Cubature Rule Research Articles

Consensus cubature filtering based on Gaussian process for distributed sensor network with model uncertainty

In this paper, we consider a distributed sensor network without accurate knowledge of the state and observation models or even the form of system functions. Traditionally, consensus filtering approaches are widely used for distributed state estimation, but they require an accurate parametric model of the system. We propose a novel consensus cubature filtering algorithm based on Gaussian process (CCF-GP) for distributed state estimation with model uncertainty. The parameters in distributed sensor networks can be obtained from training data by Gaussian process regression (GPR). The approximate parametric model can be incorporated into Gaussian process (GP) to enhance the distributed estimation performance. In the Bayesian filtering frameworks, we derive the prediction and update steps of the hybrid consensus on measurements and consensus on information (HCMCI) method based on GPR. And then every node in the distributed network cooperates the information pair with its neighbors. For nonlinear systems, we derive the GP-based consensus filtering based on the spherical-radial cubature rule. Comparing with the standard nonlinear consensus filtering methods, the validity of our proposed algorithm is confirmed by a series of simulations.

RBFCUB: A numerical package for near-optimal meshless cubature on general polygons

In this paper we improve the cubature rules discussed in Sommariva and Vianello (2021) for the computation of integrals by radial basis functions (RBFs). More precisely, we introduce in the context of meshless cubature a leave-one-out cross validation criterion for the optimization of the RBF shape parameter. This choice allows us to get highly reliable and accurate results for any kind of both infinity and finite regularity RBF. The efficacy of this approximation scheme is tested by numerical experiments on complicated polygonal regions. The related Matlab software is provided to the scientific community in [1].

An anti-outliers SRCKF algorithm for fixed single observer passive location

Outliers often appear in measurement noise of single observer passive location system, which has a negative effect on filtering accuracy and stability, and may even lead to filter divergence. Aiming at the effect of outliers, an anti-outliers square-root cubature Kalman filter algorithm is proposed based on Bayes theorem by using the normalized polluted normal model. In this method, the cubature rule is used to calculate the mean value and variance of the nonlinear function, the normalized contaminated normal model is used to deal with the measurement error, and the variance matrix of the measurement prediction residual is adjusted in real time according to the posterior probability of outliers. The simulation results in the fixed single observer passive location model show that the proposed algorithm is robust and can eliminate the adverse effects of discrete or continuous outliers in the measurement noise.

MDFEM: Multivariate decomposition finite element method for elliptic PDEs with lognormal diffusion coefficients using higher-order QMC and FEM

We introduce the multivariate decomposition finite element method (MDFEM) for elliptic PDEs with lognormal diffusion coefficients, that is, when the diffusion coefficient has the form a = exp(Z) where Z is a Gaussian random field defined by an infinite series expansion Z(y) = ∑j≥1yjϕj with yj ~ N(0,1) and a given sequence of functions {ϕj}j≥1. We use the MDFEM to approximate the expected value of a linear functional of the solution of the PDE which is an infinite-dimensional integral over the parameter space. The proposed algorithm uses the multivariate decomposition method (MDM) to compute the infinite-dimensional integral by a decomposition into finite-dimensional integrals, which we resolve using quasi-Monte Carlo (QMC) methods, and for which we use the finite element method (FEM) to solve different instances of the PDE. We develop higher-order quasi-Monte Carlo rules for integration over the finite-dimensional Euclidean space with respect to the Gaussian distribution by use of a truncation strategy. By linear transformations of interlaced polynomial lattice rules from the unit cube to a multivariate box of the Euclidean space we achieve higher-order convergence rates for functions belonging to a class of anchored Gaussian Sobolev spaces while taking into account the truncation error. These cubature rules are then used in the MDFEM algorithm. Under appropriate conditions, the MDFEM achieves higher-order convergence rates in terms of error versus cost, i.e., to achieve an accuracy of O(ε) the computational cost is O(ε-1/λ-d′/λ)=O(ε-(p*+d′/τ)/(1-p*)) where ε−1/λ and ε−d′/λ) are respectively the cost of the quasi-Monte Carlo cubature and the finite element approximations, with d′= d (1+δ′) for some δ′ ≥ 0 and d the physical dimension, and 0<p*≤(2+d′/τ)-1 is a parameter representing the sparsity of {ϕj}j≥1.

Lattice meets lattice: Application of lattice cubature to models in lattice gauge theory

High dimensional integrals are abundant in many fields of research including quantum physics. The aim of this paper is to develop efficient recursive strategies to tackle a class of high dimensional integrals having a special product structure with low order couplings, motivated by models in lattice gauge theory from quantum field theory. A novel element of this work is the potential benefit in using lattice cubature rules. The group structure within lattice rules combined with the special structure in the physics integrands may allow efficient computations based on Fast Fourier Transforms. Applications to the quantum mechanical rotor and compact U(1) lattice gauge theory in two and three dimensions are considered.

Maximum mixture correntropy based outlier-robust nonlinear filter and smoother

In this paper, we are dedicated to studying the robust filtering and smoothing problem for a nonlinear non-Gaussian system. Considering the advantage of mixture correntropy with two kernel bandwidths in dealing with non-Gaussian noise, we propose the novel robust recursive filter and smoother based on the cost functions induced by the maximum mixture correntropy criterion, where the nonlinear dynamic model function and measurement model function are linearized by using the statistical linear regression (SLR) method. In the proposed robust recursive filter and smoother, we apply a third-order spherical cubature rule to obtain the prior estimation of the state and covariance matrix, and approximate the multi-dimensional Gaussian integrals encountered in the SLR method. Furthermore, two additional weights are introduced into the proposed robust recursive filter and smoother to modify the filtering and smoothing gains, respectively. The simulation results of manoeuvring target tracking under different non-Gaussian noise scenarios illustrate the effectiveness of the proposed robust recursive filter and smoother.

Consensus‐based nonlinear smoother for distributed continuous‐discrete systems

AbstractIn order to estimate the state of general distributed continuous‐discrete nonlinear system, a new consensus on the measurement and information cubature smoothing method is proposed in this article. The distributed continuous‐discrete nonlinear dynamic system refers to a system whose process is described as a stochastic differential equation (SDE) and whose measurements are provided by a wireless sensor network (WSN) at discrete times. First, the cubature rule is embedded into the consensus on measurements and information extended Kalman filter (CMIEKF) architecture by using the statistical linear regression (SLR) method, and then the presented consensus‐based method is obtained according to the Rauch–Tung–Striebel (RTS) smoothing methodology. In a numerical simulation, the performances of the presented smoothing method and the conventional nonlinear estimating methods are compared and analyzed. The numerical results show that, compared with other distributed estimation methods, the newly proposed smoothing algorithm enjoys not only higher accuracy but also stronger robustness.

RBF Moment computation and meshless cubature on general polygonal regions

We obtain analytical formulas for the computation of integrals (moments) of all the most common Radial Basis Functions (usually shortened as RBFs) on polygonal regions that may be nonconvex or even multiply connected. With RBFs of finite regularity, such as Thin-Plate Splines, Wendland functions and Radial Powers, our Matlab codes, based on standard linear solvers for the corresponding moment matching systems, provide cubature rules that are reasonably accurate and numerically stable.

Cubature rules based on a bivariate degree-graded alternative orthogonal basis and their applications

The purpose of this paper is to derive cubature formulas using bivariate degree-graded alternative shifted Jacobi polynomials and present some of their applications through a method for fast and explicit construction of operational integration matrices in this polynomial basis. First, we take the non-degree-graded alternative shifted Jacobi polynomial basis over the interval [0,1] and introduce a corresponding degree-graded polynomial basis. We then construct sparse and well-structured operational integration matrices in that basis. The main advantage of the proposed idea is the low computational complexity which is derived through vector–matrix multiplications without change of bases. As an application, a fast and accurate numerical algorithm based on the obtained cubature formula is developed for the approximate solution of nonlinear multi-dimensional integral equations which arise in the theory of nonlinear parabolic boundary value problems as well as the mathematical modeling of the spatio-temporal development of an epidemic. It is shown that such a matrix representation of cubature formula in terms of the bivariate degree-graded basis gives an approximate solution with a higher order accuracy. The experimental results also illustrate that smaller size and lower orders of the operational matrix and basis functions can obtain a favorable approximate solution.

Scaled boundary cubature scheme for numerical integration over planar regions with affine and curved boundaries

This paper introduces the scaled boundary cubature (SBC) scheme for accurate and efficient integration of functions over polygons and two-dimensional regions bounded by parametric curves. Over two-dimensional domains, the SBC method reduces integration over a region bounded by m curves to integration over m regions (referred to as curved triangular regions), where each region is bounded by two line segments and a curve. With proper (counterclockwise) orientation of the boundary curves, the scheme is applicable to convex and nonconvex domains. Additionally, for star-convex domains, a tensor-product cubature rule with positive weights and integration points in the interior of the domain is obtained. If the integrand is homogeneous, we show that this new method reduces to the homogeneous numerical integration scheme; however, the SBC scheme is more versatile since it is equally applicable to both homogeneous and non-homogeneous functions. This paper also introduces several methods for smoothing integrands with point singularities and near-singularities. When these methods are used, highly efficient integration of weakly singular functions is realized. The SBC method is applied to a number of benchmark problems, which reveal its broad applicability and superior performance (in terms of time to generate a rule and accuracy per cubature point) when compared to existing methods for integration.

Cubature rules for weakly and fully compressible off-lattice Boltzmann methods

Off-lattice Boltzmann methods increase the flexibility and applicability of lattice Boltzmann methods by decoupling the discretizations of time, space, and particle velocities. However, the velocity sets that are mostly used in off-lattice Boltzmann simulations were originally tailored to on-lattice Boltzmann methods. In this contribution, we show how the accuracy and efficiency of weakly and fully compressible semi-Lagrangian off-lattice Boltzmann simulations are increased by velocity sets derived from cubature rules, i.e., multivariate quadratures not produced by the Gauß-product rule. In particular, simulations of 2D shock-vortex interactions indicate that the cubature-derived degree-nine D2Q19 velocity set is capable of replacing the Gauß-product rule-derived D2Q25. Likewise, the degree-five velocity sets D3Q13 and D3Q21, as well as a degree-seven D3V27 velocity set were successfully tested for 3D Taylor–Green vortex flows to challenge and surpass the quality of the customary D3Q27 velocity set. In compressible 3D Taylor–Green vortex flows with Mach numbers Ma={0.5;1.0;1.5;2.0} on-lattice simulations with velocity sets D3Q103 and D3V107 showed only limited stability, while the off-lattice degree-nine D3Q45 velocity set accurately reproduced the kinetic energy provided by literature.

A NEW AND EFFICIENT CENTROIDAL MEAN DERIVATIVE-BASED TRAPEZOIDAL SCHEME FOR NUMERICAL CUBATURE

This research presents a new and efficient Centroidal mean derivative-based numerical cubature scheme which has been proposed for the accurate evaluation of double integrals under finite range. The proposed modification is based on the Trapezoidal-type quadrature and cubature rules. The approximate values can only be obtained for some important applications to evaluate the complex double integrals. Higher precision and order of accuracy could be achieved by the proposed scheme. The schemes, in basic and composite forms, with local and global error terms are presented with necessary supporting arguments with their performance evaluation against conventional Trapezoid rule through some numerical experiments. The simultaneously observed error distributions of the proposed schemes are found to be lower than the conventional Trapezoidal cubature scheme in composite form

The Cubature Kalman Filter revisited

In this paper, the construction and effectiveness of the so-called Cubature Kalman Filter (CKF) is revisited, as well as its extensions for higher degrees of precision. In this sense, some stable (with respect to the dimension) cubature rules with a quasi-optimal number of nodes are built, and their numerical performance is checked in comparison with other known formulas. All these cubature rules are suitably placed in the mathematical framework of numerical integration in several variables. A method based on the discretization of higher order partial derivatives by certain divided differences is used to provide stable rules of degrees d=5 and d=7, though it can also be applied for higher dimensions. The application of these old and new formulas to the filter algorithm is tested by means of some examples.

Square-root high-degree cubature Kalman filters for state estimation in nonlinear continuous-discrete stochastic systems

This paper addresses the problem of square-rooting in the Cubature Kalman Filtering (CKF) originated from Arasaratnam and Haykin in 2009. Presently, this technique has been accommodated to various cubature rules, including high-degree ones. Since its discovery the CKF has become one of the most powerful state estimation methods because of outstanding performance and robustness in numerous engineering applications. Its high-degree versions are shown to be accurate and even comparable to particle filters, which are considered to be among the most effective algorithms for treating nonlinear stochastic systems. However, the lack of square-root implementations within high-degree CKFs makes them vulnerable to round-off and other errors committed because of a potential covariance matrix positivity loss, which may encounter in practice. This shortcoming affects severely and fails high-degree CKFs since the Cholesky factorization of predicted and filtering covariances underlying the filters in use may not be fulfilled for indefinite matrices. Here, we resolve it by means of hyperbolic QR transforms applied for yielding J-orthogonal square roots. Our novel square-root algorithms are justified theoretically and examined and compared numerically to the existing non-square-root CKF and some other available filters in a simulated flight control scenario, including that with ill-conditioned measurements.

Addendum to the paper “High‐order symmetric cubature rules for tetrahedra and pyramids”

AbstractIn Jaśkowiec and Sukumar (Int J Numer Methods Eng, doi: 10.1002/nme.6528, 2020), we presented high‐order (p = 2–20) symmetric cubatures rules for tetrahedra and pyramids. This algorithm was sensitive to the initial location of the cubature nodes, and it did not converge for p > 11 over prisms and hexahedra (cubes). In this addendum, we resolve this issue and obtain high‐order symmetric rules over prisms and cubes. For the prism, we use the initial guess for the cubature rule as the tensor product of a cubature rule over a triangle and a univariate Gauss quadrature rule, and for the cube the initial guess is the tensor product of univariate Gauss quadrature rules. Verification and convergence tests are presented to affirm the accuracy of the cubature rules. On applying the cubature algorithm described in Jaśkowiec and Sukumar (Int J Numer Methods Eng, 121(11), 2418–2436, 2020), we also construct nonsymmetric high‐order (p = 2–20) cubature rules over prisms, cubes, and pyramids. In the supplementary materials, all cubature rules (128 digits of precision) are provided in a text file and in format.

High‐degree cubature Kalman filter for nonlinear state estimation with missing measurements

AbstractThis paper proposes high‐degree cubature Kalman filter for nonlinear systems with missing measurements. We derive out the explicit formulas for the prediction and update in the filtering. To fulfill the numerical computation, especially the numerical integrals, of these formulas, the fifth‐degree spherical‐radial cubature rule is adopted to give a high‐degree cubature Kalman filtering algorithm. Through numerical example, it is shown that the fifth‐degree cubature Kalman filter has better precision and stability than the extended Kalman filter, the unscented Kalman filter, and the fifth‐degree unscented Kalman filter.

Analyzing Probabilistic Optimal Power Flow Problem by Cubature Rules

Analyzing Probabilistic Optimal Power Flow Problem by Cubature Rules

Verified Numerical Multiple Integration Based on Composite Gauss-Legendre Quadrature and Product Cubature Rules

Verified Numerical Multiple Integration Based on Composite Gauss-Legendre Quadrature and Product Cubature Rules

Estimation Performance for the Cubature Particle Filter under Nonlinear/Non-Gaussian Environments

This paper evaluates the state estimation performance for processing nonlinear/non-Gaussian systems using the cubature particle filter (CPF), which is an estimation algorithm that combines the cubature Kalman filter (CKF) and the particle filter (PF). The CPF is essentially a realization of PF where the third-degree cubature rule based on numerical integration method is adopted to approximate the proposal distribution. It is beneficial where the CKF is used to generate the importance density function in the PF framework for effectively resolving the nonlinear/non-Gaussian problems. Based on the spherical-radial transformation to generate an even number of equally weighted cubature points, the CKF uses cubature points with the same weights through the spherical-radial integration rule and employs an analytical probability density function (pdf) to capture the mean and covariance of the posterior distribution using the total probability theorem and subsequently uses the measurement to update with Bayes’ rule. It is capable of acquiring a maximum a posteriori probability estimate of the nonlinear system, and thus the importance density function can be used to approximate the true posterior density distribution. In Bayesian filtering, the nonlinear filter performs well when all conditional densities are assumed Gaussian. When applied to the nonlinear/non-Gaussian distribution systems, the CPF algorithm can remarkably improve the estimation accuracy as compared to the other particle filter-based approaches, such as the extended particle filter (EPF), and unscented particle filter (UPF), and also the Kalman filter (KF)-type approaches, such as the extended Kalman filter (EKF), unscented Kalman filter (UKF) and CKF. Two illustrative examples are presented showing that the CPF achieves better performance as compared to the other approaches.

Extended Kalman Filter Using Orthogonal Polynomials

This paper reports a new extended Kalman filter where the underlying nonlinear functions are linearized using a Gaussian orthogonal basis of a weighted $\mathcal {L}_{2}$ space. As we are interested in computing the states’ mean and covariance with respect to Gaussian measure, it would be better to use a linearization, that is optimal with respect to the same measure. The resulting first-order polynomial coefficients are approximately calculated by evaluating the integrals using (i) third-order Taylor series expansion (ii) cubature rule of integration. Compared to direct integration-based filters, the proposed filter is far less susceptible to the accumulation of round-off errors leading to loss of positive definiteness. The proposed algorithms are applied to four nonlinear state estimation problems. We show that our proposed filter consistently outperforms the traditional extended Kalman filter and achieves a competitive accuracy to an integration-based square root filter, at a significantly reduced computing cost.