81 research outputs found
Dimensional reduction in nonlinear filtering: A homogenization approach
We propose a homogenized filter for multiscale signals, which allows us to
reduce the dimension of the system. We prove that the nonlinear filter
converges to our homogenized filter with rate . This is
achieved by a suitable asymptotic expansion of the dual of the Zakai equation,
and by probabilistically representing the correction terms with the help of
BDSDEs.Comment: Published in at http://dx.doi.org/10.1214/12-AAP901 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Almost Sure Asymptotic Stability of Parabolic SPDEs with Small Multiplicative Noise
A better understanding of the instability margin will eventually optimize the
operational range for safety-critical industries. In this paper, we investigate
the almost-sure exponential asymptotic stability of the trivial solution of a
parabolic semilinear stochastic partial differential equation (SPDE) driven by
multiplicative noise near the deterministic Hopf bifurcation point. We show the
existence and uniqueness of the invariant measure under appropriate
assumptions, and approximate the exponential growth rate via asymptotic
expansion, given that the strength of the noise is small. This approximate
quantity can readily serve as a robust indicator of the change of almost-sure
stability
Hopf Bifurcations of Moore-Greitzer PDE Model with Additive Noise
The Moore-Greitzer partial differential equation (PDE) is a commonly used mathematical model for capturing flow and pressure changes in axial-flow jet engine compressors. Determined by compressor geometry, the deterministic model is characterized by three types of Hopf bifurcations as the throttle coefficient decreases, namely surge (mean flow oscillations), stall (inlet flow disturbances) or a combination of both. Instabilities place fundamental limits on jet-engine operating range and thus limit the design space. In contrast to the deterministic PDEs, the Hopf bifurcation in stochastic PDEs is not well understood. The goal of this particular work is to rigorously develop low-dimensional approximations using a multiscale analysis approach near the deterministic stall bifurcation points in the presence of additive noise acting on the fast modes. We also show that the reduced-dimensional approximations (SDEs) contain multiplicative noise. Instability margins in the presence of uncertainties can be thus approximated, which will eventually lead to lighter and more efficient jet engine design
Normal form transforms separate slow and fast modes in stochastic dynamical systems
Modelling stochastic systems has many important applications. Normal form
coordinate transforms are a powerful way to untangle interesting long term
macroscale dynamics from detailed microscale dynamics. We explore such
coordinate transforms of stochastic differential systems when the dynamics has
both slow modes and quickly decaying modes. The thrust is to derive normal
forms useful for macroscopic modelling of complex stochastic microscopic
systems. Thus we not only must reduce the dimensionality of the dynamics, but
also endeavour to separate all slow processes from all fast time processes,
both deterministic and stochastic. Quadratic stochastic effects in the fast
modes contribute to the drift of the important slow modes. The results will
help us accurately model, interpret and simulate multiscale stochastic systems
A Homogenization Approach to Multiscale Filtering
AbstractWe present a homogenized nonlinear filter for multi-timescale systems, which allows the reduction of the dimension of filtering equation. We prove that the actual nonlinear filter converges to our homogenized filter. This is achieved by a suitable asymptotic expansion of the dual of the Zakai equation, and probabilistically representing the correction terms with the help of backward doubly-stochastic differential equations. This homogenized filter provides a rigorous mathematical basis for the development of reduced-dimension nonlinear filters for multiscale systems. A filtering scheme, based on the homogenized filtering equation and the technique of importance sampling, is applied to a chaotic multiscale system in Lingala et al. [1]
Particle filtering in high-dimensional chaotic systems
We present an efficient particle filtering algorithm for multiscale systems,
that is adapted for simple atmospheric dynamics models which are inherently
chaotic. Particle filters represent the posterior conditional distribution of
the state variables by a collection of particles, which evolves and adapts
recursively as new information becomes available. The difference between the
estimated state and the true state of the system constitutes the error in
specifying or forecasting the state, which is amplified in chaotic systems that
have a number of positive Lyapunov exponents. The purpose of the present paper
is to show that the homogenization method developed in Imkeller et al. (2011),
which is applicable to high dimensional multi-scale filtering problems, along
with important sampling and control methods can be used as a basic and flexible
tool for the construction of the proposal density inherent in particle
filtering. Finally, we apply the general homogenized particle filtering
algorithm developed here to the Lorenz'96 atmospheric model that mimics
mid-latitude atmospheric dynamics with microscopic convective processes.Comment: 28 pages, 12 figure
- …