Pacific Institute for the Mathematical Sciences - PIMS

This talk outlines multiscale problems which combine techniques of model reduction and
filtering. Multiple time scales occur in models throughout the sciences and engineering, where
the rates of change of different variables differ by orders of magnitude. The basis of this work
is a collection of limit theories for stochastic processes which model dynamical systems with
multiple time scales.
The first objective of this talk is concerned with certain methods of dimensional reduction of
nonlinear systems with symmetries and small noise. In the presence of a separation of scales,
where the noise is asymptotically small, one exploits symmetries to use recent mathematical results
concerning stochastic averaging to find an appropriate lower-dimensional description of the
system. The unique features of the problem are interactions between bifurcations, resonances,
dissipation and random perturbations. Bifurcations are where small changes in a system result
in large changes in the structure of the fast orbits. The subtleties of the interaction between
these effects lead to new and novel analytical techniques. Hence, we are developing techniques
of stochastic dimensional reduction to find a simpler model which predicts or captures relevant
dynamics of the system. One of the preeminent modern frameworks for considering convergence
of the laws of Markov processes is that of the martingale problem, which we use in deriving the
coarse-grained dynamics.
State estimation of random dynamical systems with noisy observations has been an important
problem in many areas of science and engineering. Since the true state is usually hidden and
evolves according to its own dynamics, the objective is to get an optimal estimation of the
true state via noisy observations. The theory of filtering attempts to give a recursive procedure
for estimating an evolving signal or state from a noisy observation process. The second
objective of this talk is to develop, with mathematical rigor, a lower - dimensional nonlinear
filter by combining two ingredients, namely, stochastic dimensional reduction discussed above
and nonlinear filtering. We find a reduced nonlinear filtering problem when the system dimension
can be reduced via homogenization. We approximate the complex original nonlinear
filtering equations by simpler ones with a quantifiable error. This talk is focused on some of
the theoretical aspects that deal with reduced dimensional nonlinear filters. In particular, we
show how scaling interacts with filtering. Roughly speaking, we show the efficient utilisation of
the low-dimensional models of the signal and develop a low-dimensional filtering equation. We
achieve this through the framework of homogenisation theory which enables us to average out
the effects of the fast variables.
The authors would like to acknowledge the support of the AFOSR under grant number FA9550-
08-1-0206 and the National Science Foundation under grant numbers CNS 05-40237 and CMMI
07-58569. This is a joint work with Vishal Chikkerur, Nishanth Lingala, Kristjan Onu, Jun H.
Park, Peter W. Sauer, Richard B. Sowers and Hoong Chieh Yeong.

2:00pm, CAB 239.