Strang-Type Preconditioners for Solving Linear Systems from Delay Differential Equations

We consider the solution of delay differential equations (DDEs) by using boundary value methods (BVMs). These methods require the solution of one or more nonsymmetric, large and sparse linear systems. The GMRES method with the Strang-type block-circulant preconditioner is proposed for solving these linear systems. We show that if a P _{k 1},k ₂-stable BVM is used for solving an m-by-m system of DDEs, then our preconditioner is invertible and all the eigenvalues of the preconditioned system are clustered around 1. It follows that when the GMRES method is applied to solving the preconditioned systems, the method may converge fast. Numerical results are given to illustrate the effectiveness of our methods.     

Diagonal and Toeplitz splitting iteration methods for diagonal‐plus‐Toeplitz linear systems from spatial fractional diffusion equations

The finite difference discretization of the spatial fractional diffusion equations gives discretized linear systems whose coefficient matrices have a diagonal‐plus‐Toeplitz structure. For solving these diagonal‐plus‐Toeplitz linear systems, we construct a class of diagonal and Toeplitz splitting iteration methods and establish its unconditional convergence theory. In particular, we derive a sharp upper bound about its asymptotic convergence rate and deduct the optimal value of its iteration parameter. The diagonal and Toeplitz splitting iteration method naturally leads to a diagonal and circulant splitting preconditioner. Analysis shows that the eigenvalues of the corresponding preconditioned matrix are clustered around 1, especially when the discretization step‐size h is small. Numerical results exhibit that the diagonal and circulant splitting preconditioner can significantly improve the convergence properties of GMRES and BiCGSTAB, and these preconditioned Krylov subspace iteration methods outperform the conjugate gradient method preconditioned by the approximate inverse circulant‐plus‐diagonal preconditioner proposed recently by Ng and Pan (M.K. Ng and J.‐Y. Pan, SIAM J. Sci. Comput. 2010;32:1442‐1464). Moreover, unlike this preconditioned conjugate gradient method, the preconditioned GMRES and BiCGSTAB methods show h‐independent convergence behavior even for the spatial fractional diffusion equations of discontinuous or big‐jump coefficients.     

Preconditioners for block Toeplitz systems based on circulant preconditioners

We study the numerical solution of a block system T _m,n x=b by preconditioned conjugate gradient methods where T _m,n is an m×m block Toeplitz matrix with n×n Toeplitz blocks. These systems occur in a variety of applications, such as two-dimensional image processing and the discretization of two-dimensional partial differential equations. In this paper, we propose new preconditioners for block systems based on circulant preconditioners. From level-1 circulant preconditioner we construct our first preconditioner q ₁(T _m,n ) which is the sum of a block Toeplitz matrix with Toeplitz blocks and a sparse matrix with Toeplitz blocks. By setting selected entries of the inverse of level-2 circulant preconditioner to zero, we get our preconditioner q ₂(T _m,n ) which is a (band) block Toeplitz matrix with (band) Toeplitz blocks. Numerical results show that our preconditioners are more efficient than circulant preconditioners.     

Preconditioners for non‐Hermitian Toeplitz systems

In this paper, we construct new ω‐circulant preconditioners for non‐Hermitian Toeplitz systems, where we allow the generating function of the sequence of Toeplitz matrices to have zeros on the unit circle. We prove that the eigenvalues of the preconditioned normal equation are clustered at 1 and that for (N, N)‐Toeplitz matrices with spectral condition number 𝒪(N^α) the corresponding PCG method requires at most 𝒪(N log² N) arithmetical operations. If the generating function of the Toeplitz sequence is a rational function then we show that our preconditioned original equation has only a fixed number of eigenvalues which are not equal to 1 such that preconditioned GMRES needs only a constant number of iteration steps independent of the dimension of the problem. Numerical tests are presented with PCG applied to the normal equation, GMRES, CGS and BICGSTAB. In particular, we apply our preconditioners to compute the stationary probability distribution vector of Markovian queuing models with batch arrival. Copyright © 2001 John Wiley & Sons, Ltd.     

Quasi‐Toeplitz splitting iteration methods for unsteady space‐fractional diffusion equations

We construct a class of quasi‐Toeplitz splitting iteration methods to solve the two‐sided unsteady space‐fractional diffusion equations with variable coefficients. By making full use of the structural characteristics of the coefficient matrix, the method only requires computational costs of O(n log n) with n denoting the number of degrees of freedom. We develop an appropriate circulant matrix to replace the Toeplitz matrix as a preconditioner. We discuss the spectral properties of the quasi‐circulant splitting preconditioned matrix. Numerical comparisons with existing approaches show that the present method is both effective and efficient when being used as matrix splitting preconditioners for Krylov subspace iteration methods.     

Block ILU Preconditioners for a Nonsymmetric Block-Tridiagonal M-Matrix

We propose block ILU (incomplete LU) factorization preconditioners for a nonsymmetric block-tridiagonal M-matrix whose computation can be done in parallel based on matrix blocks. Some theoretical properties for these block ILU factorization preconditioners are studied and then we describe how to construct them effectively for a special type of matrix. We also discuss a parallelization of the preconditioner solver step used in nonstationary iterative methods with the block ILU preconditioners. Numerical results of the right preconditioned BiCGSTAB method using the block ILU preconditioners are compared with those of the right preconditioned BiCGSTAB using a standard ILU factorization preconditioner to see how effective the block ILU preconditioners are.     

Strang‐type preconditioners applied to ordinary and neutral differential‐algebraic equations

This paper deals with boundary‐value methods (BVMs) for ordinary and neutral differential‐algebraic equations. Different from what has been done in Lei and Jin (Lecture Notes in Computer Science, vol. 1988. Springer: Berlin, 2001; 505–512), here, we directly use BVMs to discretize the equations. The discretization will lead to a nonsymmetric large‐sparse linear system, which can be solved by the GMRES method. In order to accelerate the convergence rate of GMRES method, two Strang‐type block‐circulant preconditioners are suggested: one is for ordinary differential‐algebraic equations (ODAEs), and the other is for neutral differential‐algebraic equations (NDAEs). Under some suitable conditions, it is shown that the preconditioners are invertible, the spectra of the preconditioned systems are clustered, and the solution of iteration converges very rapidly. The numerical experiments further illustrate the effectiveness of the methods. Copyright © 2011 John Wiley & Sons, Ltd.     

Circulant-block preconditioners for solving ordinary differential equations

Boundary value methods for solving ordinary differential equations require the solution of non-symmetric, large and sparse linear systems. In this paper, these systems are solved by using the generalized minimal residual (GMRES) method. A circulant-block preconditioner is proposed to speed up the convergence rate of the GMRES method. Theoretical and practical arguments are given to show that this preconditioner is more efficient than some other circulant-type preconditioners in some cases. A class of waveform relaxation methods is also proposed to solve the linear systems.     

Analysis of a novel preconditioner for a class of p‐level lower rank extracted systems

This paper proposes and studies the performance of a preconditioner suitable for solving a class of symmetric positive definite systems, Âx=b, which we call p‐level lower rank extracted systems (p‐level LRES), by the preconditioned conjugate gradient method. The study of these systems is motivated by the numerical approximation of integral equations with convolution kernels defined on arbitrary p‐dimensional domains. This is in contrast to p‐level Toeplitz systems which only apply to rectangular domains. The coefficient matrix, Â, is a principal submatrix of a p‐level Toeplitz matrix, A, and the preconditioner for the preconditioned conjugate gradient algorithm is provided in terms of the inverse of a p‐level circulant matrix constructed from the elements of A. The preconditioner is shown to yield clustering in the spectrum of the preconditioned matrix which leads to a substantial reduction in the computational cost of solving LRE systems. Copyright © 2006 John Wiley & Sons, Ltd.     

Limited memory preconditioners for symmetric indefinite problems with application to structural mechanics

This paper presents a class of limited memory preconditioners (LMP) for solving linear systems of equations with symmetric indefinite matrices and multiple right‐hand sides. These preconditioners based on limited memory quasi‐Newton formulas require a small number k of linearly independent vectors and may be used to improve an existing first‐level preconditioner. The contributions of the paper are threefold. First, we derive a formula to characterize the spectrum of the preconditioned operator. A spectral analysis of the preconditioned matrix shows that the eigenvalues are all real and that the LMP class is able to cluster at least k eigenvalues at 1. Secondly, we show that the eigenvalues of the preconditioned matrix enjoy interlacing properties with respect to the eigenvalues of the original matrix provided that the k linearly independent vectors have been prior projected onto the invariant subspaces associated with the eigenvalues of the original matrix in the open right and left half‐plane, respectively. Third, we focus on theoretical properties of the Ritz‐LMP variant, where Ritz information is used to determine the k vectors. Finally, we illustrate the numerical behaviour of the Ritz limited memory preconditioners on realistic applications in structural mechanics that require the solution of sequences of large‐scale symmetric saddle‐point systems. Numerical experiments show the relevance of the proposed preconditioner leading to a significant decrease in terms of computational operations when solving such sequences of linear systems. A saving of up to 43% in terms of computational effort is obtained on one of these applications. Copyright © 2016 John Wiley & Sons, Ltd.     

AMLI preconditioner for the p‐version of the FEM

From the literature it is known that the conjugate gradient method with domain decomposition preconditioners is one of the most efficient methods for solving systems of linear algebraic equations resulting from p‐version finite element discretizations of elliptic boundary value problems. One ingredient of such a preconditioner is a preconditioner related to the Dirichlet problems. In the case of Poisson's equation, we present a preconditioner for the Dirichlet problems which can be interpreted as the stiffness matrix K_h,k resulting from the h‐version finite element discretization of a special degenerated problem. We construct an AMLI preconditioner C_h,k for the matrix K_h,k and show that the condition number of C K_h,k is independent of the discretization parameter. This proof is based on the strengthened Cauchy inequality. The theoretical result is confirmed by numerical examples. Copyright © 2003 John Wiley & Sons, Ltd.     

Block-circulant preconditioners for systems arising from discretization of the three-dimensional convection–diffusion equation

We consider the system of equations arising from finite difference discretization of a three-dimensional convection–diffusion model problem. This system is typically nonsymmetric. The GMRES method with the Strang block-circulant preconditioner is proposed for solving this linear system. We show that our preconditioners are invertible and study the spectra of the preconditioned matrices. Numerical results are reported to illustrate the effectiveness of our methods.     

Hermite collocation solution of partial differential equations via preconditioned Krylov methods

We are concerned with the numerical solution of partial differential equations (PDEs) in two spatial dimensions discretized via Hermite collocation. To efficiently solve the resulting systems of linear algebraic equations, we choose a Krylov subspace method. We implement two such methods: Bi‐CGSTAB [1] and GMRES [2]. In addition, we utilize two different preconditioners: one based on the Gauss–Seidel method with a block red‐black ordering (RBGS); the other based upon a block incomplete LU factorization (ILU). Our results suggest that, at least in the context of Hermite collocation, the RBGS preconditioner is superior to the ILU preconditioner and that the Bi‐CGSTAB method is superior to GMRES. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:120–136, 2001     

Product-type block preconditioners for electromagnetic cavity problems

In a radar detecting system, the radar cross section (RCS) of a target is an important application in electromagnetics computation. Cavity prediction is usually required in the computation process. When wavelengths of electromagnetic field are large, cavity calculation is a challenging problem. The main aim of this paper is to employ the preconditioned GMRES method to solve linear systems arising in cavity calculation of the transverse magnetic and transverse electric problems. We develop new preconditioners based on fast transforms and Toeplitz solvers. Our numerical results show that the proposed preconditioners are quite efficient and effective, especially when the wavelength is large.     

Generalization of Strang's Preconditioner with Applications to Toeplitz Least Squares Problems

In this paper, we propose a method to generalize Strang's circulant preconditioner for arbitrary n-by-n matrices A_n. The th column of our circulant preconditioner Sn is equal to the th column of the given matrix A_n. Thus if A_n is a square Toeplitz matrix, then Sn is just the Strang circulant preconditioner. When Sn is not Hermitian, our circulant preconditioner can be defined as . This construction is similar to the forward-backward projection method used in constructing preconditioners for tomographic inversion problems in medical imaging. We show that if the matrix A_n has decaying coefficients away from the main diagonal, then is a good preconditioner for An. Comparisons of our preconditioner with other circulant-based preconditioners are carried out for some 1-D Toeplitz least squares problems: min ∥ b - Ax∥₂. Preliminary numerical results show that our preconditioner performs quite well, in comparison to other circulant preconditioners. Promising test results are also reported for a 2-D deconvolution problem arising in ground-based atmospheric imaging.     

Structured preconditioners for nonsingular matrices of block two-by-two structures

For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations. For the specific versions such as modified block Jacobi-type, modified block Gauss-Seidel-type, and modified block unsymmetric (symmetric) Gauss-Seidel-type preconditioners, we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices. Also, we show that when these structured preconditioners are employed to precondition the Krylov subspace methods such as GMRES and restarted GMRES, fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two-by-two coefficient matrices. In particular, these structured preconditioners can lead to efficient and high-quality preconditioning matrices for some typical matrices from the real-world applications.

     

Variable-step multilevel preconditioning methods,I: Self-adjoint and positive definite elliptic problems

When solving large size systems of equations by preconditioned iterative solution methods, one normally uses a fixed preconditioner which may be defined by some eigenvalue information, such as in a Chebyshev iteration method. In many problems, however, it may be more effective to use variable preconditioners, in particular when the eigenvalue information is not available. In the present paper, a recursive way of constructing variable-step of, in general, nonlinear multilevel preconditioners for selfadjoint and coercive second-order elliptic problems, discretized by the finite element method is proposed. The preconditioner is constructed recursively from the coarsest to finer and finer levels. Each preconditioning step requires only block-diagonal solvers at all levels except at every k₀, k₀ ≥ 1 level where we perform a sufficient number ν, ν ≥ 1 of GCG-type variable-step iterations that involve the use again of a variable-step preconditioning for that level. It turns out that for any sufficiently large value of k₀ and, asymptotically, for ν sufficiently large, but not too large, the method has both an optimal rate of convergence and an optimal order of computational complexity, both for two and three space dimensional problem domains. The method requires no parameter estimates and the convergence results do not depend on the regularity of the elliptic problem.     

BTTB preconditioners for BTTB least squares problems

In this paper, we consider solving the least squares problem min_x‖b-Tx‖₂ by using preconditioned conjugate gradient (PCG) methods, where T is a large rectangular matrix which consists of several square block-Toeplitz-Toeplitz-block (BTTB) matrices and b is a column vector. We propose a BTTB preconditioner to speed up the PCG method and prove that the BTTB preconditioner is a good preconditioner. We then discuss the construction of the BTTB preconditioner. Numerical examples, including image restoration problems, are given to illustrate the efficiency of our BTTB preconditioner. Numerical results show that our BTTB preconditioner is more efficient than the well-known Level-1 and Level-2 circulant preconditioners.     

A comparison of abstract versions of deflation,balancing and additive coarse grid correction preconditioners

In this paper we consider various preconditioners for the conjugate gradient (CG) method to solve large linear systems of equations with symmetric positive definite system matrix. We continue the comparison between abstract versions of the deflation, balancing and additive coarse grid correction preconditioning techniques started in (SIAM J. Numer. Anal. 2004; 42 :1631–1647; SIAM J. Sci. Comput. 2006; 27 :1742–1759). There the deflation method is compared with the abstract additive coarse grid correction preconditioner and the abstract balancing preconditioner. Here, we close the triangle between these three methods. First of all, we show that a theoretical comparison of the condition numbers of the abstract additive coarse grid correction and the condition number of the system preconditioned by the abstract balancing preconditioner is not possible. We present a counter example, for which the condition number of the abstract additive coarse grid correction preconditioned system is below the condition number of the system preconditioned with the abstract balancing preconditioner. However, if the CG method is preconditioned by the abstract balancing preconditioner and is started with a special starting vector, the asymptotic convergence behavior of the CG method can be described by the so‐called effective condition number with respect to the starting vector. We prove that this effective condition number of the system preconditioned by the abstract balancing preconditioner is less than or equal to the condition number of the system preconditioned by the abstract additive coarse grid correction method. We also provide a short proof of the relationship between the effective condition number and the convergence of CG. Moreover, we compare the A‐norm of the errors of the iterates given by the different preconditioners and establish the orthogonal invariants of all three types of preconditioners. Copyright © 2008 John Wiley & Sons, Ltd.