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Petr Plecháč

University of Delaware, Mathematical Sciences, Faculty Member

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E. Kalligiannaki

Serge Prudhomme

École Polytechnique de Montréal

Markos Katsoulakis

University of Massachusetts Amherst

University of Massachusetts Amherst

Sorbonne University

Christoph Junghans

Los Alamos National Laboratory

Alexander Gorban

University of Leicester

Giuseppe D'Adamo

Università Degli Studi Dell'Aquila

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Papers by Petr Plecháč

Multilevel coarse graining and nano--pattern discovery in many particle stochastic systems

In this work we propose a hierarchy of Monte Carlo methods for sampling equilibrium properties of... more In this work we propose a hierarchy of Monte Carlo methods for sampling equilibrium properties of stochastic lattice systems with competing short and long range interactions. Each Monte Carlo step is composed by two or more sub - steps efficiently coupling coarse and microscopic state spaces. The method can be designed to sample the exact or controlled-error approximations of the

Error analysis of coarse-grained kinetic Monte Carlo method

In this paper we investigate the approximation properties of the coarse-graining procedure applie... more 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

$Research paper thumbnail of Hierarchical fractional-step approximations and parallel kinetic Monte Carlo algorithms$

Hierarchical fractional-step approximations and parallel kinetic Monte Carlo algorithms

We present a mathematical framework for constructing and analyzing parallel algorithms for lattic... more We present a mathematical framework for constructing and analyzing parallel algorithms for lattice Kinetic Monte Carlo (KMC) simulations. The resulting algorithms have the capacity to simulate a wide range of spatio-temporal scales in spatially distributed, non-equilibrium physiochemical processes with complex chemistry and transport micro-mechanisms. The algorithms can be tailored to specific hierarchical parallel architectures such as multi-core processors or clusters of Graphical Processing Units (GPUs). The proposed parallel algorithms are controlled-error approximations of kinetic Monte Carlo algorithms, departing from the predominant paradigm of creating parallel KMC algorithms with exactly the same master equation as the serial one.

Multibody Interactions in Coarse-Graining Schemes for Extended Systems

by Markos Katsoulakis, Luc Rey-bellet, and Petr Plecháč

SIAM Journal on Scientific Computing, 2009

Page 1. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. SIAM J. SCI... more

Error Analysis of Coarse‐Graining for Stochastic Lattice Dynamics

SIAM Journal on Numerical Analysis, 2006

Page 1. SIAM J. NUMER. ANAL. c 2006 Society for Industrial and Applied Mathematics Vol. 44, No. 6... more

Coarse-graining schemes for stochastic lattice systems with short and long-range interactions

by Luc Rey-bellet, Markos Katsoulakis, and Petr Plecháč

Mathematics of Computation, 2014

We develop coarse-graining schemes for stochastic many-particle microscopic models with competing... more We develop coarse-graining schemes for stochastic many-particle microscopic models with competing short-and long-range interactions on a d-dimensional lattice. We focus on the coarsegraining of equilibrium Gibbs states and using cluster expansions we analyze the corresponding renormalization group map. We quantify the approximation properties of the coarse-grained terms arising from different types of interactions and present a hierarchy of correction terms. We derive semi-analytical numerical schemes that are accompanied with a posteriori error estimates for coarsegrained lattice systems with short and long-range interactions.

Mesoscopic Modeling for Continuous Spin Lattice Systems: Model Problems and Micromagnetics Applications

by Petr Plecháč and M. Katsoulakis

Journal of Statistical Physics, 2005

In this paper we derive deterministic mesoscopic theories for model continuous spin lattice syste... more In this paper we derive deterministic mesoscopic theories for model continuous spin lattice systems both at equilibrium and non-equilibrium in the presence of thermal fluctuations. The full magnetic Hamiltonian that includes singular integral (dipolar) interactions is also considered at equilibrium. The non-equilibrium microscopic models we consider are relaxation-type dynamics arising in kinetic Monte Carlo or Langevin-type simulations of lattice systems. In this context we also employ the derived mesoscopic models to study the relaxation of such algorithms to equilibrium.

Numerical and Statistical Methods for the Coarse-Graining of Many-Particle Stochastic Systems

by Markos Katsoulakis, Luc Rey-bellet, and Petr Plecháč

Journal of Scientific Computing, 2008

In this article we discuss recent work on coarse-graining methods for microscopic stochastic latt... more 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.

Mathematical strategies in the coarse-graining of extensive systems: Error quantification and adaptivity

by Luc Rey-bellet, Markos Katsoulakis, and Petr Plecháč

Journal of Non-Newtonian Fluid Mechanics, 2008

In this paper we continue our study of coarse-graining schemes for stochastic many-body microscop... more 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. (M.A. Katsoulakis), [email protected] (P. Plecháč), [email protected] (L. Rey-Bellet), [email protected] (D.K. Tsagkarogiannis).

Symmetry Breaking Hopf Bifurcations in Equations with O(2) Symmetry with Application to the Kuramoto–Sivashinsky Equation

Journal of Computational Physics, 1997

In problems with O(2) symmetry, the Jacobian matrix at nontrivial steady state solutions with D n... more In problems with O(2) symmetry, the Jacobian matrix at nontrivial steady state solutions with D n symmetry always has a zero eigenvalue due to the group orbit of solutions. We consider bifurcations which occur when complex eigenvalues also cross the imaginary axis and develop a numerical method which involves the addition of a new variable, namely the velocity of solutions drifting round the group orbit, and another equation, which has the form of a phase condition for isolating one solution on the group orbit. The bifurcating branch has a particular type of spatiotemporal symmetry which can be broken in a further bifurcation which gives rise to modulated travelling waves solutions which drift round the group orbit. Multiple Hopf bifurcations are also considered. The methods derived are applied to the Kuramoto-Sivashinsky equation and we give results at two different bifurcations, one of which is a multiple Hopf bifurcation. Our results give insight into the numerical results of Hyman, Nicolaenko and Zaleski (Physica D 23, 265-292).

How accurate is molecular dynamics?

by Petr Plecháč and Christian Bayer

Arxiv preprint arXiv: …, 2011

Abstract. Born-Oppenheimer dynamics is shown to provide an accurate approximation of time-indepen... more

Multilevel coarse graining and nano--pattern discovery in many particle stochastic systems

In this work we propose a hierarchy of Monte Carlo methods for sampling equilibrium properties of... more In this work we propose a hierarchy of Monte Carlo methods for sampling equilibrium properties of stochastic lattice systems with competing short and long range interactions. Each Monte Carlo step is composed by two or more sub - steps efficiently coupling coarse and microscopic state spaces. The method can be designed to sample the exact or controlled-error approximations of the

Error analysis of coarse-grained kinetic Monte Carlo method

In this paper we investigate the approximation properties of the coarse-graining procedure applie... more 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

$Research paper thumbnail of Hierarchical fractional-step approximations and parallel kinetic Monte Carlo algorithms$

Hierarchical fractional-step approximations and parallel kinetic Monte Carlo algorithms

We present a mathematical framework for constructing and analyzing parallel algorithms for lattic... more We present a mathematical framework for constructing and analyzing parallel algorithms for lattice Kinetic Monte Carlo (KMC) simulations. The resulting algorithms have the capacity to simulate a wide range of spatio-temporal scales in spatially distributed, non-equilibrium physiochemical processes with complex chemistry and transport micro-mechanisms. The algorithms can be tailored to specific hierarchical parallel architectures such as multi-core processors or clusters of Graphical Processing Units (GPUs). The proposed parallel algorithms are controlled-error approximations of kinetic Monte Carlo algorithms, departing from the predominant paradigm of creating parallel KMC algorithms with exactly the same master equation as the serial one.

Multibody Interactions in Coarse-Graining Schemes for Extended Systems

by Markos Katsoulakis, Luc Rey-bellet, and Petr Plecháč

SIAM Journal on Scientific Computing, 2009

Page 1. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. SIAM J. SCI... more

Error Analysis of Coarse‐Graining for Stochastic Lattice Dynamics

SIAM Journal on Numerical Analysis, 2006

Page 1. SIAM J. NUMER. ANAL. c 2006 Society for Industrial and Applied Mathematics Vol. 44, No. 6... more

Coarse-graining schemes for stochastic lattice systems with short and long-range interactions

by Luc Rey-bellet, Markos Katsoulakis, and Petr Plecháč

Mathematics of Computation, 2014

We develop coarse-graining schemes for stochastic many-particle microscopic models with competing... more We develop coarse-graining schemes for stochastic many-particle microscopic models with competing short-and long-range interactions on a d-dimensional lattice. We focus on the coarsegraining of equilibrium Gibbs states and using cluster expansions we analyze the corresponding renormalization group map. We quantify the approximation properties of the coarse-grained terms arising from different types of interactions and present a hierarchy of correction terms. We derive semi-analytical numerical schemes that are accompanied with a posteriori error estimates for coarsegrained lattice systems with short and long-range interactions.

Mesoscopic Modeling for Continuous Spin Lattice Systems: Model Problems and Micromagnetics Applications

by Petr Plecháč and M. Katsoulakis

Journal of Statistical Physics, 2005

In this paper we derive deterministic mesoscopic theories for model continuous spin lattice syste... more In this paper we derive deterministic mesoscopic theories for model continuous spin lattice systems both at equilibrium and non-equilibrium in the presence of thermal fluctuations. The full magnetic Hamiltonian that includes singular integral (dipolar) interactions is also considered at equilibrium. The non-equilibrium microscopic models we consider are relaxation-type dynamics arising in kinetic Monte Carlo or Langevin-type simulations of lattice systems. In this context we also employ the derived mesoscopic models to study the relaxation of such algorithms to equilibrium.

Numerical and Statistical Methods for the Coarse-Graining of Many-Particle Stochastic Systems

by Markos Katsoulakis, Luc Rey-bellet, and Petr Plecháč

Journal of Scientific Computing, 2008

In this article we discuss recent work on coarse-graining methods for microscopic stochastic latt... more 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.

Mathematical strategies in the coarse-graining of extensive systems: Error quantification and adaptivity

by Luc Rey-bellet, Markos Katsoulakis, and Petr Plecháč

Journal of Non-Newtonian Fluid Mechanics, 2008

In this paper we continue our study of coarse-graining schemes for stochastic many-body microscop... more 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. (M.A. Katsoulakis), [email protected] (P. Plecháč), [email protected] (L. Rey-Bellet), [email protected] (D.K. Tsagkarogiannis).

Symmetry Breaking Hopf Bifurcations in Equations with O(2) Symmetry with Application to the Kuramoto–Sivashinsky Equation

Journal of Computational Physics, 1997

In problems with O(2) symmetry, the Jacobian matrix at nontrivial steady state solutions with D n... more In problems with O(2) symmetry, the Jacobian matrix at nontrivial steady state solutions with D n symmetry always has a zero eigenvalue due to the group orbit of solutions. We consider bifurcations which occur when complex eigenvalues also cross the imaginary axis and develop a numerical method which involves the addition of a new variable, namely the velocity of solutions drifting round the group orbit, and another equation, which has the form of a phase condition for isolating one solution on the group orbit. The bifurcating branch has a particular type of spatiotemporal symmetry which can be broken in a further bifurcation which gives rise to modulated travelling waves solutions which drift round the group orbit. Multiple Hopf bifurcations are also considered. The methods derived are applied to the Kuramoto-Sivashinsky equation and we give results at two different bifurcations, one of which is a multiple Hopf bifurcation. Our results give insight into the numerical results of Hyman, Nicolaenko and Zaleski (Physica D 23, 265-292).

How accurate is molecular dynamics?

by Petr Plecháč and Christian Bayer

Arxiv preprint arXiv: …, 2011

Abstract. Born-Oppenheimer dynamics is shown to provide an accurate approximation of time-indepen... more