Search

One of the primary goal of quantum information theory is to understand and quantify the various correlations that exist in quantum systems. Quantum correlation is broader than quantum entanglement, and quantum discord has always attracted more attention. In this paper, we investigate the hidden properties of quantum discord. In particular, for a class of two-qubit states, we prove the existence of hidden quantum discord under local filtering operations and give sufficient conditions for the absence of hidden quantum discord. For a class of three-qubit states, we investigate hidden geometric discord and analyze its dynamic behavior under several conditions. Furthermore, by comparing two- and three-qubit systems, we discover the inconsistency of hidden discord for ρ = 1 2 N ( I + ∑ j = 1 3 c j σ j ⊗ … ⊗ σ j ) ( c 3 = 1 , N = 2 , 3 ) .

In this paper, we first establish the necessary and sufficient conditions for the existence and the explicit expressions of the Hermitian { P , k + 1 } -(anti-)reflexive solutions of the matrix equation A X = B , and meanwhile the best approximation solution is considered. Then, if the solvability conditions are not satisfied, the least squares Hermitian { P , k + 1 } -(anti-)reflexive solutions and the least squares Hermitian { P , k + 1 } -(anti-)reflexive solutions with the minimum norm of the above matrix equation are respectively derived. In addition, two algorithms are shown to compute the least squares Hermitian { P , k + 1 } -(anti-)reflexive solutions, and the corresponding numerical examples are also given to illustrate the feasibility of the algorithms.

Batch-mode least squares support vector machine (LSSVM) is often associated with unbounded number of support vectors (SVs’), making it unsuitable for applications involving large-scale streaming data. Limited-scale LSSVM, which allows efficient updating, seems to be a good solution to tackle this issue. In this paper, to train the limited-scale LSSVM dynamically, we present a budget online LSSVM (BOLSSVM) algorithm. Methodologically, by setting a fixed budget for SVs’, we are able to update the LSSVM model according to the updated SVs’ set dynamically without retraining from scratch. In particular, when a new small chunk of SVs’ substitute for the old ones, the proposed algorithm employs a low rank correction technology and the Sherman–Morrison–Woodbury formula to compute the inverse of saddle point matrix derived from the LSSVM’s Karush-Kuhn-Tucker (KKT) system, which, in turn, updates the LSSVM model efficiently. In this way, the proposed BOLSSVM algorithm is especially useful for online prediction tasks. Another merit of the proposed BOLSSVM is that it can be used for $k$ -fold cross validation. Specifically, compared with batch-mode learning methods, the computational complexity of the proposed BOLSSVM method is significantly reduced from $\mathcal {O}(n^{4})$ to $\mathcal {O}(n^{3})$ for leave-one-out cross validation with $n$ training samples. The experimental results of classification and regression on benchmark data sets and real-world applications show the validity and effectiveness of the proposed BOLSSVM algorithm.

Two augmentation block triangular preconditioners were introduced by Shen et?al. (2012) for the regularized saddle point problem. However, the spectral analysis of the preconditioner based on the augmentation of the ( 2 , 2 ) block was not throughly derived there. In this short paper, we give a detailed spectral analysis on this preconditioner. A numerical example is employed to validate the efficiency of the presented theoretical results.

Combining the modified method of characteristics with adjusted advection with a splitting positive definite mixed element scheme, we establish a new mixed finite element procedure for solving compressible miscible displacement in porous media. This procedure can preserve the mass conservation globally, the coefficient matrix of the mixed system is symmetric positive definite and the flux equation is separated from the pressure equation. We analyse the convergence and give an optimal L 2 -norm error estimate. Finally we present some numerical results to confirm our theoretical analysis.

SUMMARY In this paper, we derive bounds for the complex eigenvalues of a nonsymmetric saddle point matrix with a symmetric positive semidefinite (2,2) block, that extend the corresponding previous bounds obtained by Bergamaschi. For the nonsymmetric saddle point problem, we propose a block diagonal preconditioner for the conjugate gradient method in a nonstandard inner product. Numerical experiments are also included to test the performance of the presented preconditioner. Copyright © 2013 John Wiley & Sons, Ltd.

The solution of least squares support vector machines (LS-SVMs) is characterized by a specific linear system, that is, a saddle point system. Approaches for its numerical solutions such as conjugate methods Sykens and Vandewalle (1999) and null space methods Chu et al. (2005) have been proposed. To speed up the solution of LS-SVM, this paper employs the minimal residual (MINRES) method to solve the above saddle point system directly. Theoretical analysis indicates that the MINRES method is more efficient than the conjugate gradient method and the null space method for solving the saddle point system. Experiments on benchmark data sets show that compared with mainstream algorithms for LS-SVM, the proposed approach significantly reduces the training time and keeps comparable accuracy. To heel, the LS-SVM based on MINRES method is used to track a practical problem originated from blast furnace iron-making process: changing trend prediction of silicon content in hot metal. The MINRES method-based LS-SVM can effectively perform feature reduction and model selection simultaneously, so it is a practical tool for the silicon trend prediction task.