Award Abstract #0413864
Multiscale Stochastic Modeling, Analysis and Computation

NSF Org:	DMS Division of Mathematical Sciences
Initial Amendment Date:	September 23, 2004
Latest Amendment Date:	September 26, 2007
Award Number:	0413864
Award Instrument:	Standard Grant
Program Manager:	Junping Wang DMS Division of Mathematical Sciences MPS Directorate for Mathematical & Physical Sciences
Start Date:	September 15, 2004
Expires:	August 31, 2008 (Estimated)
Awarded Amount to Date:	$314454
Investigator(s):	Markos Katsoulakis markos@math.umass.edu (Principal Investigator)
Sponsor:	University of Massachusetts Amherst Research Administration Building AMHERST, MA 01003 413/545-0698
NSF Program(s):	MSPA-INTERDISCIPLINARY, COMPUTATIONAL MATHEMATICS, STATISTICS, APPLIED MATHEMATICS
Field Application(s):	0000099 Other Applications NEC
Program Reference Code(s):	OTHR, 9263, 7454, 7303, 7237, 1271, 1269, 1266, 0000
Program Element Code(s):	7454, 1271, 1269, 1266

ABSTRACT

Problems in diverse scientific disciplines ranging from materials science to macromolecular dynamics, to atmosphere/ocean science and climate modeling involve the nonlinear interaction of physical processes across many length and time scales ranging from the microscopic to the macroscopic. From a modeling and computational perspective, microscopic simulation methods such as Molecular Dynamics (MD) and Monte

Carlo (MC) algorithms can describe complex, out of equilibrium interactions at small scales (e.g. between atoms or molecules). Although there is substantial progress in improving aspects of these computational methods, they are still limited to short length and time scales, while on the other hand device sizes and morphological features observed in experiments often involve much larger scales; at the same time stochastic

fluctuations--inherited from the microscopics--can be important, for instance in self-organization problems characterized by small coherent structures such as pattern formation in nanotechnology applications. In addition to such challenges posed by the disparity in scales within the same model, in many instances we are faced with an additional disparity in models: for example in phenomena with detailed fluid/surface or boundary layer interactions it is necessary to couple microscopic, possibly stochastic models describing the dynamics of atoms or molecules on a surface, along with continuum PDE for species, fluid and thermodynamic variables on the overlying to the surface gas phase. It is therefore inevitable that features of the microscopic model will essentially enter as a subgrid effect in the coupling with the coarse computational grid of the macroscopic PDE models. In this case, the proper incorporation and simulation of stochastic effects from the subgrid microscale is a critical element in the modeling and simulations. The proposed projects focus on aspects of the aforementioned issues by putting forward a combination of interconnected modeling, computational and analysis questions, roughly divided in two categories: (i) Mathematical strategies for the coarse-graining of microscopic models and the corresponding

simulators, addressing problems which are currently intractable with conventional MD/MC due to scale limitations. Here, it is not directly attempted to speed up microscopic simulation algorithms; instead, a hierarchy of new coarse-grained stochastic models (referred to as Coarse Grained Monte Carlo methods) is derived, ordered by the magnitude of space/time scales. This new set of models involves a reduced set of observables over the original microscopic models incorporating microscopic details and noise, as well as

the interaction of the unresolved degrees of freedom. (ii) Hybrid stochastic/deterministic systems describing detailed fluid-surface interactions arise in applications that range from deposition process and catalysis to fuel cell design and microreactors, to biology and atmosphere and ocean science. Here two such applications are addressed, namely catalytic reactors and stochastic parametrizations of tropical convection.

Due to their inherent complexity it is also necessary to develop simpler, test bed problems that capture significant features of the multi-scale nature of the physical models, but are still amenable to asymptotics, mathematical analysis and tractable computations.

The modeling and simulation of problems with multiple interrelating length and time scales is one of the preeminent issues in essentially all timely scientific and engineering challenges, ranging from the design of nanodevices, to biomolecular dynamics, to the spread of epidemics and climate modeling. In spite of a continuously increasing computing power many of these problems remain intractable, at least in realistic conditions, and new modeling and simulation strategies need to be developed. As several paradigms

have recently demonstrated, the critical step in this process is the use of (the limited in number and flexibility) existing multiscale mathematical and statistical tools as well as the development of new ones, that enable the creation of new algorithms for complex systems. In the proposed work an array of such novel multiscale mathematics and computing methods is developed. The research is motivated by and targeted to a

number of the aforementioned applications. Two such particular examples are, (i) the surface processes in catalytic reactors and fuel cells, and (ii) the tropical and open ocean convection.

PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH

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A. Sopasakis and M. A. Katsoulakis.  "Stochastic modeling and simulation of traffic flow: ASEP with Arrhenius look-ahead dynamics,"  SIAM J. Appl. Math.,  v.66,  2006,  p. 921.

A.~Chatterjee, D.~Vlachos, and M.~A. Katsoulakis,.  "Numerical assessment of theoretical error estimates in coarse-grained kinetic Monte Carlo simulations: Application to surface diffusion,"  Int. J. Multiscale Comp. Eng,  v.3,  2005,  p. 1.

A.~Chatterjee, D.~Vlachos, and M.~A. Katsoulakis. .  "Spatially adaptive lattice coarse-grained monte carlo simulations for diffusion of interacting molecules. ,"  J. Chem. Phys.,  v.121,  2004,  p. 11420.

A.~Chatterjee, M.~Katsoulakis, and D.~Vlachos. \.  "Spatially adaptive grand canonical ensemble monte carlo simulations.,"  Phys. Rev. E,  v.71,  2005,  p. 0267021.

M. A. Katsoulakis and J. Trashorras.  "Information loss in coarse-graining of stochastic particle dynamics,"  J. Stat. Phys.,  v.122,  2006,  p. 115.

M. A. Katsoulakis, A. J. Majda and A. Sopasakis.  "Intermittency, metastability and coarse-graining in deterministic/stochastic lattice systems,"  Nonlinearity,  v.19,  2006,  p. 1021.

M. A. Katsoulakis, A. J. Majda and A. Sopasakis.  "Multiscale couplings in prototype hybrid deterministic/stochastic systems: Part I, deterministic closures,"  Comm. Math. Sci.,  v.2,  2004,  p. 255.

M. A. Katsoulakis, A. J. Majda and A. Sopasakis.  "Multiscale couplings in prototype hybrid deterministic/stochastic systems: Part II: Stochastic closures,"  Comm. Math. Sci.,  v.3,  2005,  p. 453.

M. A. Katsoulakis, P. Plechac and D. Tsagkarogiannis.  "Mesoscopic modeling for continuous spin lattice systems: Model problems and micromagnetics applications,"  J. Stat. Phys.,  v.118,  2005,  p. 347.

M.~A. Katsoulakis and A.~Szepessy. .  "Stochastic hydrodynamical limits of particle systems.,"  Commun. Math. Sci.,  v.4,  2006,  p. 513.

M.~A. Katsoulakis, G.~T. Kossioris, and O.~Lakkis. \.  "A finite element method via noise regularization for the stochastic allen-cahn problem,"  J. Interfaces and Free Boundaries,  v.9,  2007,  p. 1.

M.~A. Katsoulakis, P.~Plech{c, and A.~Sopasakis. .  "Error analysis of coarse-graining for stochastic lattice dynamics. ,"  SIAM J Num. Anal.,  v.44,  2006,  p. 2270.

T.~M. Davis, T.~O. Drews, H.~Ramanan, C.~He, J.~Dong, H.~Schnablegger, M.~A. Katsoulakis, E.~Kokkoli, A.~V. McCormick, R.~L. Penn, and M.~Tsapatsis..  "Mechanistic principles of nanoparticle evolution to zeolite crystals.,"  Nature-Materials,  v.5,  2006,  p. 400.

T.~O. Drews, M.~A. Katsoulakis, and M.~Tsapatsis. .  "A mathematical model for crystal growth by oriented aggregation of precursor metastable nanoparticles.,"  J. Phys. Chem. B,  v.109,  2005,  p. 23879.


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