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An introduction to quantum filtering
, 2006
"... This paper provides an introduction to quantum filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation as a ..."
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Cited by 78 (19 self)
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This paper provides an introduction to quantum filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation as a
Exponential Stability for Nonlinear Filtering
, 1996
"... We study the a.s. exponential stability of the optimal filter w.r.t. its initial conditions. A bound is provided on the exponential rate (equivalently, on the memory length of the filter) for a general setting both in discrete and in continuous time, in terms of Birkhoff's contraction coefficie ..."
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Cited by 57 (2 self)
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We study the a.s. exponential stability of the optimal filter w.r.t. its initial conditions. A bound is provided on the exponential rate (equivalently, on the memory length of the filter) for a general setting both in discrete and in continuous time, in terms of Birkhoff's contraction coefficient. Criteria for exponential stability and explicit bounds on the rate are given in the specific cases of a diffusion process on a compact manifold, and discrete time Markov chains on both continuous and discretecountable state spaces. R'esum'e Nous 'etudions la stabilit'e du filtre optimal par raport `a ses conditions initiales. Le taux de d'ecroissance exponentielle est calcul'e dans un cadre g'en'eral, pour temps discret et temps continu, en terme du coefficient de contraction de Birkhoff. Des crit`eres de stabilit'e exponentielle et des bornes explicites sur le taux sont calcul'ees pour les cas particuliers d'une diffusion sur une vari'ete compacte, ainsi que pour des chaines de Markov sur ...
On Unscented Kalman Filtering for State Estimation of ContinuousTime Nonlinear Systems
, 2007
"... This article considers the application of the unscented Kalman filter (UKF) to continuoustime filtering problems, where both the state and measurement processes are modeled as stochastic differential equations. The mean and covariance differential equations which result in the continuoustime lim ..."
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Cited by 32 (8 self)
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This article considers the application of the unscented Kalman filter (UKF) to continuoustime filtering problems, where both the state and measurement processes are modeled as stochastic differential equations. The mean and covariance differential equations which result in the continuoustime limit of the UKF are derived. The continuousdiscrete unscented Kalman filter is derived as a special case of the continuoustime filter, when the continuoustime prediction equations are combined with the update step of the discretetime unscented Kalman filter. The filter equations are also transformed into sigmapoint differential equations, which can be interpreted as matrix square root versions of the filter equations.
A Discrete Invitation to Quantum Filtering and Feedback Control
, 2009
"... The engineering and control of devices at the quantum mechanical level—such as those consisting of small numbers of atoms and photons—is a delicate business. The fundamental uncertainty that is inherently present at this scale manifests itself in the unavoidable presence of noise, making this a nov ..."
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Cited by 25 (4 self)
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The engineering and control of devices at the quantum mechanical level—such as those consisting of small numbers of atoms and photons—is a delicate business. The fundamental uncertainty that is inherently present at this scale manifests itself in the unavoidable presence of noise, making this a novel field of application for stochastic estimation and control theory. In this expository paper we demonstrate estimation and feedback control of quantum mechanical systems in what is essentially a noncommutative version of the binomial model that is popular in mathematical finance. The model is extremely rich and allows a full development of the theory while remaining completely within the setting of finitedimensional Hilbert spaces (thus avoiding the technical complications of the continuous theory). We introduce discretized models of an atom in interaction with the electromagnetic field, obtain filtering equations for photon counting and homodyne detection, and solve a stochastic control problem using dynamic programming and Lyapunov function methods.
Weighted Stochastic Sobolev Spaces and Bilinear SPDE's Driven by Spacetime White Noise
"... In this paper we develop basic elements of Malliavin calculus on a weighted L 2(\Omega\Gamma2 This class of generalized Wiener functionals is a Hilbert space. It turns out to be substantially smaller than the space of Hida distributions while large enough to accommodate solutions of bilinear stoch ..."
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Cited by 20 (5 self)
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In this paper we develop basic elements of Malliavin calculus on a weighted L 2(\Omega\Gamma2 This class of generalized Wiener functionals is a Hilbert space. It turns out to be substantially smaller than the space of Hida distributions while large enough to accommodate solutions of bilinear stochastic PDE's. As an example, we consider a stochastic advectiondiffusion equation driven by spacetime white noise in IR d . It is known that for d ? 1, this equation has no solutions in L 2(\Omega\Gamma4 In contrast, it is shown in the paper that in an appropriately weighted L 2(\Omega\Gamma there is a unique solution to the stochastic advectiondiffusion equation for any d 1. In addition we present explicit formulas for the HermiteFourier coefficients in the Wiener chaos expansion of the solution.
Implicit particle filters for data assimilation
, 2010
"... Implicit particle filters for data assimilation update the particles by first choosing probabilities and then looking for particle locations that assume them, guiding the particles one by one to the high probability domain. We provide a detailed description of these filters, with illustrative exampl ..."
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Cited by 19 (13 self)
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Implicit particle filters for data assimilation update the particles by first choosing probabilities and then looking for particle locations that assume them, guiding the particles one by one to the high probability domain. We provide a detailed description of these filters, with illustrative examples, together with new, more general, methods for solving the algebraic equations and with a new algorithm for parameter identification. 1
3D Object Tracking Using ShapeEncoded Particle Propagation
 In: International Conference on Computer Vision
, 2001
"... We present a comprehensive treatment of 3D object tracking by posing it as a nonlinear state estimation problem. The measurements are derived using the outputs of shapeencoded filters. The nonlinear state estimation is performed by solving the Zakai equation, and we use the branching particle propa ..."
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Cited by 14 (2 self)
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We present a comprehensive treatment of 3D object tracking by posing it as a nonlinear state estimation problem. The measurements are derived using the outputs of shapeencoded filters. The nonlinear state estimation is performed by solving the Zakai equation, and we use the branching particle propagation method for computing the solution. The unnormalized conditional density for the solution to the Zakai equation is realized by the weight of the particle. We first sample a set of particles approximating the initial distribution of the state vector conditioned on the observations, where each particle encodes the set of geometric parameters of the object. The weight of the particle represents geometric and temporal fit, which is computed bottomup from the raw image using a shapeencoded filter. The particles branch so that the mean number of offspring is proportional to the weight. Time update is handled by employing a secondorder motion model, combined with local stochastic search to minimize the prediction error. The prediction adjustment suggested by system identification theory is empirically verified to contribute to global stability. The amount of diffusion is effectively adjusted using a Kalman updating of the covariance matrix. We have successfully applied this method to human head tracking, where we estimate head motion and compute structure using simple head and facial feature models. 1
ON THE SEPARATION PRINCIPLE OF QUANTUM CONTROL
, 2005
"... It is well known that continuous quantum measurements and nonlinear filtering can be developed within the framework of the quantum stochastic calculus of HudsonParthasarathy. The addition of realtime feedback control has been discussed by many authors, but never in a rigorous way. Here we introdu ..."
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Cited by 11 (2 self)
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It is well known that continuous quantum measurements and nonlinear filtering can be developed within the framework of the quantum stochastic calculus of HudsonParthasarathy. The addition of realtime feedback control has been discussed by many authors, but never in a rigorous way. Here we introduce the notion of a controlled quantum flow, where feedback is taken into account by allowing the coefficients of the quantum stochastic differential equation to be adapted processes in the observation algebra. We then prove a separation theorem for quantum control: the admissible control that minimizes a given cost function is only a function of the filter, provided that the associated Bellman equation has a sufficiently regular solution. Along the way we obtain results on the innovations problem in the quantum setting.
Finitedimensional filters with nonlinear drift XIII: Classification of . . .
, 2000
"... The idea of using estimation algebras to construct finite dimensional nonlinear filters was first proposed by Brockett and Mitter independently. For this approach, one needs to know explicitly the structure of these estimation algebras in order to construct finite dimensional nonlinear filters. Th ..."
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Cited by 10 (1 self)
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The idea of using estimation algebras to construct finite dimensional nonlinear filters was first proposed by Brockett and Mitter independently. For this approach, one needs to know explicitly the structure of these estimation algebras in order to construct finite dimensional nonlinear filters. Therefore Brockett proposed to classify all finite dimensional estimation algebras. Chiou and Yau [ChYa1] classify all finite dimensional estimation algebras of maximal rank with dimension of the state space less than or equal to two. The purpose of this paper is to give a new result on classification of all finite dimensional estimation algebras of maximal rank with state space dimension less than or equal to five.
Quantum filtering: a reference probability approach
, 2005
"... These notes are intended as an introduction to noncommutative (quantum) filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation as the least squares estimate, and culminating in the construction of Wiener and Poisson ..."
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Cited by 9 (1 self)
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These notes are intended as an introduction to noncommutative (quantum) filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation as the least squares estimate, and culminating in the construction of Wiener and Poisson processes on the Fock space. Next we describe the HudsonParthasarathy quantum Itô calculus and its use in the modelling of physical systems. Finally, we use a reference probability method to obtain quantum filtering equations, in the BelavkinZakai (unnormalized) form, for several systemobservation models from quantum optics. The normalized (BelavkinKushnerStratonovich) form is obtained through a noncommutative analogue of the KallianpurStriebel formula.