Search Results (2)

Nonlinear state estimation plays an important role in aerospace sensing applications, where estimation accuracy must be balanced against computational efficiency. In this paper, a sparse-grid Gaussian kernel quadrature Kalman filter (SGKQKF) is proposed for discrete-time nonlinear state estimation by combining Gaussian kernel quadrature (GKQ) weighting with a Smolyak sparse-grid construction. The univariate GKQ rule is constructed on scaled Gauss–Hermite nodes through a truncated Mercer eigendecomposition of the Gaussian kernel and is then extended to multivariate cases via the Smolyak construction to alleviate the curse of dimensionality associated with tensor-product rules. The proposed method is positioned within the established sparse-grid filtering framework, with the specific contribution of integrating kernel-adapted quadrature weights into sparse-grid structures for discrete-time nonlinear Gaussian filtering. For fixed nodes, the exact kernel-quadrature weights minimize the worst-case integration error in the reproducing kernel Hilbert space (RKHS) induced by the Gaussian kernel, whereas the closed-form weights used in the implementation are interpreted as a Mercer-based practical approximation to this exact rule, with the approximation error characterized through the Mercer spectral-tail expression of the Gaussian kernel. For sparse grids, where a closed-form RKHS optimality result is not available, numerical maximum mean discrepancy (MMD) evaluations are presented as empirical diagnostics in the tested configurations. Numerical experiments demonstrate that the proposed filter achieves a favorable accuracy–efficiency trade-off compared with conventional deterministic Gaussian filters. Full article

This paper is concerned with developing an efficient numerical algorithm for the fast implementation of the sparse grid method for computing the d-dimensional integral of a given function. The new algorithm, called the MDI-SG (multilevel dimension iteration sparse grid) method, implements the sparse grid method based on a dimension iteration/reduction procedure. It does not need to store the integration points, nor does it compute the function values independently at each integration point; instead, it reuses the computation for function evaluations as much as possible by performing the function evaluations at all integration points in a cluster and iteratively along coordinate directions. It is shown numerically that the computational complexity (in terms of CPU time) of the proposed MDI-SG method is of polynomial order $O\left(d^3{N}^b\right)(b\le 2)$ or better, compared to the exponential order $O\left(N{(logN)}^{d-1}\right)$ for the standard sparse grid method, where N denotes the maximum number of integration points in each coordinate direction. As a result, the proposed MDI-SG method effectively circumvents the curse of dimensionality suffered by the standard sparse grid method for high-dimensional numerical integration. Full article