Multiscale analysis: modeling and computations
Systems evolving on widely separated time-scales represent a challenge for numerical simulations. Standard computational schemes fail due to the wide separation between the fastest time-scale in the system one must compute with, and the slowest time-scales one is typically interested in analyzing the solutions.
Traditionally in the context of ODE, the method of choice for integrating stiff systems has been to use implicit schemes. Unfortunately, such schemes become ineffective when the dynamics of the system is stochastic in some way, e.g. if it is governed by a stochastic differential equation, a Markov chain as in kinetic Monte Carlo methods, or even by a large set of ODEs whose solutions are chaotic. In these situations, alternative numerical techniques must be developed.
Recently a new kind of numerical methods for multiscale dynamical systems with stochastic effects has been introduced. These methods build on asymptotic techniques and limit theorems for singularly perturbed Markov processes, originally developed in the 70s by Khasminskii, Kurtz, Papanicolaou, etc. These limit theorems provide one with closed effective equations for the slow variables in the system, and the coefficients in these equations are given by expectations over the statistics of the fast variables conditional on the value of the slow variables. In general, these expectations cannot be computed analytically, but it is possible to estimate them on-the-fly when needed via short runs of the fast variables. Once this is done, the slow variables can be evolved using the effective equations by one macro-time-step, and the procedure can be repeated.
This section contains our works in this area. There are papers describing the general methodolgy behind multiscale computation of the type above; papers applying this methodology to kinetic Monte-Carlo schemes used e.g. in chemical kinetics, to climate modeling, and to other applications. Finally, this section contains papers on parameter estimation, where the coefficients in the effective equations for the slow variables are estimated directly from the timeseries rather than by the multiscale procedure described above.