7,426 research outputs found
Nonparametric Uncertainty Quantification for Stochastic Gradient Flows
This paper presents a nonparametric statistical modeling method for
quantifying uncertainty in stochastic gradient systems with isotropic
diffusion. The central idea is to apply the diffusion maps algorithm to a
training data set to produce a stochastic matrix whose generator is a discrete
approximation to the backward Kolmogorov operator of the underlying dynamics.
The eigenvectors of this stochastic matrix, which we will refer to as the
diffusion coordinates, are discrete approximations to the eigenfunctions of the
Kolmogorov operator and form an orthonormal basis for functions defined on the
data set. Using this basis, we consider the projection of three uncertainty
quantification (UQ) problems (prediction, filtering, and response) into the
diffusion coordinates. In these coordinates, the nonlinear prediction and
response problems reduce to solving systems of infinite-dimensional linear
ordinary differential equations. Similarly, the continuous-time nonlinear
filtering problem reduces to solving a system of infinite-dimensional linear
stochastic differential equations. Solving the UQ problems then reduces to
solving the corresponding truncated linear systems in finitely many diffusion
coordinates. By solving these systems we give a model-free algorithm for UQ on
gradient flow systems with isotropic diffusion. We numerically verify these
algorithms on a 1-dimensional linear gradient flow system where the analytic
solutions of the UQ problems are known. We also apply the algorithm to a
chaotically forced nonlinear gradient flow system which is known to be well
approximated as a stochastically forced gradient flow.Comment: Find the associated videos at: http://personal.psu.edu/thb11
Universal Nonlinear Filtering Using Feynman Path Integrals II: The Continuous-Continuous Model with Additive Noise
In this paper, the Feynman path integral formulation of the
continuous-continuous filtering problem, a fundamental problem of applied
science, is investigated for the case when the noise in the signal and
measurement model is additive. It is shown that it leads to an independent and
self-contained analysis and solution of the problem. A consequence of this
analysis is Feynman path integral formula for the conditional probability
density that manifests the underlying physics of the problem. A corollary of
the path integral formula is the Yau algorithm that has been shown to be
superior to all other known algorithms. The Feynman path integral formulation
is shown to lead to practical and implementable algorithms. In particular, the
solution of the Yau PDE is reduced to one of function computation and
integration.Comment: Interdisciplinary, 41 pages, 5 figures, JHEP3 class; added more
discussion and reference
Ensemble Kalman methods for high-dimensional hierarchical dynamic space-time models
We propose a new class of filtering and smoothing methods for inference in
high-dimensional, nonlinear, non-Gaussian, spatio-temporal state-space models.
The main idea is to combine the ensemble Kalman filter and smoother, developed
in the geophysics literature, with state-space algorithms from the statistics
literature. Our algorithms address a variety of estimation scenarios, including
on-line and off-line state and parameter estimation. We take a Bayesian
perspective, for which the goal is to generate samples from the joint posterior
distribution of states and parameters. The key benefit of our approach is the
use of ensemble Kalman methods for dimension reduction, which allows inference
for high-dimensional state vectors. We compare our methods to existing ones,
including ensemble Kalman filters, particle filters, and particle MCMC. Using a
real data example of cloud motion and data simulated under a number of
nonlinear and non-Gaussian scenarios, we show that our approaches outperform
these existing methods
Convergence of discrete time Kalman filter estimate to continuous time estimate
This article is concerned with the convergence of the state estimate obtained
from the discrete time Kalman filter to the continuous time estimate as the
temporal discretization is refined. We derive convergence rate estimates for
different systems, first finite dimensional and then infinite dimensional with
bounded or unbounded observation operators. Finally, we derive the convergence
rate in the case where the system dynamics is governed by an analytic
semigroup. The proofs are based on applying the discrete time Kalman filter on
a dense numerable subset of a certain time interval .Comment: Author's version of the manuscript accepted for publication in
International Journal of Contro
Recent advances on filtering and control for nonlinear stochastic complex systems with incomplete information: A survey
This Article is provided by the Brunel Open Access Publishing Fund - Copyright @ 2012 Hindawi PublishingSome recent advances on the filtering and control problems for nonlinear stochastic complex systems with incomplete information are surveyed. The incomplete information under consideration mainly includes missing measurements, randomly varying sensor delays, signal quantization, sensor saturations, and signal sampling. With such incomplete information, the developments on various filtering and control issues are reviewed in great detail. In particular, the addressed nonlinear stochastic complex systems are so comprehensive that they include conventional nonlinear stochastic systems, different kinds of complex networks, and a large class of sensor networks. The corresponding filtering and control technologies for such nonlinear stochastic complex systems are then discussed. Subsequently, some latest results on the filtering and control problems for the complex systems with incomplete information are given. Finally, conclusions are drawn and several possible future research directions are pointed out.This work was supported in part by the National Natural Science Foundation of China under Grant nos. 61134009, 61104125, 61028008, 61174136, 60974030, and 61074129, the Qing Lan Project of Jiangsu Province of China, the Project sponsored by SRF for ROCS of SEM of China, the Engineering and Physical Sciences Research Council EPSRC of the UK under Grant GR/S27658/01, the Royal Society of the UK, and the Alexander von Humboldt Foundation of Germany
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