On the development of numerical methods driven by atomistic phase space statistics

Recent trends in chemical engineering have led researchers to study systems characterized by length and time scales that necessitate atomistic resolution because traditional continuous descriptions fail to accurately describe the system dynamics. The atomistic models used typically consist of complex N-body systems, however the relevant information (from an engineering standpoint) can often be adequately summarized through a small set of "reaction coordinates". The reaction coordinates can be observables of the system that come from physical intuition or through transformations motivated by data mining techniques; in this thesis, we deal exclusively with the former case.; Time series of the reaction coordinate(s) are generated by detailed atomistic simulations. Surrogate models of the reaction coordinate dynamics are then extracted from this information. This thesis focuses mainly on issues associated with efficiently estimating simple low-dimensional diffusion models when one has both stationary and nonstationary data. If a finite dimensional parametric model is used, efficient estimation can sometimes be achieved by appealing to maximum likelihood estimation (MLE). Unfortunately, parametric models describing the global dynamics are rarely known a priori in our systems.; The main contribution is the merging of a local estimation strategy (that utilizes the advantages of a finite dimensional parametric model while retaining the flexibility associated with a piecewise polynomial) into the so-called "equation-free" computational framework. A variety of illustrative applications relevant to small scale chemical reactor engineering are presented. Extensions of the basic idea are also used to estimate the equilibrium free energy difference via nonequilibrium molecular dynamics simulation data.