robust feti dp methods for heterogeneous three dimensional elasticity problems iterative substructuring methods with lagrange multipliers are considered for heterogeneous linear elasticity problems with large discontinuities in the material stiffnesses in particular results for algorithms belonging to the family of dual primal feti methods are presented the core issue of these algorithms is the construction of an appropriate global problem in order to obtain a robust method which converges independently of the material discontinuities in this article several necessary and sufficient conditions arising from the theory are numerically tested and confirmed furthermore results of numerical experiments are presented for situations which are not covered by the theory such as curved edges and material discontinuities not aligned with the interface and an attempt is made to develop rules for these cases

adaptive ode solvers in extended kalman filtering algorithms this paper studies numerical methods that are used to treat ordinary differential equations arising in extended kalman filtering algorithms such problems are called the moment equations and constitute an important subclass of differential equations they all possess a number of specific properties that make the standard available software ineffective for some mathematical models in this area for instance we show that built in matlab ode solvers are not able to cope with a number of tests considered here in contrast the hybrid method developed by mazzoni 2007 exhibits much better performance on the same examples the latter means that specially designed numerical schemes are needed for efficient solution of the moment equations instead of the standard software designed for general systems of ordinary differential equations

sparse grid quadrature nonlinear filtering in this paper a novel nonlinear filter named sparse grid quadrature filter sgqf is proposed the filter utilizes weighted sparse grid quadrature points to approximate the multi dimensional integrals in the nonlinear bayesian estimation algorithm the locations and weights of the univariate quadrature points with a range of accuracy levels are determined by the moment matching method then the univariate quadrature point sets are extended to form a multi dimensional grid using the sparse grid theory compared with the conventional point based methods the estimation accuracy level of the sgqf can be flexibly controlled and the number of sparse grid quadrature points for the sgqf is a polynomial of the dimension of the system which alleviates the curse of dimensionality for high dimensional problems the unscented kalman filter ukf is proven to be a subset of the sgqf at the level 2 accuracy the performance of this filter is demonstrated by an orbit estimation problem the simulation results show that the sgqf achieves higher accuracy than the extended kalman filter ekf the ukf and the cubature kalman filter ckf in addition the sgqf is computationally much more efficient than the multi dimensional gauss hermite quadrature filter ghqf with the same performance

truncated unscented kalman filtering we devise a filtering algorithm to approximate the first two moments of the posterior probability density function pdf the novelties of the algorithm are in the update step if the likelihood has a bounded support we can use a modified prior distribution that meets bayes rule exactly applying a kalman filter kf to the modified prior distribution referred to as truncated kalman filter tkf can vastly improve the performance of the conventional kalman filter particularly when the measurements are informative relative to the prior the application of the tkf to practical problems in which the measurement noise pdf has unbounded support is achieved by imposing several approximating assumptions which are valid only when the measurements are informative this implies that we adaptively choose between an approximation to the kf or the tkf according to the information provided by the measurement the resulting algorithm based on the unscented transformation is referred to as truncated unscented kf

exponential decay for maxwell equations with a boundary memory condition abstract we study the asymptotic behavior of the solution of the maxwell equations with the following boundary condition of memory type 0 1 e t 0 h t n 0 s h t s n d s we consider a graffi type free energy and we prove that if the kernel satisfies the condition 0 and the domain is strongly star shaped then the energy of the solution exponentially decays we also prove that the exponential decay of is a necessary condition for the exponential decay of the solution

gaussian particle filtering sequential bayesian estimation for nonlinear dynamic state space models involves recursive estimation of filtering and predictive distributions of unobserved time varying signals based on noisy observations this paper introduces a new filter called the gaussian particle filter it is based on the particle filtering concept and it approximates the posterior distributions by single gaussians similar to gaussian filters like the extended kalman filter and its variants it is shown that under the gaussianity assumption the gaussian particle filter is asymptotically optimal in the number of particles and hence has much improved performance and versatility over other gaussian filters especially when nontrivial nonlinearities are present simulation results are presented to demonstrate the versatility and improved performance of the gaussian particle filter over conventional gaussian filters and the lower complexity than known particle filters

general equivalence between two kinds of noise correlation filters a recent comment chang 2014 theoretically demonstrated that for the linear system with the correlated process and measurement noises the gasf framework proposed in the paper wang et al 2012 was equivalent to the conventional de coupling filtering framework in this note we would further show that such equivalence between the two frameworks can be justified in a more general way even for the nonlinear system

inexact feti dp methods inexact feti dp domain decomposition methods are considered preconditioners based on formulations of feti dp as a saddle point problem are used which allow for an inexact solution of the coarse problem a positive definite reformulation of the preconditioned saddle point problem which also allows for approximate solvers is considered as well in the formulation that iterates on the original feti dp saddle point system it is also possible to solve the local neumann subdomain problems inexactly given good approximate solvers for the local and coarse problems convergence bounds of the same quality as for the standard feti dp methods are obtained numerical experiments which compare the convergence of the inexact methods with that of standard feti dp are shown for 2d and 3d elasticity using gmres and cg as krylov space methods based on parallel computations a comparison of one variant of the inexact feti dp algorithms and the standard feti dp method is carried out and similar parallel performance is achieved parallel scalability of the inexact variant is also demonstrated it is shown that for a very large number of subdomains and a very large coarse problem the inexact method can be superior copyright 2006 john wiley sons ltd

belief condensation filtering inferring a sequence of variables from observations is prevalent in a multitude of applications traditional techniques such as kalman filters kfs and particle filters pfs are widely used for such inference problems however these techniques fail to provide satisfactory performance in many important nonlinear or non gaussian scenarios in addition there is a lack of a unified methodology for the design and analysis of different filtering techniques to address these problems in this paper we propose a new filtering methodology called belief condensation bc filtering first we establish a general framework for filtering techniques and propose an optimality criterion that leads to bc filtering we then propose efficient bc algorithms that can best represent the complex distributions arising in the filtering process the performance of the proposed techniques is evaluated for two representative nonlinear non gaussian problems showing that the bc filtering can provide accuracy approaching the theoretical bounds and outperform existing techniques in terms of the accuracy versus complexity tradeoff

dual primal feti methods for three dimensional elliptic problems with heterogeneous coefficients in this paper certain iterative substructuring methods with lagrange multipliers are considered for elliptic problems in three dimensions the algorithms belong to the family of dual primal finite element tearing and interconnecting feti methods which recently have been introduced and analyzed successfully for elliptic problems in the plane the family of algorithms for three dimensions is extended and a full analysis is provided for the new algorithms particular attention is paid to finding algorithms with a small primal subspace since that subspace represents the only global part of the dual primal preconditioner it is shown that the condition numbers of several of the dual primal feti methods can be bounded polylogarithmically as a function of the dimension of the individual subregion problems and that the bounds are otherwise independent of the number of subdomains the mesh size and jumps in the coefficients these results closely parallel those of other successful iterative substructuring methods of primal as well as dual type

mixture densities maximum likelihood and the em algorithm the problem of estimating the parameters which determine a mixture density has been the subject of a large diverse body of literature spanning nearly ninety years during the last two decades the

a preconditioner for substructuring based on constrained energy minimization a preconditioner for substructuring based on constrained energy minimization concepts is presented the preconditioner is applicable to both structured and unstructured meshes and offers a straightforward approach for the iterative solution of second and fourth order structural mechanics problems the approach involves constraints associated with disjoint sets of nodes on substructure boundaries these constraints provide the means for preconditioning at both the substructure and global levels numerical examples are presented that demonstrate the good performance of the method in terms of iterations compute time and condition numbers of the preconditioned equations

on the exponential decay in thermoelasticity without energy dissipation and of type iii in presence of an absorbing boundary we study the asymptotic behavior of the solution of a 3d hyperbolic system arising in the green naghdi models of thermoelasticity of type ii and iii with a dissipative boundary condition for the displacement and prove that the energy exponentially decays in time

feti dp methods with an adaptive coarse space a coarse space is constructed for the dual primal finite element tearing and interconnecting feti dp domain decomposition method applied to highly heterogeneous problems by solving local generalized eigenvalue problems for certain problems with highly varying coefficients e g from multiscale simulations the coefficient jump will appear in the condition number bound even if standard techniques such as scaling and the weighting of constraints are used the feti dp theory is revisited and two central estimates are identified where the dependency on the coefficient contrast can enter the condition number bound the first is a poincare inequality and the second an extension theorem these estimates are replaced by local eigenvalue problems enriching the feti dp coarse space by a few numerically computed eigenvectors yields independence of the contrast of the coefficients even in challenging situations

three level bddc in three dimensions balancing domain decomposition by constraints bddc methods are nonoverlapping iterative substructuring domain decomposition methods for the solution of large sparse linear algebraic systems arising from the discretization of elliptic boundary value problems their coarse problems are given in terms of a small number of continuity constraints for each subdomain which are enforced across the interface the coarse problem matrix is generated and factored by a direct solver at the beginning of the computation and it can ultimately become a bottleneck if the number of subdomains is very large in this paper two three level bddc methods are introduced for solving the coarse problem approximately for problems in three dimensions this is an extension of previous work for the two dimensional case edge constraints are considered in this work since vertex constraints alone which work well in two dimensions result in a noncompetitive algorithm in three dimensions some new technical tools are then needed in the analysis and this makes the three dimensional case more complicated estimates of the condition numbers are provided for two three level bddc methods and numerical experiments are also discussed

adaptive gaussian sum filter for nonlinear bayesian estimation a nonlinear filter is developed by representing the state probability density function by a finite sum of gaussian density kernels whose mean and covariance are propagated from one time step to the next using linear system theory methods such as extended kalman filter or unscented kalman filter the novelty in the proposed method is that the weights of the gaussian kernels are updated at every time step by solving a convex optimization problem posed by requiring the gaussian sum approximation to satisfy the fokker planck kolmogorov equation for continuous time dynamical systems and the chapman kolmogorov equation for discrete time dynamical systems the numerical simulation results show that updating the weights of different mixture components during propagation mode of the filter not only provides us with better state estimates but also with a more accurate state probability density function

gaussian filters for nonlinear filtering problems we develop and analyze real time and accurate filters for nonlinear filtering problems based on the gaussian distributions we present the systematic formulation of gaussian filters and develop efficient and accurate numerical integration of the optimal filter we also discuss the mixed gaussian filters in which the conditional probability density is approximated by the sum of gaussian distributions a new update rule of weights for gaussian sum filters is proposed our numerical tests demonstrate that new filters significantly improve the extended kalman filter with no additional cost and the new gaussian sum filter has a nearly optimal performance

on principles for the selection of shape functions for the generalized finite element method abstract effective shape functions for the generalized finite element method should reflect the available information on the solution this information is partially fuzzy because the solution is of course unknown and typically the only available information is the solution s inclusion in a variety of function spaces it is desirable to choose shape functions that perform robustly over a family of relevant situations quantitative notions of robustness are introduced and discussed we show in particular that in one dimension polynomials are robust when the available information consists in inclusions in sobolev type spaces that are x independent

ukf based nonlinear filtering using minimum entropy criterion a novel filter for nonlinear and non gaussian systems is proposed in this paper the unscented kalman filter is designed to give a preliminary estimation of the state an additional rbf network is added to the ukf innovation term to compensate for the non gaussianity of the whole system the renyi s entropy of the innovation is introduced and parameters of the rbf network are updated using minimum entropy criterion at each time step it has been shown that the proposed algorithm has a high accuracy in estimation because entropy can characterize all the randomness of the residual while ukf only cares for the mean and the covariance it has been proved that with properly chosen bandwidth the minimum entropy problem of the innovation is convex therefore the proposed adaptive nonlinear filter will be globally convergent and the misadjustment will be proportional to the step size the effectiveness of the proposed method is shown by simulation

accurate numerical implementation of the continuous discrete extended kalman filter this paper addresses an accurate and effective implementation of the continuous discrete extended kalman filtering method the technique under discussion is grounded in numerical solution of the moment differential equations to predict the state mean of the stochastic dynamical system and the corresponding error covariance matrix here we apply an efficient embedded runge kutta pair possessing superior stability and many other attractive features including automatic global error control in order to improve performance of the complex computational procedure consisting of the extended kalman filter and the underlying adaptive ode solver thus we introduce a new continuous discrete adaptive extended kalman filter and show its advantage over the standard variant on two test examples in practice this technique allows for much longer sampling intervals without any loss of accuracy and that improves the applied potential of the extended kalman filtering method significantly

a model updating approach based on the component mode synthesis method and perturbation analysis abstract a model updating approach based on an improved free interface component mode synthesis method is proposed first the equivalent higher order matrix of the system is developed using a set of linearly independent vectors to capture the effects of the neglected higher order modes second the perturbation method is applied to derive the change for both eigenvalues and eigenvectors of each component while implementing a modification to the updated finite element model the perturbation for both the eigenvalues and eigenvectors of each component are used to formulate the synthesis equations containing the perturbation expression next the model updating process is conducted using the synthesis model for which the degrees of freedom have been greatly reduced the reduction in the total degree of freedom speeds up the solution process for the model updating problem finally the computational efficiency and accuracy of the presented method is demonstrated using a wing structure and by modal testing of a bolted plate

square root receding horizon information filters for nonlinear dynamic system models new nonlinear filtering algorithms are designed based on a receding horizon strategy i e a finite impulse response fir structure and square root information filtering to achieve high accuracy and good performance in empirical error covariance tests the new nonlinear receding horizon filters reduce approximation errors in nonlinear filtering by considering a set of recent observations with non informative initial conditions by applying information filtering we are able to manage the non informative initial conditions and thus propose the square root version of the algorithm as a means of retaining the positive definiteness of the error covariance based on the proposed strategy we then implement known nonlinear filtering frameworks simulation results confirm that the new nonlinear receding horizon filters outperform existing nonlinear filters in well known nonlinear examples

some relations between extended and unscented kalman filters the unscented kalman filter ukf has become a popular alternative to the extended kalman filter ekf during the last decade ukf propagates the so called sigma points by function evaluations using the unscented transformation ut and this is at first glance very different from the standard ekf algorithm which is based on a linearized model the claimed advantages with ukf are that it propagates the first two moments of the posterior distribution and that it does not require gradients of the system model we point out several less known links between ekf and ukf in terms of two conceptually different implementations of the kalman filter the standard one based on the discrete riccati equation and one based on a formula on conditional expectations that does not involve an explicit riccati equation first it is shown that the sigma point function evaluations can be used in the classical ekf rather than an explicitly linearized model second a less cited version of the ekf based on a second order taylor expansion is shown to be quite closely related to ukf the different algorithms and results are illustrated with examples inspired by core observation models in target tracking and sensor network applications

non gaussian state space modeling of nonstationary time series abstract a non gaussian state space approach to the modeling of nonstationary time series is shown the model is expressed in state space form where the system noise and the observational noise are not necessarily gaussian recursive formulas of prediction filtering and smoothing for the state estimation and identification of the non gaussian state space model are given also given is a numerical method based on piecewise linear approximation to the density functions for realizing these formulas significant merits of non gaussian modeling and the wide range of applicability of the method are illustrated by some numerical examples a typical application of this non gaussian modeling is the smoothing of a time series that has mean value function with both abrupt and gradual changes simple gaussian state space modeling is not adequate for this situation here the model with small system noise variance cannot detect jump whereas the one with large system noise variance yields unfavorable wiggle to work

improper complex valued multiple model adaptive estimation motivated by the problem of estimating the discrete and continuous states of an improper complex valued stochastic hybrid system the paper proposes a class of widely linear augmented multiple model adaptive estimation algorithms referred to as the c mmae we show that for an improper complex valued signal pseudo covariance of the innovation sequence is not zero and therefore carries useful statistical information regarding the unknown behaviour mode of the hybrid system a new bayesian law is therefore derived as a function of the pseudo covariance of the innovation sequence and used to compute the probability that a hypothesized model is in effect at a certain time we show that the c mmae which uses the new bayesian law and utilizes the complete second order statistical characterization of the complex valued innovation sequence convergences faster than its counterpart which only uses the conventional covariance of the innovation sequence in order to simplify the computational complexity we develop two circularized versions of the c mmae using a preprocessing step referred to as the circularizing filter cf the cf is incorporated to convert the improper observations innovations into the proper ones in order to reduce the computational complexity of the hypothesis testing step finally an interacting version of the c mmae referred to as c imm is developed for improper complex valued systems with markovian switching coefficients simulation results indicate that the proposed hybrid estimators provide improved performance and convergence properties over their traditional counterparts

a finite difference method for linearization in nonlinear estimation algorithms abstract linearizations of nonlinear functions that are based on jacobian matrices often cannot be applied in practical applications of nonlinear estimation techniques an alternative linearization method is presented in this paper the method assumes that covariance matrices are determined on a square root factored form a factorization of the output covariance from a nonlinear vector function is directly determined by perturbing the nonlinear function with the columns of the factored input covariance without explicitly calculating the linearization and with no differentiations involved the output covariance is more accurate than that obtained with the ordinary jacobian linearization method it also has an advantage that jacobian matrices do not have to be derived symbolically

fully symmetric interpolatory rules for multiple integrals over infinite regions with gaussian weight fully symmetric interpolatory integration rules are constructed for multidimensional integrals over infinite integration regions with a gaussian weight function the points for these rules are determined by successive extensions of the one dimensional three point gauss hermite rule the new rules are shown to be efficient and only moderately unstable

box gaussian mixture filter this note presents the box gaussian mixture filter bgmf which is an efficient filter for the systems with mainly linear measurements but enables utilizing highly nonlinear measurements bgmf contains a new way to approximate the prior distributions with a gaussian mixture whose components have small covariances in this note we present results on the weak convergence of bgmf in simulations we see that in our hybrid position example bgmf outperforms the conventional particle filter

the approximate component mode synthesis special finite element method in two dimensions a special finite element method based on approximate component mode synthesis acms was introduced in hetmaniuk and lehoucq 2010 acms was developed for second order elliptic partial differential equations with rough or highly varying coefficients here a parallel implementation of acms is presented and parallel scalability issues are discussed for representative examples additionally a parallel domain decomposition preconditioner feti dp is applied to solve the acms finite element system weak parallel scalability results for acms are presented for up to 1024 cores our numerical results also suggest a quadratic logarithmic condition number bound for the preconditioned feti dp method applied to acms discretizations

nonlinear feti dp and bddc methods new nonlinear feti dp dual primal finite element tearing and interconnecting and bddc balancing domain decomposition by constraints domain decomposition methods are introduced in all these methods in each iteration local nonlinear problems are solved on the subdomains the new approaches can significantly reduce communication and show a significantly improved performance especially for problems with localized nonlinearities compared to a standard newton krylov feti dp or bddc approach moreover the coarse space of the nonlinear feti dp methods can be used to accelerate the newton convergence it is also found that the new nonlinear feti dp and nonlinear bddc methods are not as closely related as in the linear context numerical results for the p laplace operator are presented

an iterative kalman like algorithm ignoring noise and initial conditions we address a p shift finite impulse response fir unbiased estimator ue for linear discrete time varying filtering p 0 p step prediction p 0 and p lag smoothing p 0 in state space with no requirements for initial conditions and zero mean noise a solution is found in a batch form and represented in a computationally efficient iterative kalman like one it is shown that the kalman like fir ue is able to outperform the kalman filter if the noise covariances and initial conditions are not known exactly noise is not white and both the system and measurement noise components need to be filtered out otherwise the errors are similar extensive numerical studies of the fir ue are provided in gaussian and non gaussian environments with outliers and temporary uncertainties

marginalized particle filters for mixed linear nonlinear state space models the particle filter offers a general numerical tool to approximate the posterior density function for the state in nonlinear and non gaussian filtering problems while the particle filter is fairly easy to implement and tune its main drawback is that it is quite computer intensive with the computational complexity increasing quickly with the state dimension one remedy to this problem is to marginalize out the states appearing linearly in the dynamics the result is that one kalman filter is associated with each particle the main contribution in this paper is the derivation of the details for the marginalized particle filter for a general nonlinear state space model several important special cases occurring in typical signal processing applications will also be discussed the marginalized particle filter is applied to an integrated navigation system for aircraft it is demonstrated that the complete high dimensional system can be based on a particle filter using marginalization for all but three states excellent performance on real flight data is reported

independent particle filters sequential monte carlo methods especially the particle filter pf and its various modifications have been used effectively in dealing with stochastic dynamic systems the standard pf samples the current state through the underlying state dynamics then uses the current observation to evaluate the sample s importance weight however there is a set of problems in which the current observation provides significant information about the current state but the state dynamics are weak and thus sampling using the current observation often produces more efficient samples than sampling using the state dynamics in this article we propose a new variant of the pf the independent particle filter ipf to deal with these problems the ipf generates exchangeable samples of the current state from a sampling distribution that is conditionally independent of the previous states a special case of which uses only the current observation each sample can then be matched with multiple samples of the previous states in

existence and uniform decay rates of solutions to a degenerate system with memory conditions at the boundary in this article we study the degenerate system r 1 r 2 0 subject to memory conditions on the boundary given email protected 1 x u t email protected email protected u v email protected 0 r 2 x v t email protected a u v email protected 0 email protected 0 email protected 0 tg 1 t s email protected email protected s email protected 1x 0 email protected 0 email protected 0 tg 2 t s email protected email protected s email protected 1x 0 u 0 v 0 u 0 v 0 r 1u t 0 r 2v t 0 r 1u 1 r 2v 1 email protected where w is a bounded region in r n whose boundary is partitioned into disjoint sets c 0 c 1 we prove that the dissipations given by the memory terms are strong enough to guarantee exponential or polynomial decay provided the relaxation functions also decay exponentially or polynomially and with the same rate of decay

nirk based accurate continuous discrete extended kalman filters for estimating continuous time stochastic target tracking models abstract this paper presents three state estimators grounded in the variable stepsize gauss and lobatto type nested implicit runge kutta nirk formulas of orders 4 and 6 and designed for treating continuous time stochastic systems arisen in radar tracking our filters are built within the extended kalman filtering ekf framework and based on accurate numerical integrations of the corresponding moment differential equations mdes automatic local and global error regulation mechanisms implemented in these methods allow the committed discretization error to be under control and made negligible in automatic mode the latter raises the state estimation accuracy of the constructed filters significantly this also leads to the advanced notion of accurate continuous discrete extended kalman filtering acd ekf developed by kulikov and kulikova in 2013 2016 our novel methods are constructed within the same approach but possess the improved accuracy and efficiency in comparison to their predecessors due to both more effective error control mechanisms implemented for integrating mdes and more accurate iterations used for treating arisen nonlinear equations in the revised filters numerical experiments with the updated state estimators and their comparison to the cited earlier designed acd ekfs are fulfilled in severe conditions of tackling a seven dimensional radar tracking problem where an aircraft executes a coordinated turn in matlab this examination suggests that the novel state estimation algorithms outperform their predecessors and possess a promising potential for solving target tracking tasks in real world applications

saturated particle filter almost sure convergence and improved resampling nonlinear stochastic dynamical systems are widely used to model physical processes in many practical applications the state variables are defined on a compact set of the state space i e they are bounded or saturated to estimate the states of systems with saturated variables the saturated particle filter spf has recently been developed this filter exploits the structure of the saturated system using a specific importance sampling distribution in this paper we investigate the asymptotic properties of the filter in particular its almost sure convergence to the true posterior pdf furthermore an improved spf is developed that uses a novel resampling procedure to overcome the practical shortcomings of the original spf we prove that this new filter also converges almost surely to the true posterior pdf both versions of the spf are presented in easy to implement algorithmic forms

the augmented complex particle filter complex gaussian signals in engineering applications are typically noncircular that is they have rotation dependent distributions however current complex valued particle filters pfs have assumed implicitly or explicitly circular signal distributions which for noncircular signals leads to suboptimal performance in this correspondence we employ a recent development in the so called augmented complex statistics and propose the augmented complex pf acpf and the augmented complex gaussian pf acgpf for the sequential estimation of complex states in both circular and noncircular noise the analysis and simulations illuminate theoretical and practical advantages for the proposed solutions

adaptive sparse grid gauss hermite filter abstract in this paper a new nonlinear filter based on sparse grid quadrature method has been proposed the proposed filter is named as adaptive sparse grid gauss hermite filter asghf ordinary sparse grid technique treats all the dimensions equally whereas the asghf assigns a fewer number of points along the dimensions with lower nonlinearity it uses adaptive tensor product to construct multidimensional points until a predefined error tolerance level is reached the performance of the proposed filter is illustrated with two nonlinear filtering problems simulation results demonstrate that the new algorithm achieves a similar accuracy as compared to sparse grid gauss hermite filter sghf and gauss hermite filter ghf with a considerable reduction in computational load further in the conventional ghf and sghf any increase in the accuracy level may result in an unacceptably high increase in the computational burden however in asghf a little increase in estimation accuracy is possible with a limited increase in computational burden by varying the error tolerance level and the error weighting parameter this enables the online estimator to operate near full efficiency with a predefined computational budget

particle filtering for large dimensional state spaces with multimodal observation likelihoods we study efficient importance sampling techniques for particle filtering pf when either a the observation likelihood ol is frequently multimodal or heavy tailed or b the state space dimension is large or both when the ol is multimodal but the state transition pdf stp is narrow enough the optimal importance density is usually unimodal under this assumption many techniques have been proposed but when the stp is broad this assumption does not hold we study how existing techniques can be generalized to situations where the optimal importance density is multimodal but is unimodal conditioned on a part of the state vector sufficient conditions to test for the unimodality of this conditional posterior are derived our result is directly extendable to testing for unimodality of any posterior the number of particles n to accurately track using a pf increases with state space dimension thus making any regular pf impractical for large dimensional tracking problems but in most such problems most of the state change occurs in only a few dimensions while the change in the rest of the dimensions is small using this property we propose to replace importance sampling from a large part of the state space whose conditional posterior is narrow enough by just tracking the mode of the conditional posterior this introduces some extra error but it also greatly reduces the importance sampling dimension the net effect is much smaller error for a given n especially when the available n is small an important class of large dimensional problems with multimodal ol is tracking spatially varying physical quantities such as temperature or pressure in a large area using a network of sensors which may be nonlinear and or may have nonnegligible failure probabilities improved performance of our proposed algorithms over existing pfs is demonstrated for this problem

square root quadrature kalman filtering the quadrature kalman filter qkf is a recursive nonlinear filtering algorithm developed in the kalman filtering framework it computes the mean and covariance of all conditional densities using the gauss hermite quadrature rule in this correspondence we develop a square root extension of the quadrature kalman filter using matrix triangularizations the square root quadrature kalman filter sqkf propagates the mean and the square root of the covariance although equivalent to the qkf algebraically the sqkf exhibits excellent numerical characteristics but at the expense of increased computational complexity we also present possible refinements of the generic sqkf

dual primal feti methods for linear elasticity dual primal feti methods are nonoverlapping domain decomposition methods where some of the continuity constraints across subdomain boundaries are required to hold throughout the iterations as in primal iterative substructuring methods while most of the constraints are enforced by lagrange multipliers as in one level feti methods these methods are used to solve the large algebraic systems of equations that arise in elliptic finite element problems the purpose of this article is to develop strategies for selecting these constraints which are enforced throughout the iterations such that good convergence bounds are obtained that are independent of even large changes in the stiffness of the subdomains across the interface between them the algorithms are described in terms of a change of basis that has proven to be quite robust in practice a theoretical analysis is provided for the case of linear elasticity and condition number bounds are established that are uniform with respect to arbitrarily large jumps in the young s modulus of the material and otherwise depend only polylogarithmically on the number of unknowns of a single subdomain the strategies have already proven quite successful in large scale implementations of these iterative methods 2006 wiley periodicals inc

gaussian smoothers for nonlinear systems with one step randomly delayed measurements this technical note is concerned with the nonlinear state smoothing problem in the case that the measurements are randomly delayed by one sampling time under gaussian domain two general gaussian approximation ga and gaussian mixture approximation gma smoothers are proposed in minimum mean square error mmse sense the smoothing implementation is transformed into computing some special posterior covariances which triggers the development of the new unscented kalman smoother uks by applying unscented transformation ut simulation results demonstrate the superior performance of the proposed ga uks and gma uks algorithms

three level bddc in two dimensions balancing domain decomposition by constraints bddc methods are non overlapping iterative substructuring domain decomposition methods for the solution of large sparse linear algebraic systems arising from discretization of elliptic boundary value problems they are similar to the balancing neumann neumann algorithm however in bddc methods a small number of continuity constraints are enforced across the interface and these constraints form a new coarse global component an important advantage of using such constraints is that the schur complements that arise in the computation will all be strictly positive definite the matrix of the coarse problem is generated and factored by direct solvers at the beginning of the computation however this problem can ultimately become a bottleneck if the number of subdomains is very large in this paper two three level bddc methods are introduced for solving the coarse problem approximately in two dimensional space while still maintaining a good convergence rate estimates of the condition numbers are provided for the two three level bddc methods and numerical experiments are also discussed copyright 2006 john wiley sons ltd

generalized finite element methods their performance and their relation to mixed methods the notion of a generalized finite element method is introduced this class of methods is analyzed and their relation to mixed methods is discussed the class of generalized finite element methods offers a wide variety of computational procedures from which particular procedures can be selected for particular problems a particular generalized finite element method which is very effective for problems with rough coefficients is discussed in detail

mixture kalman filters in treating dynamic systems sequential monte carlo methods use discrete samples to represent a complicated probability distribution and use rejection sampling importance sampling and weighted resampling to complete the on line filtering task we propose a special sequential monte carlo method the mixture kalman filter which uses a random mixture of the gaussian distributions to approximate a target distribution it is designed for on line estimation and prediction of conditional and partial conditional dynamic linear models which are themselves a class of widely used non linear systems and also serve to approximate many others compared with a few available filtering methods including monte carlo methods the gain in efficiency that is provided by the mixture kalman filter can be very substantial another contribution of the paper is the formulation of many non linear systems into conditional or partial conditional linear form to which the mixture kalman filter can be applied examples in target tracking and digital communications are given to demonstrate the procedures proposed

gaussian sum particle filtering we use the gaussian particle filter to build several types of gaussian sum particle filters these filters approximate the filtering and predictive distributions by weighted gaussian mixtures and are basically banks of gaussian particle filters then we extend the use of gaussian particle filters and gaussian sum particle filters to dynamic state space dss models with non gaussian noise with non gaussian noise approximated by gaussian mixtures the non gaussian noise models are approximated by banks of gaussian noise models and gaussian mixture filters are developed using algorithms developed for gaussian noise dss models as a result problems involving heavy tailed densities can be conveniently addressed simulations are presented to exhibit the application of the framework developed herein and the performance of the algorithms is examined

advanced point mass method for nonlinear state estimation state estimation of discrete time nonlinear non gaussian stochastic systems by point mass approach which is based on discretization of state space by a regular grid and numerical solution of bayesian recursive relations is treated the stress is laid to grid design which is crucial for estimator quality and significantly affects the computational demands of the estimator boundary based grid design thrifty convolution and multigrid design with grid splitting and merging are proposed the main advantages of these techniques are nonnegligible support delimitation time saving computation of convolution and effective processing of multimodal probability density functions respectively the techniques are involved into the basic point mass approach and a new general purpose more sophisticated point mass algorithm is designed computational demands and estimation quality of the designed algorithm are presented and compared with the particle filter in a numerical example

advances in multidimensional integration this paper gives a personal bird s eye view of some aspects of multivariate numerical integration it will sketch what happened during the past 50 years and point the reader s attention to what did not happen

a class of stable square root nonlinear information filters information filters can process nonlinear systems with uncertain prior knowledge and the particular square root form of adaptive filters can improve numerical stability based on a square root decomposition of information matrix and an extra positive definite matrix the unscented transform and the cubature rule are applied to the information filtering architecture for nonlinear estimation a class of stable square root nonlinear information filters is then proposed in this technical note in addition the boundedness of their estimation errors is also proven results from simulations of filtering a chaotic map demonstrate that the proposed square root nonlinear filters can improve numerical stability and has better filtering performance than other information filters

on mmse estimation a linear model under gaussian mixture statistics in a bayesian linear model suppose observation y hx n stems from independent inputs x and n which are gaussian mixture gm distributed with known matrix h the minimum mean square error mmse estimator for x has analytical form however its performance measure the mmse itself has no such closed form because existing bayesian mmse bounds prove to have limited practical value under these settings we instead seek analytical bounds for the mmse both upper and lower this paper provides such bounds and relates them to the signal to noise ratio snr

likelihood consensus and its application to distributed particle filtering we consider distributed state estimation in a wireless sensor network without a fusion center each sensor performs a global estimation task based on the past and current measurements of all sensors using only local processing and local communications with its neighbors in this estimation task the joint all sensors likelihood function jlf plays a central role as it epitomizes the measurements of all sensors we propose a distributed method for computing at each sensor an approximation of the jlf by means of consensus algorithms this likelihood consensus method is applicable if the local likelihood functions of the various sensors viewed as conditional probability density functions of the local measurements belong to the exponential family of distributions we then use the likelihood consensus method to implement a distributed particle filter and a distributed gaussian particle filter each sensor runs a local particle filter or a local gaussian particle filter that computes a global state estimate the weight update in each local gaussian particle filter employs the jlf which is obtained through the likelihood consensus scheme for the distributed gaussian particle filter the number of particles can be significantly reduced by means of an additional consensus scheme simulation results are presented to assess the performance of the proposed distributed particle filters for a multiple target tracking problem

feti dp a dual primal unified feti method part i a faster alternative to the two level feti method the feti method and its two level extension feti 2 are two numerically scalable domain decomposition methods with lagrange multipliers for the iterative solution of second order solid mechanics and fourth order beam plate and shell structural problems respectively the feti 2 method distinguishes itself from the basic or one level feti method by a second set of lagrange multipliers that are introduced at the subdomain cross points to enforce at each iteration the exact continuity of a subset of the displacement field at these specific locations in this paper we present a dual primal formulation of the feti 2 concept that eliminates the need for that second set of lagrange multipliers and unifies all previously developed one level and two level feti algorithms into a single dual primal feti dp method we show that this new feti dp method is numerically scalable for both second order and fourth order problems we also show that it is more robust and more computationally efficient than existing feti solvers particularly when the number of subdomains and or processors is very large copyright 2001 john wiley sons ltd

analysis of feti dp and bddc for linear elasticity in 3d with almost incompressible components and varying coefficients inside subdomains feti dp dual primal finite element tearing and interconnecting methods are nonoverlapping domain decomposition methods which are used to solve large algebraic systems of equations that arise e g from problems in linear elasticity good convergence bounds for problems of compressible linear elasticity are well known for two and three dimensional problems more recently feti dp and bddc balancing domain decomposition by constraints methods have been developed that are robust also in the regime of homogeneous almost incompressible linear elasticity the coarse space of such methods is large especially in 3d three dimensions and its implementation needs knowledge of geometrical information here the convergence of feti dp methods for problems in 3d with almost incompressible inclusions or compressible inclusions with different material parameters embedded in a compressible matrix material is analyzed for such problems where the material is compressible in the vicinity of the subdomain interface

a new method for the nonlinear transformation of means and covariances in filters and estimators this paper describes a new approach for generalizing the kalman filter to nonlinear systems a set of samples are used to parametrize the mean and covariance of a not necessarily gaussian probability distribution the method yields a filter that is more accurate than an extended kalman filter ekf and easier to implement than an ekf or a gauss second order filter its effectiveness is demonstrated using an example

brief paper a quadrature based method of moments for nonlinear filtering according to the nonlinear filtering theory optimal estimates of a general continuous discrete nonlinear filtering problem can be obtained by solving the fokker planck equation coupled with a bayesian update rule this procedure does not rely on linearizations of the dynamical and or measurement models however the lack of fast and efficient methods for solving the fokker planck equation presents challenges in real time nonlinear filtering problems in this paper a direct quadrature method of moments is introduced to solve the fokker planck equation efficiently and accurately this approach involves representation of the state conditional probability density function in terms of a finite collection of dirac delta functions the weights and locations abscissas in this representation are determined by moment constraints and modified using the bayes rule according to measurement updates as demonstrated by numerical examples this approach appears to be promising in the field of nonlinear filtering

robust filtering and smoothing with gaussian processes we propose a principled algorithm for robust bayesian filtering and smoothing in nonlinear stochastic dynamic systems when both the transition function and the measurement function are described by non parametric gaussian process gp models gps are gaining increasing importance in signal processing machine learning robotics and control for representing unknown system functions by posterior probability distributions this modern way of system identification is more robust than finding point estimates of a parametric function representation our principled filtering smoothing approach for gp dynamic systems is based on analytic moment matching in the context of the forward backward algorithm our numerical evaluations demonstrate the robustness of the proposed approach in situations where other state of the art gaussian filters and smoothers can fail

linear optimal fir estimation of discrete time invariant state space models this paper addresses a general p shift linear optimal finite impulse response fir estimator intended for solving universally the problems of filtering p 0 smoothing p 0 of discrete time invariant models in state space an optimal solution is found in the batch form with the initial mean square state function self determined by solving the discrete algebraic riccati equation an unbiased solution represented both in the batch and recursive forms does not involve any knowledge about noise and initial state the mean square errors in both the optimal and unbiased estimates are found via the noise power gain npg and a recursive algorithm for fast computation of the npg is supplied applications are given for fir filtering with fixed receding and full averaging horizons

new developments in state estimation for nonlinear systems based on an interpolation formula accurate state estimators for nonlinear systems can be derived the estimators do not require derivative information which makes them simple to implement

improved cubature kalman filter for gnss ins based on transformation of posterior sigma points error tightly coupled gnss ins has been widely approved as a promising substitute for standalone gnss in urban areas navigation however due to the frequent gnss signal outages the filter used in gnss ins should be insensitive to the less informative observations in this paper a novel sigma points update method is proposed to enhance the robustness of cubature kalman filter ckf under the circumstance of unavailable observations first the problems of existing sampling based filters are analyzed then by transforming the posterior sigma points error matrix from prediction phase of filtering to the posterior domain of update the updated sigma points are expected to capture the covariance more precisely than traditional sigma points finally an improved ckf ickf is developed by embedding these points into the bayesian estimation framework and the upper bounds of error covariance matrices are analyzed theoretically signal outages with different durations are simulated and results demonstrate that ickf outperforms state of the art methods

detection of random signals in gaussian mixture noise a locally optimal detection algorithm for random signals in dependent noise is derived and applied to independent identically distributed complex valued gaussian mixture noise the resulting detector is essentially a weighted sum of power detectors the power detector is the locally optimal detector for random signals in gaussian noise the weighting functions are modified to enhance the detection performance for small sample sizes an implementation of the mixture detector using the expectation maximization algorithm is described moments of these detectors are calculated from piecewise polynomial approximations of the weighting functions the sum of sufficiently many independent identically distributed detector outputs is then approximated by a normal distribution probability distributions are also derived for the power detector in gaussian mixture noise for a particular set of noise parameters the theoretical distributions are compared with those obtained from monte carlo simulation and seen to be quite close the theoretical distributions are then used to compare the performance of the mixture and power detectors in gaussian mixture noise over a range of parameters and to assess the impact of parameter error on detection performance in this study the signal gain of the mixture detectors varies from 15 to 38 db and the degradation of the probability of detection due to parameter estimation error is relatively minor

a receding horizon kalman fir filter for discrete time invariant systems a receding horizon kalman fir filter is presented that combines the kalman filter and the receding horizon strategy when the horizon initial state is assumed to be unknown the suggested filter is a fir filter form which has many good inherent properties it can always be defined irrespective of singularity problems caused by unknown information about the horizon initial state the suggested filter can be represented in either an iterative form or a standard fir form it is also shown that the suggested filter possesses the unbiasedness property and the remarkable deadbeat property irrespective of any horizon initial condition the validity of the suggested filter is illustrated by numerical examples

cubature kalman filters in this paper we present a new nonlinear filter for high dimensional state estimation which we have named the cubature kalman filter ckf the heart of the ckf is a spherical radial cubature rule which makes it possible to numerically compute multivariate moment integrals encountered in the nonlinear bayesian filter specifically we derive a third degree spherical radial cubature rule that provides a set of cubature points scaling linearly with the state vector dimension the ckf may therefore provide a systematic solution for high dimensional nonlinear filtering problems the paper also includes the derivation of a square root version of the ckf for improved numerical stability the ckf is tested experimentally in two nonlinear state estimation problems in the first problem the proposed cubature rule is used to compute the second order statistics of a nonlinearly transformed gaussian random variable the second problem addresses the use of the ckf for tracking a maneuvering aircraft the results of both experiments demonstrate the improved performance of the ckf over conventional nonlinear filters

constrained state estimation for nonlinear discrete time systems stability and moving horizon approximations state estimator design for a nonlinear discrete time system is a challenging problem further complicated when additional physical insight is available in the form of inequality constraints on the state variables and disturbances one strategy for constrained state estimation is to employ online optimization using a moving horizon approximation we propose a general theory for constrained moving horizon estimation sufficient conditions for asymptotic and bounded stability are established we apply these results to develop a practical algorithm for constrained linear and nonlinear state estimation examples are used to illustrate the benefits of constrained state estimation our framework is deterministic

quasi monte carlo filtering in nonlinear dynamic systems we develop a new framework for bayesian filtering in general nonlinear dynamic systems based on the quasi monte carlo qmc numerical techniques we first propose a general approach to deterministic filtering called the quasi monte carlo kalman filter qmc kf which unifies several existing advanced filtering methods in the literature such as the unscented kalman filter ukf and the quadrature kalman filter qkf the computationally expensive step of calculating the jacobian matrix involved in the extended kalman filter ekf is avoided in the proposed qmc kf approach we also propose sequential quasi monte carlo sqmc filtering techniques which is analogous to the sequential monte carlo smc or particle filtering methods in the literature we show in particular how to effectively combine deterministic filtering and adaptive importance sampling schemes which lead to powerful sqmc filtering strategies the properties of the proposed sqmc and sqmc is methods in terms of almost sure convergence and numerical error propagation behavior are analyzed finally numerical examples are provided to demonstrate the effectiveness of the proposed new qmc based filtering algorithms

particle filter with a mode tracker for visual tracking across illumination changes in this correspondence our goal is to develop a visual tracking algorithm that is able to track moving objects in the presence of illumination variations in the scene and that is robust to occlusions we treat the illumination and motion x y translation and scale parameters as the unknown state sequence the observation is the entire image and the observation model allows for occasional occlusions modeled as outliers the nonlinearity and multimodality of the observation model necessitate the use of a particle filter pf due to the inclusion of illumination parameters the state dimension increases thus making regular pfs impractically expensive we show that the recently proposed approach using a pf with a mode tracker can be used here since even in most occlusion cases the posterior of illumination conditioned on motion and the previous state is unimodal and quite narrow the key idea is to importance sample on the motion states while approximating importance sampling by posterior mode tracking for estimating illumination experiments demonstrate the advantage of the proposed algorithm over existing pf based approaches for various face and vehicle tracking we are also able to detect illumination model changes e g those due to transition from shadow to sunlight or vice versa by using the generalized expected log likelihood statistics and successfully compensate for it without ever loosing track

high degree cubature kalman filter the cubature kalman filter ckf which is based on the third degree spherical radial cubature rule is numerically more stable than the unscented kalman filter ukf but less accurate than the gauss hermite quadrature filter ghqf to improve the performance of the ckf a new class of ckfs with arbitrary degrees of accuracy in computing the spherical and radial integrals is proposed the third degree ckf is a special case of the class the high degree ckfs of the class can achieve the accuracy and stability performances close to those of the ghqf but at lower computational cost a numerical integration problem and a target tracking problem are utilized to demonstrate the necessity of using the high degree cubature rules to improve the performance the target tracking simulation shows that the fifth degree ckf can achieve higher accuracy than the extended kalman filter the ukf the third degree ckf and the particle filter and is computationally much more efficient than the ghqf

visual tracking by continuous density propagation in sequential bayesian filtering framework particle filtering is frequently used for visual tracking problems since it provides a general framework for estimating and propagating probability density functions for nonlinear and non gaussian dynamic systems however this algorithm is based on a monte carlo approach and the cost of sampling and measurement is a problematic issue especially for high dimensional problems we describe an alternative to the classical particle filter in which the underlying density function has an analytic representation for better approximation and effective propagation the techniques of density interpolation and density approximation are introduced to represent the likelihood and the posterior densities with gaussian mixtures where all relevant parameters are automatically determined the proposed analytic approach is shown to perform more efficiently in sampling in high dimensional space we apply the algorithm to real time tracking problems and demonstrate its performance on real video sequences as well as synthetic examples

a linearized inverse boundary value problem for maxwell s equations abstract in this paper we consider the problem of determining the electromagnetic state of a body from measurements made on the surface of the body we study the full set of maxwell s equations that govern time harmonic electric and magnetic fields we seek to reconstruct the magnetic permeability the electric permitivity and the electric conductivity in the interior of the body from measurements made on the surface we exhibit appropriate boundary measurements in the form of a boundary mapping specifically the mapping from the tangential components of the electric field to the tangential components of the magnetic field this data can be used to reconstruct and approximately provided they deviate only slightly from known constants we also estimate the reconstruction errors

fuzzy particle filtering for uncertain systems in this paper we propose a novel fuzzy particle filtering method for online estimation of nonlinear dynamic systems with fuzzy uncertainties this approach uses a sequential fuzzy simulation to approximate the possibilities of the state intervals in the state space and estimates the state by fuzzy expected value operator to solve the degeneracy problem of the fuzzy particle filter one corresponding resampling technique is introduced in addition we compare the fuzzy particle filter with ordinary particle filter in both aspects of the theoretical basis and algorithm design and demonstrate that the proposed filter outperforms standard particle filters especially when the number of the particles is small the numerical simulations of two continuous state nonlinear systems and a jump markov system are employed to show the effectiveness and robustness of the proposed fuzzy particle filter

accurate state estimation in continuous discrete stochastic state space systems with nonlinear or nondifferentiable observations this paper presents a novel method of nonlinear kalman filtering which unifies the best features of the accurate continuous discrete extended and cubature kalman filters more precisely the time updates in the discussed state estimator are done as those in the first filter whereas the measurement updates are conducted with use of the third degree spherical radial cubature rule applied for approximating the arisen gaussian weighted integrals all this allows accurate predictions of the state mean and covariance matrix to be combined with accurate measurement updates moreover the new filter is particularly effective for continuous discrete stochastic systems with nonlinear and or nondifferentiable observations the efficiency of this mixed type method is shown in comparison to the performance of the original accurate continuous discrete extended and cubature kalman filters in severe conditions of tackling a seven dimensional radar tracking problem where an aircraft executes a coordinated turn with sufficiently long sampling times

asymptotic optimality of the maximum likelihood kalman filter for bayesian tracking with multiple nonlinear sensors bayesian tracking is a general technique for state estimation of nonlinear dynamic systems but it suffers from the drawback of computational complexity this paper is concerned with a class of wiener systems with multiple nonlinear sensors such a system consists of a linear dynamic system followed by a set of static nonlinear measurements we study a maximum likelihood kalman filtering mlkf technique which involves maximum likelihood estimation of the nonlinear measurements followed by classical kalman filtering this technique permits a distributed implementation of the bayesian tracker and guarantees the boundedness of the estimation error the focus of this paper is to study the extent to which the mlkf technique approximates the theoretically optimal bayesian tracker we provide conditions to guarantee that this approximation becomes asymptotically exact as the number of sensors becomes large two case studies are analyzed in detail

feti dp bddc and block cholesky methods the feti dp and bddc algorithms are reformulated using block cholesky factorizations an approach which can provide a useful framework for the design of domain decomposition algorithms for solving symmetric positive definite linear system of equations instead of introducing lagrange multipliers to enforce the coarse level primal continuity constraints in these algorithms a change of variables is used such that each primal constraint corresponds to an explicit degree of freedom with the new formulation of these algorithms a simplified proof is provided that the spectra of a pair of feti dp and bddc algorithms with the same set of primal constraints are essentially the same numerical experiments for a two dimensional laplace s equation also confirm this result copyright 2005 john wiley sons ltd