A Spectral Analysis of Subspace Enchanced Preconditioners

It is well-known that the convergence of Krylov subspace methods to solve linear system depends on the spectrum of the coefficient matrix, moreover, it is widely accepted that for both symmetric and unsymmetric systems Krylov subspace methods will converge fast if the spectrum of the coefficient matrix is clustered. In this paper we investigate the spectrum of the system preconditioned by the deflation, coarse correction and adapted deflation preconditioners. Our analysis shows that the spectrum of the preconditioned system is highly impacted by the angle between the coarse space for the construction of the three preconditioners and the subspace spanned by the eigenvectors associated with the small eigenvalues of the coefficient matrix. Furthermore, we prove that the accuracy of the inverse of projection matrix also impacts the spectrum of the preconditioned system. Numerical experiments emphasized the theoretical analysis.Comment: 18 pages, 18 figures. arXiv admin note: substantial text overlap with arXiv:1110.210

High order numerical schemes for second-order FBSDEs with applications to stochastic optimal control

This is one of our series papers on multistep schemes for solving forward backward stochastic differential equations (FBSDEs) and related problems. Here we extend (with non-trivial updates) our multistep schemes in [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751.] to solve the second order FBSDEs (2FBSDEs). The key feature of the multistep schemes is that the Euler method is used to discrete the forward SDE, which dramatically reduces the entire computational complexity. Moreover, it is shown that the usual quantities of interest (e.g., the solution tuple $(Y_t, Z_t, A_t, \Gamma_t)$ in the 2FBSDEs) are still of high order accuracy. Several numerical examples are given to show the effective of the proposed numerical schemes. Applications of our numerical schemes for stochastic optimal control problems are also presented

A Global Algorithm for Training Multilayer Neural Networks

We present a global algorithm for training multilayer neural networks in this Letter. The algorithm is focused on controlling the local fields of neurons induced by the input of samples by random adaptations of the synaptic weights. Unlike the backpropagation algorithm, the networks may have discrete-state weights, and may apply either differentiable or nondifferentiable neural transfer functions. A two-layer network is trained as an example to separate a linearly inseparable set of samples into two categories, and its powerful generalization capacity is emphasized. The extension to more general cases is straightforward

One-Dimensional Compressible Heat-Conducting Gas with Temperature-Dependent Viscosity

We consider the one-dimensional compressible Navier--Stokes system for a viscous and heat-conducting ideal polytropic gas when the viscosity $\mu$ and the heat conductivity $\kappa$ depend on the specific volume $v$ and the temperature $\theta$ and are both proportional to $h(v)\theta^{\alpha}$ for certain non-degenerate smooth function $h$. We prove the existence and uniqueness of a global-in-time non-vacuum solution to its Cauchy problem under certain assumptions on the parameter $\alpha$ and initial data, which imply that the initial data can be large if $|\alpha|$ is sufficiently small. Our result appears to be the first global existence result for general adiabatic exponent and large initial data when the viscosity coefficient depends on both the density and the temperature.Comment: To appear in "Mathematical Models and Methods in Applied Sciences"; Contact [email protected] for any comment

Efficient spectral sparse grid approximations for solving multi-dimensional forward backward SDEs

This is the second part in a series of papers on multi-step schemes for solving coupled forward backward stochastic differential equations (FBSDEs). We extend the basic idea in our former paper [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751] to solve high-dimensional FBSDEs, by using the spectral sparse grid approximations. The main issue for solving high dimensional FBSDEs is to build an efficient spatial discretization, and deal with the related high dimensional conditional expectations and interpolations. In this work, we propose the sparse grid spatial discretization. We use the sparse grid Gaussian-Hermite quadrature rule to approximate the conditional expectations. And for the associated high dimensional interpolations, we adopt an spectral expansion of functions in polynomial spaces with respect to the spatial variables, and use the sparse grid approximations to recover the expansion coefficients. The FFT algorithm is used to speed up the recovery procedure, and the entire algorithm admits efficient and high accurate approximations in high-dimensions, provided that the solutions are sufficiently smooth. Several numerical examples are presented to demonstrate the efficiency of the proposed methods

The classical capacity for continuous variable teleportation channel

The process of quantum teleportation can be considered as a quantum channel. The exact classical capacity of the continuous variable teleportation channel is given. Also, the channel fidelity is derived. Consequently, the properties of the continuous variable quantum teleportation are discussed and interesting results are obtained.Comment: The formula of fidelity corrected; the figure of fidelity replotted; submitted for publicatio

Numerical method for hyperbolic conservation laws via forward backward SDEs

It is well known that for solutions of semi-linear parabolic PDEs, there are equivalent probabilistic interpretations, which yields the so called nonlinear Feymman-Kac formula. By adopting such formula, we consider in this work a novel numerical approach for solutions of hyperbolic conservation laws. Our numerical method consists in efficiently computing the viscosity solutions of conservation laws. However, instead of solving the viscosity problem directly (which is difficult), we find its equivalent probabilistic solution by adopting the Feymman-Kac formula, which relies on solving the equivalent forward backward stochastic differential equations. It is noticed that such framework possesses the following advantages: (i) the viscosity parameter can be chosen sufficiently small (say $10^{-10}$); (ii) the computational procedure on each discretized time level can be \textit{completely parallel}; (iii) the traditional CFL condition is dramatically weakened; (iv) one does not need to handle the transition layers and discertizations of derivatives. Thus, high accuracy viscosity solutions can be efficiently found. Several numerical examples are given to demonstrate the effectiveness of the proposed numerical method.Comment: This paper has been withdrawn by the authors due to some wrong plots in numerical test

The classical capacity for the quantum Markov channel of continuous variables

Quantum communications using continuous variables are quite mature experimental techniques and the relevant theories have been extensively investigated with various methods. In this paper, we study the continuous variable quantum channels from a different angle, i.e., by exploring master equations. And we finally give explicitly the capacity of the channel we are studying. By the end of this paper, we derive the criterion for the optimal capacities of the Gaussian channel versus its fidelity.Comment: 7 pages, 3 figures, some corrections have been made, figures replotte

Global Spherical Symmetric Flows for a Viscous Radiative and Reactive Gas in an Exterior Domain with Large Initial Data

In this paper, we study the global existence, uniqueness and large-time behavior of spherically symmetric solution of a viscous radiative and reactive gas in an unbounded domain exterior to the unit sphere in $\mathbb{R}^{n}$ for $n\geq 2$. The key point in the analysis is to deduce certain uniform estimates on the solutions, especially on the uniform positive lower and upper bounds on the specific volume and the temperature.Comment: arXiv admin note: text overlap with arXiv:1705.0127

Parametric Channel Estimation by Exploiting Hopping Pilots in Uplink OFDMA

This paper proposes a parametric channel estimation algorithm applicable to uplink of OFDMA systems with pseudo-random subchannelization. It exploits the hopping pilots to facilitate ESPRIT to estimate the delay subspace of the multipath fading channel, and utilizes the global pilot tones to interpolate on data subcarriers. Hence, it outperforms the traditional local channel interpolators considerably.Comment: 5 pages, 3 figures, Appeared in IEEE PIMRC'0