An Efficient Block Circulant Preconditioner For Simulating Fracture Using Large Fuse Networks

{\it Critical slowing down} associated with the iterative solvers close to the critical point often hinders large-scale numerical simulation of fracture using discrete lattice networks. This paper presents a block circlant preconditioner for iterative solvers for the simulation of progressive fracture in disordered, quasi-brittle materials using large discrete lattice networks. The average computational cost of the present alorithm per iteration is $O(rs log s) + delops$, where the stiffness matrix ${\bf A}$ is partioned into $r$-by-$r$ blocks such that each block is an $s$-by-$s$ matrix, and $delops$ represents the operational count associated with solving a block-diagonal matrix with $r$-by-$r$ dense matrix blocks. This algorithm using the block circulant preconditioner is faster than the Fourier accelerated preconditioned conjugate gradient (PCG) algorithm, and alleviates the {\it critical slowing down} that is especially severe close to the critical point. Numerical results using random resistor networks substantiate the efficiency of the present algorithm.Comment: 16 pages including 2 figure

Superoptimal Preconditioned Conjugate Gradient Iteration for Image Deblurring

We study the superoptimal Frobenius operators in the two-level circulant algebra. We consider two specific viewpoints: ( 1) the regularizing properties in imaging and ( 2) the computational effort in connection with the preconditioned conjugate gradient method. Some numerical experiments illustrating the effectiveness of the proposed technique are given and discussed

A preconditioned MINRES method for nonsymmetric Toeplitz matrices

Circulant preconditioning for symmetric Toeplitz linear systems is well established; theoretical guarantees of fast convergence for the conjugate gradient method are descriptive of the convergence seen in computations. This has led to robust and highly efficient solvers based on use of the fast Fourier transform exactly as originally envisaged in [G. Strang, Stud. Appl. Math., 74 (1986), pp. 171--176]. For nonsymmetric systems, the lack of generally descriptive convergence theory for most iterative methods of Krylov type has provided a barrier to such a comprehensive guarantee, though several methods have been proposed and some analysis of performance with the normal equations is available. In this paper, by the simple device of reordering, we rigorously establish a circulant preconditioned short recurrence Krylov subspace iterative method of minimum residual type for nonsymmetric (and possibly highly nonnormal) Toeplitz systems. Convergence estimates similar to those in the symmetric case are established

smt: a Matlab structured matrices toolbox

We introduce the smt toolbox for Matlab. It implements optimized storage and fast arithmetics for circulant and Toeplitz matrices, and is intended to be transparent to the user and easily extensible. It also provides a set of test matrices, computation of circulant preconditioners, and two fast algorithms for Toeplitz linear systems.Comment: 19 pages, 1 figure, 1 typo corrected in the abstrac

A fast normal splitting preconditioner for attractive coupled nonlinear Schr\"odinger equations with fractional Laplacian

A linearly implicit conservative difference scheme is applied to discretize the attractive coupled nonlinear Schr\"odinger equations with fractional Laplacian. Complex symmetric linear systems can be obtained, and the system matrices are indefinite and Toeplitz-plus-diagonal. Neither efficient preconditioned iteration method nor fast direct method is available to deal with these systems. In this paper, we propose a novel matrix splitting iteration method based on a normal splitting of an equivalent real block form of the complex linear systems. This new iteration method converges unconditionally, and the quasi-optimal iteration parameter is deducted. The corresponding new preconditioner is obtained naturally, which can be constructed easily and implemented efficiently by fast Fourier transform. Theoretical analysis indicates that the eigenvalues of the preconditioned system matrix are tightly clustered. Numerical experiments show that the new preconditioner can significantly accelerate the convergence rate of the Krylov subspace iteration methods. Specifically, the convergence behavior of the related preconditioned GMRES iteration method is spacial mesh-size-independent, and almost fractional order insensitive. Moreover, the linearly implicit conservative difference scheme in conjunction with the preconditioned GMRES iteration method conserves the discrete mass and energy in terms of a given precision

Diagonal and normal with Toeplitz-block splitting iteration method for space fractional coupled nonlinear Schr\"odinger equations with repulsive nonlinearities

By applying the linearly implicit conservative difference scheme proposed in [D.-L. Wang, A.-G. Xiao, W. Yang. J. Comput. Phys. 2014;272:670-681], the system of repulsive space fractional coupled nonlinear Schr\"odinger equations leads to a sequence of linear systems with complex symmetric and Toeplitz-plus-diagonal structure. In this paper, we propose the diagonal and normal with Toeplitz-block splitting iteration method to solve the above linear systems. The new iteration method is proved to converge unconditionally, and the optimal iteration parameter is deducted. Naturally, this new iteration method leads to a diagonal and normal with circulant-block preconditioner which can be executed efficiently by fast algorithms. In theory, we provide sharp bounds for the eigenvalues of the discrete fractional Laplacian and its circulant approximation, and further analysis indicates that the spectral distribution of the preconditioned system matrix is tight. Numerical experiments show that the new preconditioner can significantly improve the computational efficiency of the Krylov subspace iteration methods. Moreover, the corresponding preconditioned GMRES method shows space mesh size independent and almost fractional order parameter insensitive convergence behaviors