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. Keywords—Estimation, Kalman filtering, nonlinear systems, target tracking. I.

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

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.

Abstract. An error bound for multidimensional quadrature is derived that includes the Koksma-Hlawka 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 p-star discrepancy and Pα that arises in the study of lattice rules.

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 Monte-Carlo or quasirandom methods. We propose a meta-algorithm 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 computational-intensive 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...

We survey some of the recent developments on quasi-Monte Carlo (QMC) methods, which, in their basic form, are a deterministic counterpart to the Monte Carlo (MC) method. Our main focus is the applicability of these methods to practical problems that involve the estimation of a high-dimensional integral. We review several QMC constructions and dierent randomizations that have been proposed to provide unbiased estimators and for error estimation. Randomizing QMC methods allows us to view them as variance reduction techniques. New and old results on this topic are used to explain how these methods can improve over the MC method in practice. We also discuss how this methodology can be coupled with clever transformations of the integrand in order to reduce the variance further. Additional topics included in this survey are the description of gures of merit used to measure the quality of the constructions underlying these methods, and other related techniques for multidimensional integration. 1 2 1.

Many real-world systems are naturally modeled as hybrid stochastic processes, i.e., stochastic processes that contain both discrete and continuous variables. Examples include speech recognition, target tracking, and monitoring of physical systems. The task is usually to perform probabilistic inference, i.e., infer the hidden state of the system given some noisy observations. For example, we can ask what is the probability that a certain word was pronounced given the readings of our microphone, what is the probability that a submarine is trying to surface given our sonar data, and what is the probability of a valve being open given our pressure and flow readings. Bayesian networks are

We analyze multiscale Galerkin methods for strongly elliptic boundary integral equations of order zero on closed surfaces in R³. Piecewise polynomial, discontinuous multiwavelet bases of any polynomial degree are constructed explicitly. We show that optimal convergence rates in the boundary energy norm and in certain negative norms can be achieved with "compressed" stiffness matrices containing O(N (log N)²) nonvanishing entries where N denotes the number of degrees of freedom on the boundary manifold. We analyze a quadrature scheme giving rise to fully discrete methods. We show that the fully discrete scheme preserves the asymptotic accuracy of the Galerkin scheme with exact integration and without compression. The overall computational complexity of our algorithm is O(N (log N) 4 ) kernel evaluations. The implications of the results for the numerical solution of elliptic boundary value problems in or exterior to bounded, three-dimensional domains are discussed.

This paper compares methods for the numerical computation of multivariate t-probabilities for hyperrectangular integration regions. Methods based on acceptance-rejection, spherical-radial transformations and separation-of-variables transformations are considered. Tests using randomly chosen problems show that the most efficient numerical methods use a transformation developed by Genz (1992) for multivariate normal probabilities. These methods allow moderately accurate multivariate t-probabilities to be quickly computed for problems with as many as twenty variables. Methods for the non-central multivariate t-distribution are also described. Key Words: multivariate t-distribution, non-central distribution, numerical integration, statistical computation. 1 Introduction A common problem in many statistics applications is the numerical computation of the multivariate t (MVT) distribution function (see Tong, 1990) defined by T(a; b; \Sigma; ) = \Gamma( +m 2 ) \Gamma( 2 ) p j\Sigma...

We present and review algorithms for the numerical integration of multivariate functions defined over d-dimensional cubes using several variants of the sparse grid method first introduced by Smolyak [51]. In this approach, multivariate quadrature formulas are constructed using combinations of tensor products of suited one--dimensional formulas. The computing cost is almost independent of the dimension of the problem if the function under consideration has bounded mixed derivatives. We suggest the usage of extended Gauss (Patterson) quadrature formulas as the one-dimensional basis of the construction and show their superiority in comparison to previously used sparse grid approaches based on the trapezoidal, Clenshaw-Curtis and Gauss rules in several numerical experiments and applications.