1,181 research outputs found
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
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