifurcation diagram for the stationary McKean-Vlasov equation in a two-scale potential, before passing to the homogenization limit, and we analyze the effect of the multiple local minima in the confining potential on the number and the stability of stationary solutions.
Noronha Moreira Antunes Gomes ST, Tate SJ, 2017, On the numerical solution of a T-Sylvester type matrix equation arising in the control of stochastic partial differential equations, IMA Journal of Applied Mathematics, Vol: 82, Pages: 1192-1208, ISSN: 0272-4960
We outline a derivation of a nonlinear system of equations, which finds the entries of an m×N matrix K, giventhe eigenvalues of a matrix D, a diagonal N ×N matrix A and an N ×m matrix B. These matrices are related throughthe matrix equation D = 2A + BK + KtBt, which is sometimes called a t-Sylvester equation. The need to prescribethe eigenvalues of the matrix D is motivated by the control of the surface roughness of certain nonlinear SPDEs (e.g.,the stochastic Kuramoto-Sivashinsky equation) using nontrivial controls. We implement the methodology to solvenumerically the nonlinear system for various test cases, including matrices related to the control of the stochasticKuramoto-Sivashinsky equation and for randomly generated matrices. We study the effect of increasing the dimensionsof the system and changing the size of the matrices B and K (which correspond to using more or less controls)and find good convergence of the solutions.
Noronha Moreira Antunes Gomes ST, 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
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.
Gomes SN, Papageorgiou DT, Pavliotis GA, 2016, 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
Thompson AB, Gomes SN, Pavliotis, et al., 2016, Stabilising falling liquid film flows using feedback control, Physics of Fluids, Vol: 28, ISSN: 1089-7666
Falling liquid films become unstable due to inertial effects when the fluid layer is sufficiently thick or the slopesufficiently steep. This free surface flow of a single fluid layer has industrial applications including coating andheat transfer, which benefit from smooth and wavy interfaces, respectively. Here we discuss how the dynamicsof the system are altered by feedback controls based on observations of the interface height, and supplied tothe system via the perpendicular injection and suction of fluid through the wall. In this study, we modelthe system using both Benney and weighted-residual models that account for the fluid injection throughthe 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 thequalitative effects are independent of the details of the flow modelling. Furthermore, we show that the systemcan still be successfully controlled when the feedback is applied via a set of localised actuators and only asmall number of system observations are available, and that this is possible using both static (where thecontrols are based on only the most recent set of observations) and dynamic (where the controls are based onan approximation of the system which evolves over time) control schemes. This study thus provides a solidtheoretical foundation for future experimental realisations of the active feedback control of falling liquid films.
Gomes SN, Pradas M, Kalliadasis S, et al., 2015, Controlling spatiotemporal chaos in active dissipative-dispersive nonlinear systems, Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, Vol: 92, ISSN: 1539-3755
We present an alternative methodology for the stabilization and control of infinite-dimensional dynamical systems exhibiting low-dimensional spatiotemporal chaos. We show that with an appropriate choice of time-dependent controls we are able to stabilize and/or control all stable or unstable solutions, including steady solutions, traveling waves (single and multipulse ones or bound states), and spatiotemporal chaos. We exemplify our methodology with the generalized Kuramoto-Sivashinsky equation, a paradigmatic model of spatiotemporal chaos, which is known to exhibit a rich spectrum of wave forms and wave transitions and a rich variety of spatiotemporal structures.
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