Applied Mathematics and Computation Seminar
Hierarchical Pattern Discovery in Many-Body Complex Stochastic Systems
Markos Katsoulakis, UMass Amherst

We develop a hierarchical approach for pattern discovery in many-body stochastic systems, motivated by challenges in guiding engineering tasks in nanopattern formation in heteroepitaxial processes. Patterns in such systems have rich morphologies at mesoscales that change dramatically as control parameters vary; typically they form as a result of microscopic particle dynamics in a complex landscape, in the presence of stochastic fluctuations. Developing a complete understanding of the pattern formation mechanisms as functions of the control parameters of the system is a vast computational challenge which is currently intractable with conventional Kinetic Monte Carlo methods. Here we present hierarchical strategies towards this "systems' task" goal by combining mesoscopic PDE and Coarse-Grained Monte Carlo (CGMC) approximations of KMC algorithms that we have developed in earlier work. More precisely, (i) we employ deterministic mesoscopic PDE as means to obtain an approximate (and in principle rather crude) phase diagram of the system; subsequently, (ii) we employ adaptive CGMC at selected regions of the approximate phase diagram in order to refine it by including interactions and fluctuations properly. Our adaptivity framework allows us to obtain accurate and near-optimal coarse-grainings for each parameter regime, ensuring proper "knowledge representation"-in an information theory sense-of the complex system for the desired observables, e.g., spatial correlations, power spectra or scaling laws. In turn such tools can be also used in model reduction and control of the underlying complex systems.

Refreshments at 3:45 in 1634

4:00pm–5:00pm, Tuesday, September 8, 2009 in LGRT 1528

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