I am working on application of various stochastic modeling techniques to various low- and high-dimensional systems. I'm interested in applying these technique to realistic problems in fluid dynamics, while low-dimensional problems serve as prototype testbeds for new ideas.
Joint project with Dr. Azencott
Given a discrete dataset of observations, an important practical problem is to construct a dynamical system which adequately describes these data in some suitable sense. I am primarily interested in obtaining a stochastic system which reproduces the statistical properties of the dataset. Many approaches exist for reconstructing SDEs from time-series, but the parametric estimation combined with the Maximum Likelihood approach is one of the most widely used techniques.
It is typically assumed that the data and the model proposed for estimation are in a perfect agreement, i.e. the data was generated by the model of the same functional form with some unknown parameters. The difficulty arises when it is not the case, i.e. the model proposed for estimation is only approximately agrees with the equation which generated the data; and the equation which generated the data is assumed to be unknown. The Indirect Observability refers to the fact that the underlying model (which generated the data) is unknown and cannot be used for estimation. Moldeing of the observational data also fits in this context since the nature is very unlikely to follow any differential equations exactly.
Using some simple examples, we have shown that parametric estimation can fail if certain conditions on the time-step of observation is not met. Moreover, we developed a criteria to predict and understand compilations with parametric estimation under Indirect Observability. We also constructed new estimators which are unbiased and reproduce the statistical behavior of the data.
Possible projects in this area include extending the analysis of parametric estimation under Indirect Observability to more complex models and analyzing the role of the observational time-step in the estimation procedure and efficiency of the estimators.
Techniques: analysis of stationary properties of stochastic differential equations, computing leading terms in the stationary moments, numerical simulations of SDEs and computation of estimators to analyze the role of the sampling/observational time-step. Some numerical simulations can be done in Matlab, but a C/C++ code can be necessary to tackle some problems.
An important practical question is how to reduced the dimensionality of dynamical systems; i.e. how to use stochastic processes to parametrize the non-essential degrees of freedom to obtain an adequate low-dimensional description for the important (essential) degrees of freedom. Recently, Markov Chain stochastic parametrization has been proposed for this purpose. In the paper with K. Nimsaila "Markov Chain Stochastic Parametrizations of Essential Variables" (SIAM Mult. Mod. Simul. 2010) we demonstrated the applicability of this approach to several prototype models without strong scale separation.
I would like to apply this idea to more realistic models/datasets in atmospheric fluid dynamics. In particular, I would like to model the large-scale structures in the barotropic quasi-geostrophic (QG) equation and apply the Markov Chain parametrization to predict the El-Nino events.
Techniques: this project is fairly computational; I have the C-code for the barotropic QG equation in periodic geometry. Some modifications are necessary for implementing the Markov Chain parametrization into the this model. El-Nino predictions require some data-processing in Matlab and estimating a low-dimensional model for the El-Nino forecasts.