632 research outputs found
A flexible and adaptive Simpler GMRES with deflated restarting for shifted linear systems
In this paper, two efficient iterative algorithms based on the simpler GMRES
method are proposed for solving shifted linear systems. To make full use of the
shifted structure, the proposed algorithms utilizing the deflated restarting
strategy and flexible preconditioning can significantly reduce the number of
matrix-vector products and the elapsed CPU time. Numerical experiments are
reported to illustrate the performance and effectiveness of the proposed
algorithms.Comment: 17 pages. 9 Tables, 1 figure; Newly update: add some new numerical
results and correct some typos and syntax error
A note on the growth factor in Gaussian elimination for generalized Higham matrices
The Higham matrix is a complex symmetric matrix A=B+iC, where both B and C
are real, symmetric and positive definite and is the
imaginary unit. For any Higham matrix A, Ikramov et al. showed that the growth
factor in Gaussian elimination is less than 3. In this paper, based on the
previous results, a new bound of the growth factor is obtained by using the
maximum of the condition numbers of matrixes B and C for the generalized Higham
matrix A, which strengthens this bound to 2 and proves the Higham's conjecture.Comment: 8 pages, 2 figures; Submitted to MOC on Dec. 22 201
Well-posedness of the fractional Ginzburg-Landau equation
In this paper, we investigate the well-posedness of the real fractional Ginzburg-Landau equation in several different function spaces, which have been used to deal with the Burgers' equation, the semilinear heat equation, the Navier-Stokes equations, etc. The long time asymptotic behavior of the nonnegative global solutions is also studied in details
A parallel preconditioning technique for an all-at-once system from subdiffusion equations with variable time steps
Volterra subdiffusion problems with weakly singular kernel describe the
dynamics of subdiffusion processes well.The graded scheme is often chosen
to discretize such problems since it can handle the singularity of the solution
near . In this paper, we propose a modification. We first split the time
interval into and , where ()
is reasonably small. Then, the graded scheme is applied in ,
while the uniform one is used in . Our all-at-once system is derived
based on this strategy. In order to solve the arising system efficiently, we
split it into two subproblems and design two preconditioners. Some properties
of these two preconditioners are also investigated. Moreover, we extend our
method to solve semilinear subdiffusion problems. Numerical results are
reported to show the efficiency of our method.Comment: 3 figures; 3 table
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