From: Variational Integrators for Structure-Preserving Filtering Date of download: 1/14/2018 Copyright © ASME. All rights reserved. From: Variational Integrators for Structure-Preserving Filtering J. Comput. Nonlinear Dynam. 2016;12(2):021005-021005-10. doi:10.1115/1.4034728 Figure Legend: Comparison of the Kalman filter variants for the harmonic oscillator. Analytic deterministic solution is given in black, standard discrete covariance updates with explicit Euler state updates in blue, VI midpoint state updates with standard discrete covariance updates in green, and symplectic Euler state update with symplectic covariance update in red. The top left figure shows the filtered states and some covariance ellipses in the phase plane. The top right figure is a zoomed-in view of the data in the top left figure. The symplectic covariance updates increase the filter performance. The bottom left figure shows the time evolution of the eigenvalues of the covariance matrices (as in Fig. 2 there are two eigenvalues per filter). The symplectic method leads to eigenvalues that are an order of magnitude smaller. The bottom right figure shows the mechanical energy of the oscillator's filtered states for each of the filter variants; the symplectic covariance update yields a better approximation that the two other filters with standard covariance updates.