Katsoulakis, Markos

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Professor, Department of Mathematics and Statistics
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Non-linear Dynamics
Applied Stochastic Analysis, Nonlinear Partial Differential Equations and Computational Methods for Multiscale Problems.
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.

Search Results

Now showing 1 - 10 of 46
  • Publication
    Numerical and statistical methods for the coarse-graining of many-particle stochastic systems
    (2008-01-01) Katsoulakis, MA; Plechac, P; Rey-Bellet, L
    In this article we discuss recent work on coarse-graining methods for microscopic stochastic lattice systems. We emphasize the numerical analysis of the schemes, focusing on error quantification as well as on the construction of improved algorithms capable of operating in wider parameter regimes. We also discuss adaptive coarse-graining schemes which have the capacity of automatically adjusting during the simulation if substantial deviations are detected in a suitable error indicator. The methods employed in the development and the analysis of the algorithms rely on a combination of statistical mechanics methods (renormalization and cluster expansions), statistical tools (reconstruction and importance sampling) and PDE-inspired analysis (a posteriori estimates). We also discuss the connections and extensions of our work on lattice systems to the coarse-graining of polymers.
  • Publication
    (2005-01-01) Katsoulakis, MA; Plechac, P; Sopasakis, A
    In this paper we investigate the approximation properties of the coarse-graining procedure applied to kinetic Monte Carlo simulations of lattice stochastic dynamics. We provide both analytical and numerical evidence that the hierarchy of the coarse models is built in a systematic way that allows for error control in both transient and long-time simulations. We demonstrate that the numerical accuracy of the CGMC algorithm as an approximation of stochastic lattice spin flip dynamics is of order two in terms of the coarse-graining ratio and that the natural small parameter is the coarse-graining ratio over the range of particle/particle interactions. The error estimate is shown to hold in the weak convergence sense. We employ the derived analytical results to guide CGMC algorithms and we demonstrate a CPU speed-up in demanding computational regimes that involve nucleation, phase transitions and metastability.
  • Publication
    Stochastic hydrodynamical limits of particle systems
    (2006-01-01) Katsoulakis, MA; Szepessy, A
  • Publication
    Wave initiation through spatiotemporally controllable perturbations
    (2003-01-01) Wolff, J; Papathanasiou, AG; Rotermund, HH; Ertl, G; Katsoulakis, MA; Li, X; Kevrekidis, IG
    We study the initiation of pulses and fronts in a two-dimensional catalytic reaction-diffusion system: CO oxidation on Pt(110). Using a computer-controlled mobile focused laser beam, we impart various patterns (in space and time) of localized temperature “kicks” to the surface. We explore, and also rationalize through modeling, the cooperativity of such individually subcritical perturbations in both the excitable and the bistable regime.
  • Publication
    Spatially adaptive grand canonical ensemble Monte Carlo simulations
    (2005-01-01) Chatterjee, A; Katsoulakis, MA; Vlachos, DG
    A spatially adaptive Monte Carlo method is introduced directly from the underlying microscopic mechanisms, which satisfies detailed balance, gives the correct noise, and describes accurately dynamic and equilibrium states for adsorption-desorption (grand canonical ensemble) processes. It enables simulations of large scales while capturing sharp gradients with molecular resolution at significantly reduced computational cost. A posteriori estimates, in the sense used in finite-elements methods, are developed for assessing errors (information loss) in coarse-graining and guiding mesh generation.
  • Publication
    A Comparison Principle for Hamilton-Jacobi Equations Related to Controlled Gradient Flows in Infinite Dimensions
    (2009-01-01) Feng, J; Katsoulakis, MA
    We develop new comparison principles for viscosity solutions of Hamilton–Jacobi equations associated with controlled gradient flows in function spaces as well as the space of probability measures. Our examples are optimal control of Ginzburg–Landau and Fokker–Planck equations. They arise in limit considerations of externally forced non-equilibrium statistical mechanics models, or through the large deviation principle for interacting particle systems. Our approach is based on two key ingredients: an appropriate choice of geometric structure defining the gradient flow, and a free energy inequality resulting from such gradient flow structure. The approach allows us to handle Hamiltonians with singular state dependency in the nonlinear term, as well as Hamiltonians with a state space which does not satisfy the Radon–Nikodym property. In the case where the state space is a Hilbert space, the method simplifies existing theories by avoiding the perturbed optimization principle.
  • Publication
    Binomial distribution based tau-leap accelerated stochastic simulation
    (2005-01-01) Chatterjee, A; Vlachos, DG; Katsoulakis, MA
    Recently, Gillespie introduced the τ-leap approximate, accelerated stochastic Monte Carlo method for well-mixed reacting systems [J. Chem. Phys. 115, 1716 (2001)]. In each time increment of that method, one executes a number of reaction events, selected randomly from a Poisson distribution, to enable simulation of long times. Here we introduce a binomial distribution τ-leap algorithm (abbreviated as BD-τ method). This method combines the bounded nature of the binomial distribution variable with the limiting reactant and constrained firing concepts to avoid negative populations encountered in the original τ-leap method of Gillespie for large time increments, and thus conserve mass. Simulations using prototype reaction networks show that the BD-τ method is more accurate than the original method for comparable coarse-graining in time.
  • Publication
    Spatially adaptive lattice coarse-grained Monte Carlo simulations for diffusion of interacting molecules
    (2004-01-01) Chatterjee, A; Vlachos, DG; Katsoulakis, MA
    While lattice kinetic Monte Carlo (KMC) methods provide insight into numerous complex physical systems governed by interatomic interactions, they are limited to relatively short length and time scales. Recently introduced coarse-grained Monte Carlo (CGMC) simulations can reach much larger length and time scales at considerably lower computational cost. In this paper we extend the CGMC methods to spatially adaptive meshes for the case of surface diffusion (canonical ensemble). We introduce a systematic methodology to derive the transition probabilities for the coarse-grained diffusion process that ensure the correct dynamics and noise, give the correct continuum mesoscopic equations, and satisfy detailed balance. Substantial savings in CPU time are demonstrated compared to microscopic KMC while retaining high accuracy.
  • Publication
    (2008-01-01) Are, S; Katsoulakis, MA; Plechac, P; Rey-Bellet, L
    In this paper we address the role of multibody interactions for the coarse-grained approximation of stochastic lattice systems. Such interaction potentials are often not included in coarse-graining schemes, as they can be computationally expensive. The multibody interactions are obtained from the error expansion of the reference measure which is, in many cases, chosen as a Gibbs measure corresponding to a local mean-field approximation. We identify the parameter $\epsilon$ that characterizes the level of approximation and its relation to the underlying interaction potential. The error analysis suggests strategies to overcome the computational costs due to evaluations of multibody interactions by additional approximation steps with controlled errors. We present numerical examples demonstrating that the inclusion of multibody interactions shows substantial improvement in dynamical simulations, e.g., of rare events and metastability in phase transitions regimes.
  • Publication
    From microscopic interactions to macroscopic laws of cluster evolution
    (2000-01-01) Katsoulakis, MA; Vlachos, DG
    We derive macroscopic governing laws of growth velocity, surface tension, mobility, critical nucleus size, and morphological evolution of clusters, from microscopic scale master equations for a prototype surface reaction system with long range adsorbate-adsorbate interactions.