Frontiers of Engineering Reports on Leading-Edge Engineering from the 2004 NAE Symposium on Frontiers of Engineering (2005) / Chapter Skim
Currently Skimming:
Equation-Free Modeling for Complex Systems--Ioannis G. Kevrekidis
Equation-Free Modeling for Complex Systems--Ioannis G. Kevrekidis
Pages 69-76
The Chapter Skim interface presents what we've algorithmically identified as the most significant single chunk of text within every page in the chapter.
Select key terms on the right to highlight them within pages of the chapter.
From page 69... ...
Traditional modeling approaches first derive macroscopic evolution equations from the microscopic models and then bring 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.
|
From page 70... ...
For laminar Newtonian fluid mechanics, a successful coarse-grained description, the Navier-Stokes equations, was known on a phenomenological basis long before its approximate derivation from kinetic theory. Today we must frequently study systems for which the physics can be modeled at a microscopic, fine scale; but for whose macroscopic behavior explicit equations are practically impossible to derive.
|
From page 71... ...
We will argue that, in many cases, the derivation of macroscopic equations can be circumvented -- that by using short bursts of appropriately initialized microscopic simulation, one can effectively solve the macroscopic equations without ever writing them down. Thus, a direct bridge can be built between microscopic simulation (e.g., kinetic Monte Carlo [kMC]
|
From page 72... ...
We are in effect doing forward Euler again, but the coefficients of the local linear model are obtained using experimentation "on demand" (Cybenko, 1996) rather than function evaluations of an a priori available model.
|
From page 73... ...
is the same one we would use if we had the macroscopic equation, but now function evaluations are substituted by short computational experiments with the microscopic simulator, whose results are appropriately processed for local macroscopic identification and estimation. If a large separation of time scales exists between microscopic dynamics (here, the time we need to run kMC to estimate dc/dt)
|
From page 74... ...
But when these equations are not available in closed form, and such cases arise with increasing frequency in contemporary modeling, the equation-free computational enabling technology we have outlined here may hold the key to engineering effectively simple systems. ACKNOWLEDGMENTS This work was partially supported over the years by AFOSR, through an NSF/ITR grant, DARPA, and Princeton University.
|
From page 75... ...
coarse projective integration and (b) coarse time-stepper-based bifurcation computations.
|
From page 76... ...
2002a. Coarse stability and bifurcation analysis using stochastic simulators: kinetic Monte Carlo examples.
|
Key Terms
This material may be derived from roughly machine-read images, and so is provided only to facilitate research.
More
information on Chapter Skim is available.