Spectral diagonal ensemble Kalman filters

A new type of ensemble Kalman filter is developed, which is based on replacing the sample covariance in the analysis step by its diagonal in a spectral basis. It is proved that this technique improves the aproximation of the covariance when the covariance itself is diagonal in the spectral basis, as is the case, e.g., for a second-order stationary random field and the Fourier basis. The method is extended by wavelets to the case when the state variables are random fields, which are not spatially homogeneous. Efficient implementations by the fast Fourier transform (FFT) and discrete wavelet transform (DWT) are presented for several types of observations, including high-dimensional data given on a part of the domain, such as radar and satellite images. Computational experiments confirm that the method performs well on the Lorenz 96 problem and the shallow water equations with very small ensembles and over multiple analysis cycles.

On the Convergence of a Non-linear Ensemble Kalman Smoother

Ensemble methods, such as the ensemble Kalman filter (EnKF), the local ensemble transform Kalman filter (LETKF), and the ensemble Kalman smoother (EnKS) are widely used in sequential data assimilation, where state vectors are of huge dimension. Little is known, however, about the asymptotic behavior of ensemble methods. In this paper, we prove convergence in L^p of ensemble Kalman smoother to the Kalman smoother in the large-ensemble limit, as well as the convergence of EnKS-4DVAR, which is a Levenberg-Marquardt-like algorithm with EnKS as the linear solver, to the classical Levenberg-Marquardt algorithm in which the linearized problem is solved exactly.

Data Assimilation of Satellite Fire Detection in Coupled Atmosphere-Fire Simulation by WRF-SFIRE

Currently available satellite active fire detection products from the VIIRS and MODIS instruments on polar-orbiting satellites produce detection squares in arbitrary locations. There is no global fire/no fire map, no detection under cloud cover, false negatives are common, and the detection squares are much coarser than the resolution of a fire behavior model. Consequently, current active fire satellite detection products should be used to improve fire modeling in a statistical sense only, rather than as a direct input. We describe a new data assimilation method for active fire detection, based on a modification of the fire arrival time to simultaneously minimize the difference from the forecast fire arrival time and maximize the likelihood of the fire detection data. This method is inspired by contour detection methods used in computer vision, and it can be cast as a Bayesian inverse problem technique, or a generalized Tikhonov regularization. After the new fire arrival time on the whole simulation domain is found, the model can be re-run from a time in the past using the new fire arrival time to generate the heat fluxes and to spin up the atmospheric model until the satellite overpass time, when the coupled simulation continues from the modified state.

Convergence of the Square Root Ensemble Kalman Filter in the Large Ensemble Limit

Ensemble filters implement sequential Bayesian estimation by representing the probability distribution by an ensemble mean and covariance. Unbiased square root ensemble filters use deterministic algorithms to produce an analysis (posterior) ensemble with prescribed mean and covariance, consistent with the Kalman update. This includes several filters used in practice, such as the Ensemble Transform Kalman Filter (ETKF), the Ensemble Adjustment Kalman Filter (EAKF), and a filter by Whitaker and Hamill. We show that at every time index, as the number of ensemble members increases to infinity, the mean and covariance of an unbiased ensemble square root filter converge to those of the Kalman filter, in the case a linear model and an initial distribution of which all moments exist. The convergence is in $L^{p}$ and the convergence rate does not depend on the model dimension. The result holds in the infinitely dimensional Hilbert space as well.

BDDC by a frontal solver and the stress computation in a hip joint replacement

A parallel implementation of the BDDC method using the frontal solver is employed to solve systems of linear equations from finite element analysis, and incorporated into a standard finite element system for engineering analysis by linear elasticity. Results of computation of stress in a hip replacement are presented. The part is made of titanium and loaded by the weight of human body. The performance of BDDC with added constraints by averages and with added corners is compared.