Multiscale modeling of physical phenomena: Adaptive control of models

It is common knowledge that the accuracy with which computer simulations can depict physical events depends strongly on the choice of the mathematical model of the events. Perhaps less appreciated is the notion that the error due to modeling can be defined, estimated, and used adaptively to control modeling error, provided one accepts the existence of a base model that can serve as a datum with respect to which other models can be compared. In this work, it is shown that the idea of comparing models and controlling model error can be used to develop a general approach for multiscale modeling, a subject of growing importance in computational science. A posteriori estimates of modeling error in so-called quantities of interest are derived and a class of adaptive modeling algorithms is presented. Several applications of the theory and methodology are presented. These include preliminary work on random multiphase composite materials, molecular statics simulations with applications to problems in nanoindentation, and analysis of molecular dynamics models using various techniques for scale bridging.