Afrl - Osr - Va - Tr - 2013

This project outlines a collection of problems which combine techniques of model reduction and filtering. The basis of this work is a collection of limit theories for stochastic processes which model dynamical systems with multiple time scales. Multiple time scales occur in many real systems, and reflect different orders of magnitudes of rates of change of different variables. These different time scales often allow one to find effective behaviors of the fast time scales. In systems subject to both bifurcations and noise (which form one of the main components of this project), various singular perturbations problems must be understood. When the rates of change of different variables differ by orders of magnitude, efficient data assimilation can be accomplished by constructing nonlinear filtering equations for the coarse-grained signal. We consider the conditional law of a signal given the observations in a multi-scale context. In particular, we study how scaling interacts with filtering via stochastic averaging. We combine our study of stochastic dimensional reduction and nonlinear filtering to provide a rigorous framework for identifying and simulating filters which are specifically adapted to the complexities of the underlying multi-scale dynamical system. This is the final report for AFOSR GRANT NUMBER FA9550-08-1-0206, which was awarded in August 2008 and ended December 2011. This research involved the work of the PI, N. Sri Namachchivaya, Co-PI, Richard B. Sowers, and graduate students at University of Illinois at Urbana-Champaign (UIUC) in collaboration with a graduate student at the Humboldt University, Berlin, Germany. Introduction The first objective of current project is concerned with certain methods of dimensional reduction of nonlinear systems with symmetries and small noise [1,2]. 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. Reduced models can be used in place of the original complex models, either for simulation and prediction or real-time control. To this end, reduced models [3,4]. often provide qualitatively accurate and computationally feasible descriptions. The second objective is to derive a low-dimensional filtering equation, that determines conditional law of a plant, in a multi-scale environment given the observations. This project is less concerned with specific applications and more focused on some of the theoretical aspects that deal with reduced dimensional nonlinear filters. In particular, we showed the efficient utilisation of the lowdimensional models of the signal to develop a low-dimensional filtering equation. We combine two ingredients, namely, stochastic dimensional reduction discussed above and nonlinear filtering [6,7]. We achieved this through the framework of homogenisation theory which enables us to average out the effects of the fast variables. To introduce the basic idea of filtering in a multi-scale environment, let (Ω,F ,P) be a probability space. We consider nonlinear R × R-valued signal processes (Z, X) and an R-valued observation process Y ε given by the SDE’s (1) dZ t = ε −1a(Zε t , X ε t )dt+ ε −1/2γ(Zε t , X ε t )dWt, Z ε 0 = η dX t = b(Z ε t , X ε t )dt+ σ(Z ε t , X ε t )dVt, X ε 0 = ξ dY ε t = h(Z ε t , X ε t )dt+ dBt, Y ε 0 = 0 where W , V and B are independent Wiener processes and η and ξ are random initial conditions which are independent of W , V and B. Let ε be a small parameter that measures the ratio of slow and fast time scales. Hence the dynamics of (1) are separated into two scales, where Z and X represent the fast and slow variables, respectively. The generator Lε of the Markov processes (Z, X) is then of the form (2) Lεφ = ε LFφ+ LSφ, for all ε ∈ (0, 1) and all φ ∈ C∞(Rm+n), where LF and LS represent generators of fast and slow variables. The main objective of filtering theory is to estimate the statistics of the signal Xt def = (Z t , X ε t ) at time t based on the information in the observation process Y ε up to time t, that is, on the basis of the sigma-algebra, Y ε t . More precisely for each t ≥ 0, we want to find the conditional law of Xt given Y ε t , that is, we want to compute P{Xt ∈ A|Y ε t } for all A ∈ B(R). The insight of filtering theory is that one can construct this conditional measure via a stochastic PDE which is a recursive equation driven by the observation process. If the above conditional measure admits a smooth density, p(t, x) with respect to Lebesgue measure i.e., P{Xt ∈ A|Y ε t } = ∫ A p(t, x)dx for all A ∈ B(R), then u(t, x), the un-normalised density, solves the Zakai equation (3) du(t, x) = L ∗ ε u (t, x)dt+ u(t, x)h(x)dY ε t , u (0, ·) = px, where L ∗ ε is the adjoint operator of the Lε (with respect to Lebesgue measure on R) and X0 has density px. Since we are interested in the slowly-varying coordinates which usually describe the essential coarse-grained dynamics, the focus of our work is to estimate the signal X t at time t on the basis of the sigma-algebra Y ε t . More precisely for each t ≥ 0, we want to find the conditional law P{X t ∈ A|Y ε t } for all A ∈ B(R). For each t ≥ 0, define the C([0, t];R)-valued random variable ~ Y [0,t] as (~ Y [0,t])s(ω) def =