Adaptive Gaussian sum squared-root cubature Kalman filter with split-merge scheme for state estimation

The paper deals with state estimation problem of nonlinear non-Gaussian discrete dynamic systems for improvement of accuracy and consistency. An efficient new algorithm called the adaptive Gaussian-sum square-root cubature Kalman filter（AGSSCKF） with a split-merge scheme is proposed. It is developed based on the squared-root extension of newly introduced cubature Kalman filter（SCKF） and is built within a Gaussian-sum framework. Based on the condition that the probability density functions of process noises and initial state are denoted by a Gaussian sum using optimization method, a bank of SCKF are used as the sub-filters to estimate state of system with the corresponding weights respectively, which is adaptively updated. The new algorithm consists of an adaptive splitting and merging procedure according to a proposed split-decision model based on the nonlinearity degree of measurement. The results of two simulation scenarios（one-dimensional state estimation and bearings-only tracking） show that the proposed filter demonstrates comparable performance to the particle filter with significantly reduced computational cost.

Applications of Simple Estimation Scheme of Electron Correlation Energy to Strong Ionic Compounds

The calculation results of electron correlation energies of KF and （KF）2 were reported. The transferability of 1s^2 K , 1s^2 F and the inner core correlation effects of K and F in both K, K^＋, KF and F, F^-, KF systems were investigated respectively. The correlation energy contributions of K and F component to KF system were calculated. By applying the simple estimation scheme to the calculation of the correlation energy of the strong ionic compound KF and （KF）2, it was shown that such a powerful scheme could not only reach the chemical accuracy but also need little computational work.

Reduced-order extrapolation spectral-finite difference scheme based on POD method and error estimation for three-dimensional parabolic equation

In this study, a classical spectral-finite difference scheme （SFDS） for the three-dimensional （3D） parabolic equation is reduced by using proper orthogonal decomposition （POD） and singular value decomposition （SVD）. First, the 3D parabolic equation is discretized in spatial variables by using spectral collocation method and the discrete scheme is transformed into matrix formulation by tensor product. Second~ the classical SFDS is obtained by difference discretization in time-direction. The ensemble of data are comprised with the first few transient solutions of the classical SFDS for the 3D parabolic equation and the POD bases of ensemble of data are generated by using POD technique and SVD. The unknown quantities of the classical SFDS are replaced with the linear combination of POD bases and a reduced- order extrapolation SFDS with lower dimensions and sufficiently high accuracy for the 3D parabolic equation is established. Third, the error estimates between the classical SFDS solutions and the reduced-order extrapolation SFDS solutions and the implementation for solving the reduced-order extrapolation SFDS are provided. Finally, a numerical example shows that the errors of numerical computations are consistent with the theoretical results. Moreover, it is shown that the reduced-order extrapolation SFDS is effective and feasible to find the numerical solutions for the 3D parabolic equation.

AN EXPONENTIAL WAVE INTEGRATOR PSEUDOSPECTRAL METHOD FOR THE SYMMETRIC REGULARIZED-LONG-WAVE EQUATION

An efficient and accurate exponential wave integrator Fourier pseudospectral （EWI-FP） method is proposed and analyzed for solving the symmetric regularized-long-wave （SRLW） equation, which is used for modeling the weakly nonlinear ion acoustic and space-charge waves. The numerical method here is based on a Gautschi-type exponential wave integrator for temporal approximation and the Fourier pseudospectral method for spatial discretization. The scheme is fully explicit and efficient due to the fast Fourier transform. Numerical analysis of the proposed EWI-FP method is carried out and rigorous error estimates are established without CFL-type condition by means of the mathematical induction. The error bound shows that EWI-FP has second order accuracy in time and spectral accuracy in space. Numerical results are reported to confirm the theoretical studies and indicate that the error bound here is optimal.