Search

In this paper, we tackle two important problems in low-rank learning, which are partial singular value decomposition and numerical rank estimation of huge matrices. By using the concepts of Krylov subspaces such as Golub-Kahan bidiagonalization (GK-bidiagonalization) as well as Ritz vectors, we propose two methods for solving these problems in a fast and accurate way. Our experiments show the advantages of the proposed methods compared to the traditional and randomized singular value decomposition methods. The proposed methods are appropriate for applications involving huge matrices where the accuracy of the desired singular values and also all of their corresponding singular vectors are essential. As a real application, we evaluate the performance of our methods on the problem of Riemannian similarity learning between two different image datasets of MNIST and USPS. [ABSTRACT FROM AUTHOR]

The CMRH (Changing Minimal Residual method based on the Hessenberg process) is an iterative method for solving nonsymmetric linear systems. The method generates a Krylov subspace in which an approximate solution is determined. The CMRH method is generally used with restarting to reduce the storage. Restarting often slows down the convergence. In this paper we present augmentation and de ation techniques for accelerating the convergence of the restarted CMRH method. Augmentation adds a subspace to the Krylov subspace, while de ation removes certain parts from the operator. Numerical experiments show that the new algorithms can be more efficient compared with the CMRH method. [ABSTRACT FROM AUTHOR]

In this paper, a denoising PDE model based on a combination of the isotropic diffusion and total variation models is presented. The new weighted model is able to be adaptive in each region in accordance with the image's information. The model performs more diffusion in the flat regions of the image, and less diffusion in the edges of the image. The new model has more ability to restore the image in terms of peak signal to noise ratio and visual quality, compared with total variation, isotropic diffusion, and some well-known models. Experimental results show that the model is able to suppress the noise effectively while preserving texture features and edge information well. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new dynamic model for the air traffic flow prediction to estimate the traffic distribution for given airspaces in the future. Based on Lighthill-Whitham-Richards traffic flow model and the Newton's second law, we establish a nonlinear model to describe inter-relationship and influential factors of the three characteristic parameters as traffic flow, density, and velocity. The upwind scheme is applied to perform the numerical simulations. Numerical results show that the proposed model can reproduce the evolution of shockwave, rarefaction wave, and small perturbation. [ABSTRACT FROM AUTHOR]

We present an iterative method based on the Hermitian and skew-Hermitian splitting (HSS) for solving the continuous Sylvester equation. By using the HSS of the coefficient matricesAandB, we establish a method which is practically inner/outer iterations, by employing a conjugate gradient on the normal equations (CGNR)-like method as inner iteration to approximate each outer iterate, while each outer iteration is induced by a convergent splitting of the coefficient matrices. Via this method, a Sylvester equation with coefficient matricesand(which are the skew-Hermitian part ofAandB, respectively) is solved iteratively by a CGNR-like method. Convergence conditions of this method are studied and numerical examples show the efficiency of this method. In addition, we show that the quasi-Hermitian splitting can induce accurate, robust and effective preconditioned Krylov subspace methods. [ABSTRACT FROM PUBLISHER]

The major drawback of the s-step iterative methods for nonsymmetric linear systems of equations is that, in the floating-point arithmetic, a quick loss of orthogonality of s-dimensional direction subspaces can occur, and consequently slow convergence and instability in the algorithm may be observed as s gets larger than 5. In [18], Swanson and Chronopoulos have demonstrated that the value of s in the s-step Orthomin(k) algorithm can be increased beyond s=5 by orthogonalizing the s direction vectors in each iteration, and have shown that the ATA-orthogonal s-step Orthomin(k) is stable for large values of s (up to s=16). The subject of this paper is to show how by using the CADNA library, it is possible to determine a good value of s for ATA-orthogonal s-step Orthomin(k), and during the run of its code to detect the numerical instabilities and to stop the process correctly, and to restart the ATA-orthogonal s-step Orthomin(k) in order to improve the computed solution. Numerical examples are used to show the good numerical properties. [ABSTRACT FROM AUTHOR]