A mixed precision Jacobi method for the symmetric eigenvalue problem

The eigenvalue problem is a fundamental problem in scientific computing. In this paper, we propose a mixed precision Jacobi method for the symmetric eigenvalue problem. We first compute the eigenvalue decomposition of a real symmetric matrix by an eigensolver at low precision and we obtain a low-precision matrix of eigenvectors. Then by using the modified Gram-Schmidt orthogonalization process to the low-precision eigenvector matrix in high precision, a high-precision orthogonal matrix is obtained, which is used as an initial guess for the Jacobi method. We give the rounding error analysis for the proposed method and the quadratic convergence of the proposed method is established under some sufficient conditions. We also present a mixed precision one-side Jacobi method for the singular value problem and the corresponding rounding error analysis and quadratic convergence are discussed. Numerical experiments on CPUs and GPUs are conducted to illustrate the efficiency of the proposed mixed precision Jacobi method over the original Jacobi method.Comment: 31 pages, 2 figure

A Fast Alternating Minimization Algorithm for Total Variation Deblurring Without Boundary Artifacts

Recently, a fast alternating minimization algorithm for total variation image deblurring (FTVd) has been presented by Wang, Yang, Yin, and Zhang [{\em SIAM J. Imaging Sci.}, 1 (2008), pp. 248--272]. The method in a nutshell consists of a discrete Fourier transform-based alternating minimization algorithm with periodic boundary conditions and in which two fast Fourier transforms (FFTs) are required per iteration. In this paper, we propose an alternating minimization algorithm for the continuous version of the total variation image deblurring problem. We establish convergence of the proposed continuous alternating minimization algorithm. The continuous setting is very useful to have a unifying representation of the algorithm, independently of the discrete approximation of the deconvolution problem, in particular concerning the strategies for dealing with boundary artifacts. Indeed, an accurate restoration of blurred and noisy images requires a proper treatment of the boundary. A discrete version of our continuous alternating minimization algorithm is obtained following two different strategies: the imposition of appropriate boundary conditions and the enlargement of the domain. The first one is computationally useful in the case of a symmetric blur, while the second one can be efficiently applied for a nonsymmetric blur. Numerical tests show that our algorithm generates higher quality images in comparable running times with respect to the Fast Total Variation deconvolution algorithm