Katsoulakis, Markos
Loading...
Email Address
Birth Date
Research Projects
Organizational Units
Job Title
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.
Name
46 results
Search Results
Now showing 1 - 10 of 46
Publication Stochastic Ising models and anisotropic front propagation(1997) Katsoulakis, MA; Souganidis, PEWe study Ising models with general spin-flip dynamics obeying the detailed balance law. After passing to suitable macroscopic limits, we obtain interfaces moving with normal velocity depending anisotropically on their principal curvatures and direction. In addition we deduce (direction-dependent) Kubo-Green-type formulas for the mobility and the Hessian of the surface tension, thus obtaining an explicit description of anisotropy in terms of microscopic quantities. The choice of dynamics affects only the mobility, a scalar function of the direction.Publication Contractive relaxation systems and the scalar multidimensional conservation law(1997) Katsoulakis, MA; Tzavaras, AEThis Article does not have an abstract.Publication Convergence and error estimates of relaxation schemes for multidimensional conservation laws(1999) Katsoulakis, MA; Kossioris, G; Makridakis, CThis Article does not have an abstract.Publication Relaxation approximations to front propagation(1997) Jin, S; Katsoulakis, MAWe introduce a relaxation model for front propagation problems. Our proposed relaxation approximation is a semilinear hyperbolic system without singularities. It yields a direction-depedent normal velocity at the leading term and captures, in the Chapman–Enskog expansion, the higher order curvature dependent corrections, including possible anisotropies.Publication Multiscale analysis for interacting particles: Relaxation systems and scalar conservation laws(1999) Katsoulakis, MA; Tzavaras, AEWe investigate the derivation of semilinear relaxation systems and scalar conservation laws from a class of stochastic interacting particle systems. These systems are Markov jump processes set on a lattice, they satisfy detailed mass balance (but not detailed balance of momentum), and are equipped with multiple scalings. Using a combination of correlation function methods with compactness and convergence properties of semidiscrete relaxation schemes we prove that, at a mesoscopic scale, the interacting particle system gives rise to a semilinear hyperbolic system of relaxation type, while at a macroscopic scale it yields a scalar conservation law. Rates of convergence are obtained in both scalings.Publication Relaxation schemes for curvature-dependent front propagation(1999) Jin, S; Katsoulakis, MA; Xin, ZPIn this paper we study analytically and numerically a novel relaxation approximation for front evolution according to a curvature-dependent local law. In the Chapman-Enskog expansion, this relaxation approximation leads to the level-set equation for transport-dominated front propagation, which includes the mean curvature as the next-order term. This approach yields a new and possibly attractive way of calculating numerically the propagation of curvature-dependent fronts. Since the relaxation system is a symmetrizable, semilinear, and linearly convective hyperbolic system without singularities, the relaxation scheme captures the curvature-dependent front propagation without discretizing directly the complicated yet singular mean curvature term.Publication Contractive relaxation systems and interacting particles for scalar conservation laws(1996) Katsoulakis, MA; Tzavaras, AENous 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.Publication ERROR ANALYSIS OF COARSE-GRAINED KINETIC MONTE CARLO METHOD(2005-01) Katsoulakis, MA; Plechac, P; Sopasakis, AIn 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 Numerical and statistical methods for the coarse-graining of many-particle stochastic systems(2008-01) Katsoulakis, MA; Plechac, P; Rey-Bellet, LIn 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 MULTIBODY INTERACTIONS IN COARSE-GRAINING SCHEMES FOR EXTENDED SYSTEMS(2008-01) Are, S; Katsoulakis, MA; Plechac, P; Rey-Bellet, LIn 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.