75,948 research outputs found
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
A Compact Third-order Gas-kinetic Scheme for Compressible Euler and Navier-Stokes Equations
In this paper, a compact third-order gas-kinetic scheme is proposed for the
compressible Euler and Navier-Stokes equations. The main reason for the
feasibility to develop such a high-order scheme with compact stencil, which
involves only neighboring cells, is due to the use of a high-order gas
evolution model. Besides the evaluation of the time-dependent flux function
across a cell interface, the high-order gas evolution model also provides an
accurate time-dependent solution of the flow variables at a cell interface.
Therefore, the current scheme not only updates the cell averaged conservative
flow variables inside each control volume, but also tracks the flow variables
at the cell interface at the next time level. As a result, with both cell
averaged and cell interface values the high-order reconstruction in the current
scheme can be done compactly. Different from using a weak formulation for
high-order accuracy in the Discontinuous Galerkin (DG) method, the current
scheme is based on the strong solution, where the flow evolution starting from
a piecewise discontinuous high-order initial data is precisely followed. The
cell interface time-dependent flow variables can be used for the initial data
reconstruction at the beginning of next time step. Even with compact stencil,
the current scheme has third-order accuracy in the smooth flow regions, and has
favorable shock capturing property in the discontinuous regions. Many test
cases are used to validate the current scheme. In comparison with many other
high-order schemes, the current method avoids the use of Gaussian points for
the flux evaluation along the cell interface and the multi-stage Runge-Kutta
time stepping technique.Comment: 27 pages, 38 figure
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
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