Katsoulakis, Markos

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Professor, Department of Mathematics and Statistics
Last Name
Katsoulakis
First Name
Markos
Discipline
Non-linear Dynamics
Expertise
Applied Stochastic Analysis, Nonlinear Partial Differential Equations and Computational Methods for Multiscale Problems.
Introduction
Markos Katsoulakis works in Applied Stochastic Analysis, Nonlinear Partial Differential Equations and Statistical Mechanics, focusing on the mathematical and computational aspects of phenomena with multiple, interrelating scales. He has extensively worked in interdisciplinary collaborations involving Mathematics, Chemical Engineering and lately in Atmospheric and Oceanic Sciences problems. His research work is primarily in (a) deterministic and stochastic mesoscopic modeling and simulation, (b) developing rigorous mathematical methods for the derivation of macroscopic laws from microscopic stochastic models, and (c) studying relaxation mechanisms in hyperbolic conservation laws and discrete velocity kinetic models. He is currently the Director for the Center of Applied Mathematics at the University of Massachusetts and he is an Associate Editor of SIAM Journal of Mathematical Analysis.
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1996 - 199972000 - 201039
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Now showing 1 - 10 of 46
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    Mathematical strategies in the coarse-graining of extensive systems: Error quantification and adaptivity
    (2008-01-01) Katsoulakis, MA; Plechac, P; Rey-Bellet, L; Tsagkarogiannis, DK
    In this paper we continue our study of coarse-graining schemes for stochastic many-body microscopic models started in Katsoulakis et al. [M. Katsoulakis, A. Majda, D. Vlachos, Coarse-grained stochastic processes for microscopic lattice systems, Proc. Natl. Acad. Sci. 100 (2003) 782–782, M.A. Katsoulakis, L. Rey-Bellet, P. Plecháč, D. Tsagkarogiannis, Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems, M2AN Math. Model. Numer. Anal., in press], focusing on equilibrium stochastic lattice systems. Using cluster expansion techniques we expand the exact coarse-grained Hamiltonian around a first approximation and derive higher accuracy schemes by including more terms in the expansion. The accuracy of the coarse-graining schemes is measured in terms of information loss, i.e., relative entropy, between the exact and approximate coarse-grained Gibbs measures. We test the effectiveness of our schemes in systems with competing short- and long-range interactions, using an analytically solvable model as a computational benchmark. Furthermore, the cluster expansion in Katsoulakis et al. [M.A. Katsoulakis, L. Rey-Bellet, P. Plecháč, D. Tsagkarogiannis, Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems, M2AN Math. Model. Numer. Anal., in press] yields sharp a posteriori error estimates for the coarse-grained approximations that can be computed on-the-fly during the simulation. Based on these estimates we develop a numerical strategy to assess the quality of the coarse-graining and suitably refine or coarsen the simulations. We demonstrate the use of this diagnostic tool in the numerical calculation of phase diagrams.
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    Contractive relaxation systems and interacting particles for scalar conservation laws
    (1996) Katsoulakis, MA; Tzavaras, AE
    Nous considérons une classe de svstèmes hvperboliques semilinéaires avec relaxation, qui sont contractifS en norme L1 et qui possèdent des régions invariantes. Nous démontrons que, quand le paramètre de relaxation ε tend vers zéro, les solutions du système convergent vers une.solution faible d 'une loi de conservation, qui satisfait aux conditions d'entropie de Kruzhkov. Dans le cas de la dimension un d'espace, nous proposons des systèmes de particules avec interaction qui, à l'échelle mésoscopique, convergent vers les.systèmes avec relaxation; leur dynamique macroscopique est décrite par les solutions entropiques d'une loi de conservation.
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    Multiscale couplings in prototype hybrid deterministic/stochastic systems: Part II, Stochastic closures
    (2005-01-01) Katsoulakis, MA; Majda, AJ; Sopasakis, A
    Couplings of microscopic stochastic models to deterministic macroscopic ordinary and partial differential equations are commonplace in numerous applications such as catalysis, deposition processes, polymeric flows, biological networks and parametrizations of tropical and open ocean convection. In this paper we continue our study of the class of prototype hybrid systems presented in [8]. These model systems are comprised of a microscopic Arrhenius dynamics stochastic process modeling adsorption/desorption of interacting particles which is coupled to an ordinary differential equation exhibiting a variety of bifurcation profiles. Here we focus on the case where phase transitions do not occur in the microscopic stochastic system and examine the influence of noise in the overall system dynamics. Deterministic mean field and stochastic averaging closures derived in [8] are valid under stringent conditions on the range of microscopic interactions and time-scale separation respectively. Furthermore, their derivation is valid only for finite time intervals where rare events will not trigger a large deviation from the average behavior in the zero noise limit. In this paper we study such questions in the context of simple hybrid systems, demonstrating that deterministic closures based on various separation of scales arguments cannot in general capture transient and long-time dynamics. For this purpose we develop coarse grained stochastic closures for this class of hybrid systems and compare them to deterministic, mean-field and stochastic averaging closures. We show that the proposed coarse grained closures describe correctly the microscopic hybrid system solutions in all test cases examined, including rare events and random transitions between multiple stable states.
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    Statistical equilibrium measures in micromagnetics
    (2001-01-01) Katsoulakis, MA; Plechac, P
    We derive an equilibrium statistical theory for the macroscopic description of a ferromagnetic material at positive finite temperatures. Our formulation describes the most-probable equilibrium macrostates that yield a coherent deterministic large-scale picture varying at the size of the domain, as well as it captures the effect of random spin fluctuations caused by the thermal noise. We discuss connections of the proposed formulation to the Landau-Lifschitz theory and to the studies of domain formation based on Monte Carlo lattice simulations.
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    Noise regularization and computations for the 1-dimensional stochastic Allen-Cahn problem
    (2007-01-01) Katsoulakis, MA; Kossioris, GT; Lakkis, O
    We address the numerical discretization of the Allen-Cahn problem with additive white noise in one-dimensional space. Our main focus is to understand the behavior of the discretized equation with respect to a small ``interface thickness'' parameter and the noise intensity. The discretization is conducted in two stages: (1) regularize the white noise and study the regularized problem, (2) approximate the regularized problem. We address (1) by introducing a piecewise constant random approximation of the white noise with respect to a space-time mesh. We analyze the regularized problem and study its relation to both the original problem and the deterministic Allen-Cahn problem. Step (2) is then performed leading to a practical Monte-Carlo method combined with a Finite Element-Implicit Euler scheme. The resulting numerical scheme is tested against theoretical benchmark results concerning the behavior of the solution as the interface thickness goes to zero.
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    A mathematical model for crystal growth by aggregation of precursor metastable nanoparticles
    (2005-01-01) Drews, TO; Katsoulakis, MA; Tsapatsis, M
    A mathematical model is developed to describe aggregative crystal growth, including oriented aggregation, from evolving pre-existing primary nanoparticles with composition and structure that are different from that of the final crystalline aggregate. The basic assumptions of the model are based on the ideas introduced in an earlier published report [Buyanov and Krivoruchko, Kinet. Katal. 1976, 17, 666−675] to describe the growth of low-solubility metal hydroxides (e.g., iron oxides) by oriented aggregation. It is assumed that primary particles can be described as pseudo-species A, B, and C, which have the following properties: (1) fresh primary particles (colloidally stable inert nanoparticles, denoted as A), (2) mature primary particles (partially transformed nanoparticles at an optimum stage of development for attachment to a growing crystal, denoted as B), and (3) nucleated primary particles (denoted as C1). The evolution of primary particles, A → B → C1, is treated as two first-order consecutive reactions. Crystal growth via crystal−crystal aggregation (Ci + ) is described using the Smoluchowski equation. The new element of this model is the inclusion of an additional crystal growth mechanism via the addition of primary particles (B) to crystals (Ci): (B + ). Two distinct, but constant, kernels (K ≠ K‘) are used. It is shown that, when K‘ = 0, a steplike crystal size distribution (CSD) is obtained. Within a range of K‘/K values (e.g., K‘/K = 103), CSD with multiple peaks are obtained. Comparison with predictions of models that do not include the intermediate stage of primary particles (B) indicates pronounced differences. Despite its simplicity, the model is able to capture the qualitative features of CSD evolution that have been obtained from crystal growth experiments in hematite, which is a system that is believed to undergo oriented aggregation.
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    Mesoscopic modeling of transport and reaction in microporous crystalline membranes
    (2003-01-01) Snyder, MA; Vlachos, DG; Katsoulakis, MA
    A mesoscopic framework, derived from first principles via a rigorous coarse-graining of an underlying master equation, has proven to be a powerful tool in bridging the disparate scales between atomistic simulations and practical applications involving diffusion of interacting species through microporous films. This mesoscopic framework is validated here via gradient continuous time Monte Carlo (G-CTMC) simulations for realistic boundary conditions in the limit of thin, single crystal membranes. It is shown that intermolecular forces have a non-Arrhenius effect on the permeation flux, and a stationary concentration pattern develops for strong repulsive interactions. It is found that diffusion through complex multiple site lattices, such as those encountered in diffusion of benzene in Na–Y zeolite films, exhibits strongly nonlinear behavior even in the absence of interactions between molecular species. Finally, the mesoscopic framework is applied to diffusion/reaction systems, where excellent agreement between G-CTMC and mesoscopic solutions is demonstrated for the first time.
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    Spectral methods for mesoscopic models of pattern formation
    (2001-01-01) Horntrop, DJ; Katsoulakis, MA; Vlachos, DG
    In this paper we present spectral algorithms for the solution of mesoscopic equations describing a broad class of pattern formation mechanisms, focusing on a prototypical system of surface processes. These models are in principle stochastic integrodifferential equations and are derived directly from microscopic lattice models, containing detailed information on particle–particle interactions and particle dynamics. The enhanced computational efficiency and accuracy of spectral methods versus finite difference methods are also described.
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    Coarse-Grained Langevin Approximations and Spatiotemporal Acceleration for Kinetic Monte Carlo Simulations of Diffusion of Interacting Particles
    (2009-01-01) Are, S; Katsoulakis, MA; Szepessy, A
    Kinetic Monte Carlo methods provide a powerful computational tool for the simulation of microscopic processes such as the diffusion of interacting particles on a surface, at a detailed atomistic level. However such algorithms are typically computationally expensive and are restricted to fairly small spatiotemporal scales. One approach towards overcoming this problem was the development of coarse-grained Monte Carlo algorithms. In recent literature, these methods were shown to be capable of efficiently describing much larger length scales while still incorporating information on microscopic interactions and fluctuations. In this paper, a coarse-grained Langevin system of stochastic differential equations as approximations of diffusion of interacting particles is derived, based on these earlier coarse-grained models. The authors demonstrate the asymptotic equivalence of transient and long time behavior of the Langevin approximation and the underlying microscopic process, using asymptotics methods such as large deviations for interacting particles systems, and furthermore, present corresponding numerical simulations, comparing statistical quantities like mean paths, auto correlations and power spectra of the microscopic and the approximating Langevin processes. Finally, it is shown that the Langevin approximations presented here are much more computationally efficient than conventional Kinetic Monte Carlo methods, since in addition to the reduction in the number of spatial degrees of freedom in coarse-grained Monte Carlo methods, the Langevin system of stochastic differential equations allows for multiple particle moves in a single timestep.
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    Homogenization of mesoscopic theories: Effective properties of model membranes
    (2002-01-01) Lam, R; Vlachos, DG; Katsoulakis, MA
    A new mathematical framework for modeling diffusion in nanoporous materials or on surfaces exhibits heterogeneity in properties over large length scales and retains molecular scale information, typically captured only by molecular simulations (kinetic Monte Carlo). It first uses newly developed mesoscopic equations derived rigorously from underlying master equations by coarse-graining statistical mechanics techniques. Homogenization techniques are then used to derive the leading-order effective mesoscopic models that are subsequently solved by spectral methods. These solutions are also compared to direct numerical simulations for selected 2-D model membranes with defects, when attractive adsorbate-adsorbate interactions affect particle difSsion. Both the density and dispersion of defects significantly alter the macroscopic behavior in terms of fluxes and concentration patterns, especially when phase transitions can occur. In the presence of adsorbate-adsorbate interactions, permeation through a nanoporous film can depend on the face of a membrane exposed to the high-pressure side. Homogenization techniques also could offer a promising alternative to direct numerical simulations, when complex, large-scale heterogeneities are present.