Numerical Methods for Plasma Astrophysics:
From Particle Kinetics to MHD

CSIC Building (#406), Seminar Room 4122.
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Recent Developments in Equation-free Complex Systems Modeling

Dr. Yannis Kevrekidis

Chemical Engineering at Princeton University

Abstract:   In current modeling, the best available descriptions of a system often come at a fine level (atomistic, stochastic, microscopic, individual-based) while the questions asked and the tasks required by the modeler (prediction, parametric analysis, optimization and control) are at a much coarser, averaged, macroscopic level. Traditional modeling approaches start by first deriving macroscopic evolution equations from the microscopic models, and then bringing our arsenal of mathematical and algorithmic tools to bear on these macroscopic descriptions. Over the last few years, and with several collaborators, we have developed and validated a mathematically inspired, computational enabling technology that allows the modeler to perform macroscopic tasks acting on the microscopic models directly. We call this the 'equation-free' approach, since it circumvents the step of obtaining accurate macroscopic descriptions. The backbone of this approach is the design of (computational) experiments. Traditional continuum numerical algorithms can be viewed as a set protocols for experimental design (where ?experiment? means a computational experiment set up and performed with a model at a different level of description). I will discuss several examples studied over the last year, including the rheology of nematic liquid crystals, coarse molecular dynamics, individual-based modeling, and issues of bifurcation, optimization and control. Ultimately, what makes it all possible is the ability to initialize computational experiments at will. Short bursts of appropriately initialized computational experimentation ­through matrix-free numerical analysis and systems theory tools like variance reduction and estimation- bridges microscopic simulation with macroscopic modeling.