Sparsity-Based Kalman Filters for Data Assimilation

Several variations of the Kalman filter algorithm, such as the extended Kalman filter (EKF) and the unscented Kalman filter (UKF), are widely used in science and engineering applications. In this paper, we introduce two algorithms of sparsity-based Kalman filters, namely the sparse UKF and the progressive EKF. The filters are designed specifically for problems with very high dimensions. Different from various types of ensemble Kalman filters (EnKFs) in which the error covariance is approximated using a set of dense ensemble vectors, the algorithms developed in this paper are based on sparse matrix approximations of error covariance. The new algorithms enjoy several advantages. The error covariance has full rank without being limited by a set of ensembles. In addition to the estimated states, the algorithms provide updated error covariance for the next assimilation cycle. The sparsity of error covariance significantly reduces the required memory size for the numerical computation. In addition, the granularity of the sparse error covariance can be adjusted to optimize the parallelization of the algorithms

Asymptotic minimax risk of predictive density estimation for non-parametric regression

We consider the problem of estimating the predictive density of future observations from a non-parametric regression model. The density estimators are evaluated under Kullback--Leibler divergence and our focus is on establishing the exact asymptotics of minimax risk in the case of Gaussian errors. We derive the convergence rate and constant for minimax risk among Bayesian predictive densities under Gaussian priors and we show that this minimax risk is asymptotically equivalent to that among all density estimators.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ222 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Fermion correction to the mass of the scalar glueball in QCD sum rule

Contributions of fermions to the mass of the scalar glueball $0^{++}$ are calculated at two-loop level in the framework of QCD sum rules. It obviously changes the coefficients in the operator product expansion (OPE) and shifts the mass of glueball.Comment: 5 pages, 2 figure

A Third-order Compact Gas-kinetic Scheme on Unstructured Meshes for Compressible Navier-Stokes Solutions

In this paper, for the first time a compact third-order gas-kinetic scheme is proposed on unstructured meshes for the compressible viscous flow computations. The possibility to de sign such a third-order compact scheme is due to the high-order gas evolution model, where a time-dependent gas distribution function at a cell interface not only provides the fluxes across a cell interface, but also the time evolution of the flow variables at the cell interface as well. As a result, both cell averaged and cell interface flow variables can be used for the initial data reconstruction at the beginning of next time step. A weighted least-square reconstruction has been used for the construction of a third-order initial condition. Therefore, a compact third-order gas-kinetic scheme with the involvement of neighboring cells only can be developed on unstructured meshes. In comparison with other conventional high-order schemes, the current method avoids the use of Gaussian points for the flux integration along a cell interface and the multi-stage Runge-Kutta time stepping technique. The third-order compact scheme is numerically stable under CFL condition above 0.5. Due to the multidimensional gas-kinetic formulation and the coupling of inviscid and viscous terms, even with unstructured meshes the boundary layer solution and the vortex structure can be accurately captured in the current scheme. At the same time, the compact scheme can capture strong shocks as well.Comment: arXiv admin note: substantial text overlap with arXiv:1412.448

A note on directed 4-cycles in digraphs

Using some combinatorial techniques, in this note, it is proved that if $\alpha\geq 0.28866$, then any digraph on $n$ vertices with minimum outdegree at least $\alpha n$ contains a directed cycle of length at most 4