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 $\mathrm{i}=\sqrt{-1}$ 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 $L1$ scheme is often chosen to discretize such problems since it can handle the singularity of the solution near $t = 0$. In this paper, we propose a modification. We first split the time interval $[0, T]$ into $[0, T_0]$ and $[T_0, T]$, where $T_0$ ($0 < T_0 < T$) is reasonably small. Then, the graded $L1$ scheme is applied in $[0, T_0]$, while the uniform one is used in $[T_0, T]$. 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