Yannis Kevrekidis (Princeton University)
Abstract. In current modeling practice for complex systems (including agent-based models and evolving networks) 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 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. For several years now, 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.
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 — bridge microscopic simulation with macroscopic modeling.
I will discuss the approach and illustrate it through several applications in science and engineering; I will also discuss how to link the approach with recent developments in machine learning/data mining algorithms exploring large complex data sets to find good “reduction coordinates.”
In a sense, the approach could be thought of as a data-based calculus for the modeling of complex systems.