Title: Equation-Free Computations for complex/multiscale systems


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
an 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 \u201cequation-free\u201d approach, since it circumvents the step of 
obtaining accurate macroscopic descriptions. The
backbone of this approach is the design of (computational) experiments. 
  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 in 
general (and bifurcation computations in particular).

I will briefly discuss the approach and present a selection of 
illustrative examples arising in a biological modeling context, ranging 
from coarse molecular dynamics to coarse kinetic Monte Carlo and 
agent-based simulations.