Reduced-Complexity Algorithms for Data Assimilation of Large-Scale Systems.

Data assimilation is the use of measurement data to improve estimates of the state of dynamical systems using mathematical models. Estimates from models alone are inherently imperfect due to the presence of unknown inputs that affect dynamical systems and model uncertainties. Thus, data assimilation is used in many applications: from satellite tracking to biological systems monitoring. As the complexity of the underlying model increases, so does the complexity of the data assimilation technique. This dissertation considers reduced-complexity algorithms for data assimilation of large-scale systems. For linear discrete-time systems, an estimator that injects data into only a specified subset of the state estimates is considered. Bounds on the performance of the new filter are obtained, and conditions that guarantee the asymptotic stability of the new filter for linear time-invariant systems are derived. We then derive a reduced-order estimator that uses a reduced-order model to propagate the estimator state using a finite-horizon cost, and hence solutions of algebraic Riccati and Lyapunov equations are not required. Finally, a reduced-rank square-root filter that propagates only a few columns of the square root of the state-error covariance is developed. Specifically, the columns are chosen from the Cholesky factor of the state-error covariance. Next, data assimilation algorithms for nonlinear systems is considered. We first compare the performance of two suboptimal estimation algorithms, the extended Kalman filter and unscented Kalman filter. To reduce the computational requirements, variations of the unscented Kalman filter with reduced ensemble are suggested. Specifically, a reduced-rank unscented Kalman filter is introduced whose ensemble members are chosen according to the Cholesky decomposition of the square root of the pseudo-error covariance. Finally, a reduced-order model is used to propagate the pseudo-error covariance, while the full-order model is used to propagate the estimator state. To compensate for the neglected correlations, a complementary static estimator gain based on the full-order steady-state correlations is also used. We use these variations of the unscented Kalman filter for data assimilation of one-dimensional compressible flow and two-dimensional magnetohydrodynamic flow.Ph.D.Aerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/58430/1/jchandra_1.pd

Reducing "Structure From Motion": a General Framework for Dynamic Vision - Part 1: Modeling

The literature on recursive estimation of structure and motion from monocular image sequences comprises a large number of different models and estimation techniques. We propose a framework that allows us to derive and compare all models by following the idea of dynamical system reduction. The "natural" dynamic model, derived by the rigidity constraint and the perspective projection, is first reduced by explicitly decoupling structure (depth) from motion. Then implicit decoupling techniques are explored, which consist of imposing that some function of the unknown parameters is held constant. By appropriately choosing such a function, not only can we account for all models seen so far in the literature, but we can also derive novel ones

Distributing the Kalman Filter for Large-Scale Systems

This paper derives a \emph{distributed} Kalman filter to estimate a sparsely connected, large-scale, $n-$dimensional, dynamical system monitored by a network of $N$ sensors. Local Kalman filters are implemented on the ($n_l-$dimensional, where $n_l\ll n$) sub-systems that are obtained after spatially decomposing the large-scale system. The resulting sub-systems overlap, which along with an assimilation procedure on the local Kalman filters, preserve an $L$th order Gauss-Markovian structure of the centralized error processes. The information loss due to the $L$th order Gauss-Markovian approximation is controllable as it can be characterized by a divergence that decreases as $L\uparrow$. The order of the approximation, $L$, leads to a lower bound on the dimension of the sub-systems, hence, providing a criterion for sub-system selection. The assimilation procedure is carried out on the local error covariances with a distributed iterate collapse inversion (DICI) algorithm that we introduce. The DICI algorithm computes the (approximated) centralized Riccati and Lyapunov equations iteratively with only local communication and low-order computation. We fuse the observations that are common among the local Kalman filters using bipartite fusion graphs and consensus averaging algorithms. The proposed algorithm achieves full distribution of the Kalman filter that is coherent with the centralized Kalman filter with an $L$th order Gaussian-Markovian structure on the centralized error processes. Nowhere storage, communication, or computation of $n-$dimensional vectors and matrices is needed; only $n_l \ll n$ dimensional vectors and matrices are communicated or used in the computation at the sensors

Reducing “Structure from Motion”: a general framework for dynamic vision. 1. Modeling

The literature on recursive estimation of structure and motion from monocular image sequences comprises a large number of apparently unrelated models and estimation techniques. We propose a framework that allows us to derive and compare all models by following the idea of dynamical system reduction. The “natural” dynamic model, derived from the rigidity constraint and the projection model, is first reduced by explicitly decoupling structure (depth) from motion. Then, implicit decoupling techniques are explored, which consist of imposing that some function of the unknown parameters is held constant. By appropriately choosing such a function, not only can we account for models seen so far in the literature, but we can also derive novel ones

A New Distribution-Free Concept for Representing, Comparing, and Propagating Uncertainty in Dynamical Systems with Kernel Probabilistic Programming

This work presents the concept of kernel mean embedding and kernel probabilistic programming in the context of stochastic systems. We propose formulations to represent, compare, and propagate uncertainties for fairly general stochastic dynamics in a distribution-free manner. The new tools enjoy sound theory rooted in functional analysis and wide applicability as demonstrated in distinct numerical examples. The implication of this new concept is a new mode of thinking about the statistical nature of uncertainty in dynamical systems

A new framework for extracting coarse-grained models from time series with multiscale structure

In many applications it is desirable to infer coarse-grained models from observational data. The observed process often corresponds only to a few selected degrees of freedom of a high-dimensional dynamical system with multiple time scales. In this work we consider the inference problem of identifying an appropriate coarse-grained model from a single time series of a multiscale system. It is known that estimators such as the maximum likelihood estimator or the quadratic variation of the path estimator can be strongly biased in this setting. Here we present a novel parametric inference methodology for problems with linear parameter dependency that does not suffer from this drawback. Furthermore, we demonstrate through a wide spectrum of examples that our methodology can be used to derive appropriate coarse-grained models from time series of partial observations of a multiscale system in an effective and systematic fashion

Reducing "Structure From Motion": a General Framework for Dynamic Vision - Part 2: Experimental Evaluation

A number of methods have been proposed in the literature for estimating scene-structure and ego-motion from a sequence of images using dynamical models. Although all methods may be derived from a "natural" dynamical model within a unified framework, from an engineering perspective there are a number of trade-offs that lead to different strategies depending upon the specific applications and the goals one is targeting. Which one is the winning strategy? In this paper we analyze the properties of the dynamical models that originate from each strategy under a variety of experimental conditions. For each model we assess the accuracy of the estimates, their robustness to measurement noise, sensitivity to initial conditions and visual angle, effects of the bas-relief ambiguity and occlusions, dependence upon the number of image measurements and their sampling rate

Reducing “Structure from Motion”: a general framework for dynamic vision. 2. Implementation and experimental assessment

For pt.1 see ibid., p.933-42 (1998). A number of methods have been proposed in the literature for estimating scene-structure and ego-motion from a sequence of images using dynamical models. Despite the fact that all methods may be derived from a “natural” dynamical model within a unified framework, from an engineering perspective there are a number of trade-offs that lead to different strategies depending upon the applications and the goals one is targeting. We want to characterize and compare the properties of each model such that the engineer may choose the one best suited to the specific application. We analyze the properties of filters derived from each dynamical model under a variety of experimental conditions, assess the accuracy of the estimates, their robustness to measurement noise, sensitivity to initial conditions and visual angle, effects of the bas-relief ambiguity and occlusions, dependence upon the number of image measurements and their sampling rate