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538
Unscented Filtering and Nonlinear Estimation
 PROCEEDINGS OF THE IEEE
, 2004
"... The extended Kalman filter (EKF) is probably the most widely used estimation algorithm for nonlinear systems. However, more than 35 years of experience in the estimation community has shown that is difficult to implement, difficult to tune, and only reliable for systems that are almost linear on the ..."
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Cited by 555 (3 self)
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The extended Kalman filter (EKF) is probably the most widely used estimation algorithm for nonlinear systems. However, more than 35 years of experience in the estimation community has shown that is difficult to implement, difficult to tune, and only reliable for systems that are almost linear on the time scale of the updates. Many of these difficulties arise from its use of linearization. To overcome this limitation, the unscented transformation (UT) was developed as a method to propagate mean and covariance information through nonlinear transformations. It is more accurate, easier to implement, and uses the same order of calculations as linearization. This paper reviews the motivation, development, use, and implications of the UT.
A family of algorithms for approximate Bayesian inference
, 2001
"... One of the major obstacles to using Bayesian methods for pattern recognition has been its computational expense. This thesis presents an approximation technique that can perform Bayesian inference faster and more accurately than previously possible. This method, "Expectation Propagation," ..."
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Cited by 369 (11 self)
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One of the major obstacles to using Bayesian methods for pattern recognition has been its computational expense. This thesis presents an approximation technique that can perform Bayesian inference faster and more accurately than previously possible. This method, "Expectation Propagation," unifies and generalizes two previous techniques: assumeddensity filtering, an extension of the Kalman filter, and loopy belief propagation, an extension of belief propagation in Bayesian networks. The unification shows how both of these algorithms can be viewed as approximating the true posterior distribution with a simpler distribution, which is close in the sense of KLdivergence. Expectation Propagation exploits the best of both algorithms: the generality of assumeddensity filtering and the accuracy of loopy belief propagation. Loopy belief propagation, because it propagates exact belief states, is useful for limited types of belief networks, such as purely discrete networks. Expectation Propagati...
Quadraturebased methods for obtaining approximate solutions to nonlinear asset pricing models
 ECONOMETRICA
, 1991
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Cooperative control of mobile sensor networks: Adaptive gradient climbing in a distributed environment
 IEEE Transactions on Automatic Control
, 2004
"... Abstract—We present a stable control strategy for groups of vehicles to move and reconfigure cooperatively in response to a sensed, distributed environment. Each vehicle in the group serves as a mobile sensor and the vehicle network as a mobile and reconfigurable sensor array. Our control strategy d ..."
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Cited by 262 (20 self)
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Abstract—We present a stable control strategy for groups of vehicles to move and reconfigure cooperatively in response to a sensed, distributed environment. Each vehicle in the group serves as a mobile sensor and the vehicle network as a mobile and reconfigurable sensor array. Our control strategy decouples, in part, the cooperative management of the network formation from the network maneuvers. The underlying coordination framework uses virtual bodies and artificial potentials. We focus on gradient climbing missions in which the mobile sensor network seeks out local maxima or minima in the environmental field. The network can adapt its configuration in response to the sensed environment in order to optimize its gradient climb. Index Terms—Adaptive systems, cooperative control, gradient methods, mobile robots, multiagent systems, sensor networks. I.
Orthogonal Polynomials of Several Variables
 Encyclopedia of Mathematics and its Applications
, 2001
"... Abstract. We report on the recent development on the general theory of orthogonal polynomials in several variables, in which results parallel to the theory of orthogonal polynomials in one variable are established using a vectormatrix notation. 1 ..."
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Cited by 233 (45 self)
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Abstract. We report on the recent development on the general theory of orthogonal polynomials in several variables, in which results parallel to the theory of orthogonal polynomials in one variable are established using a vectormatrix notation. 1
Parallel Computation of Multivariate Normal Probabilities
"... We present methods for the computation of multivariate normal probabilities on parallel/ distributed systems. After a transformation of the initial integral, an approximation can be obtained using MonteCarlo or quasirandom methods. We propose a metaalgorithm for asynchronous sampling methods and d ..."
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Cited by 207 (9 self)
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We present methods for the computation of multivariate normal probabilities on parallel/ distributed systems. After a transformation of the initial integral, an approximation can be obtained using MonteCarlo or quasirandom methods. We propose a metaalgorithm for asynchronous sampling methods and derive efficient parallel algorithms for the computation of MVN distribution functions, including a method based on randomized Korobov and Richtmyer sequences. Timing results of the implementations using the MPI parallel environment are given. 1 Introduction The computation of the multivariate normal distribution function F (a; b) = j\Sigmaj \Gamma 1 2 (2) \Gamma n 2 Z b a e \Gamma 1 2 x \Sigma \Gamma1 x dx: (1) often leads to computationalintensive integration problems. Here \Sigma is an n \Theta n symmetric positive definite covariance matrix; furthermore one of the limits in each integration variable may be infinite. Genz [5] performs a sequence of transformations resu...
A generalized discrepancy and quadrature error bound
 Math. Comp
, 1998
"... Abstract. An error bound for multidimensional quadrature is derived that includes the KoksmaHlawka inequality as a special case. This error bound takes the form of a product of two terms. One term, which depends only on the integrand, is defined as a generalized variation. The other term, which dep ..."
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Cited by 141 (13 self)
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Abstract. An error bound for multidimensional quadrature is derived that includes the KoksmaHlawka inequality as a special case. This error bound takes the form of a product of two terms. One term, which depends only on the integrand, is defined as a generalized variation. The other term, which depends only on the quadrature rule, is defined as a generalized discrepancy. The generalized discrepancy is a figure of merit for quadrature rules and includes as special cases the L pstar discrepancy and Pα that arises in the study of lattice rules.
COMPUTING SEMICLASSICAL QUANTUM DYNAMICS WITH HAGEDORN
"... Abstract. We consider the approximation of multiparticle quantum dynamics in the semiclassical regime by Hagedorn wavepackets, which are products of complex Gaussians with polynomials that form an orthonormal L 2 basis and preserve their type under propagation in Schrödinger equations with quadrati ..."
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Cited by 134 (4 self)
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Abstract. We consider the approximation of multiparticle quantum dynamics in the semiclassical regime by Hagedorn wavepackets, which are products of complex Gaussians with polynomials that form an orthonormal L 2 basis and preserve their type under propagation in Schrödinger equations with quadratic potentials. We build a fully explicit, timereversible timestepping algorithm to approximate the solution of the Hagedorn wavepacket dynamics. The algorithm is based on a splitting between the kinetic and potential part of the Hamiltonian operator, as well as on a splitting of the potential into its local quadratic approximation and the remainder. The algorithm is robust in the semiclassical limit. It reduces to the Strang splitting of the Schrödinger equation in the limit of the full basis set, and it advances positions and momenta by the Störmer–Verlet method for the classical equations of motion. The algorithm allows for the treatment of multiparticle problems by thinning out the basis according to a hyperbolic cross approximation, and of highdimensional problems by Hartreetype approximations in a moving coordinate frame.
Models3 Community Multiscale Air Quality (CMAQ) model aerosol component. 1. Model description
 Journal of Geophysical Research
, 2003
"... [1] The aerosol component of the Community Multiscale Air Quality (CMAQ) model is designed to be an efficient and economical depiction of aerosol dynamics in the atmosphere. The approach taken represents the particle size distribution as the superposition of three lognormal subdistributions, called ..."
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Cited by 113 (4 self)
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[1] The aerosol component of the Community Multiscale Air Quality (CMAQ) model is designed to be an efficient and economical depiction of aerosol dynamics in the atmosphere. The approach taken represents the particle size distribution as the superposition of three lognormal subdistributions, called modes. The processes of coagulation, particle growth by the addition of mass, and new particle formation, are included. Time stepping is done with analytical solutions to the differential equations for the conservation of number, surface area, and species mass. The component considers both PM2.5 and PM10 and includes estimates of the primary emissions of elemental and organic carbon, dust, and other species not further specified. Secondary species considered are sulfate, nitrate, ammonium, water, and secondary organics from precursors of anthropogenic and biogenic origin. Extinction of visible light by aerosols is represented by two methods: a parametric approximation to Mie extinction and an empirical approach based upon field data. The algorithms that simulate cloud interactions with aerosols are also described. Results from box model and threedimensional simulations are
Is Gauss Quadrature Better Than Clenshaw–Curtis?
, 2008
"... We compare the convergence behavior of Gauss quadrature with that of its younger brother, Clenshaw–Curtis. Sevenline MATLAB codes are presented that implement both methods, and experiments show that the supposed factorof2 advantage of Gauss quadrature is rarely realized. Theorems are given to exp ..."
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Cited by 105 (4 self)
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We compare the convergence behavior of Gauss quadrature with that of its younger brother, Clenshaw–Curtis. Sevenline MATLAB codes are presented that implement both methods, and experiments show that the supposed factorof2 advantage of Gauss quadrature is rarely realized. Theorems are given to explain this effect. First, following O’Hara and Smith in the 1960s, the phenomenon is explained as a consequence of aliasing of coefficients in Chebyshev expansions. Then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log((z +1)/(z − 1)) in the complex plane. Gauss quadrature corresponds to Padé approximation at z = ∞. Clenshaw–Curtis quadrature corresponds to an approximation whose order of accuracy at z = ∞ is only half as high, but which is nevertheless equally accurate near [−1, 1].