A limited-memory block bi-diagonal Toeplitz preconditioner for block lower triangular Toeplitz system from time–space fractional diffusion equation

Abstract

A block lower triangular Toeplitz system arising from the time-space fractional diffusion equation is discussed. For efficient solutions of such the linear system, the preconditioned biconjugate gradient stabilized method and the flexible general minimal residual method are exploited. The main contribution of this paper has two aspects: (i) A block bi-diagonal Toeplitz preconditioner is developed for the block lower triangular Toeplitz system, whose storage is of O(N ) with N being the spatial grid number; (ii) A new skew-circulant preconditioner is designed to accelerate the inverse of the block bi-diagonal Toeplitz preconditioner multiplying a vector. Numerical experiments are given to demonstrate the effectiveness of our two proposed preconditioners.

Applications of Distributed-Order Fractional Operators: A Review

“…Further, preconditioners are often designed to accelerate matrix computations in nonlinear CO FDEs involving iterative problem solving procedures. Many studies have proposed different types of preconditioners such as, for example, preconditioned biconjugate gradient method [ 208 ] and generalized minimal residual method [ 209 ], for solving nonlinear CO FDEs. Both the above described techniques, that are parallel computing and preconditioning, present possible opportunities to reduce the computational time for solving DODEs and are hence worthy of detailed investigation in the future.…”

A fast implicit difference scheme for solving the generalized time–space fractional diffusion equations with variable coefficients

“…In order to solve Equation (3.3) effectively, we consider two specific classes of problems:When the diffusion coefficient ξ ( x , t ) ≡ ξ , the coefficient matrix of Equation (3.3) will be a time‐independent Toeplitz matrix, that is, ℳ j + 1 = ℳ; then we can compute its matrix inverse via the Gohberg–Semencul formula (GSF) [13] using only its first and last columns. Such a strategy does not need to call the preconditioned Krylov subspace solvers at each time level 0 ≤ j ≤ M − 1, and the solution at each time level (i.e., ℳ −1 u j + 1 ) can be calculated via about six FFTs, thus saving considerable computational cost; refer to [14, 18, 32, 60] for detail. When the diffusion coefficient is just a function related to both x and t , that is, ξ ( x , t ), the coefficient matrix of Equation (3.3) becomes the sum of a scalar matrix and of a diagonal‐multiply‐Toeplitz matrix, which is time‐dependent. In this case, Equation (3.3) has to be solved via a preconditioned Krylov subspace solver at each time level j . …”

“…For accelerating BiCGSTAB, we consider the following skew‐circulant and banded preconditioners: where the vector is the first column of the skew‐circulant matrix sk ( W α ) [60], and …”

“…(i) When the diffusion coefficient (x, t) ≡ , the coefficient matrix of Equation (33) will be a time-independent Toeplitz matrix, that is,  j + 1 = ; then we can compute its matrix inverse via the Gohberg-Semencul formula (GSF) [13] using only its first and last columns. Such a strategy does not need to call the preconditioned Krylov subspace solvers at each time level 0 ≤ j ≤ M − 1, and the solution at each time level (i.e.,  −1 u j + 1 ) can be calculated via about six FFTs, thus saving considerable computational cost; refer to [14,18,32,60] for detail. (ii) When the diffusion coefficient is just a function related to both x and t, that is, (x, t),…”

“…respectively. Meanwhile, the high efficiency of skew-circulant and banded preconditioners for (time-)space FDEs has been shown in [28,60,61]. In practical implementations, when P sk or P b is employed as the preconditioner, a fast preconditioned version of the BiCGSTAB method is obtained.…”