A framework for polynomial preconditioners based on fast transforms II: PDE applications

The solution of systems of equations arising from systems of time-dependent partial differential equations (PDEs) is considered. Primarily, first-order PDEs are studied, but second-order derivatives are also accounted for. The discretization is performed using a general finite difference stencil in space and an implicit method in time. The systems of equations are solved by a preconditioned Krylov subspace method. The preconditioners exploit optimal and superoptimal approximations by low-degree polynomials in a normal basis matrix, associated with a fast trigonometric transform. Numerical experiments for high-order accurate discretizations are presented. The results show that preconditioners based on fast transforms yield efficient solution algorithms, even for large quotients between the time and space steps. Utilizing a spatial grid ratio less than one, the arithmetic work per grid point is bounded by a constant as the number of grid points increases. This research was supported by the Swedish National Board for Industrial and Technical Development (NUTEK) and by the U.S. National Science Foundation under grant ASC-8958544.     

Preconditioners for nonsymmetric block toeplitz-like-plus-diagonal linear systems

Summary. We consider the system of linear equations Lu=f, where L is a nonsymmetric block Toeplitz-like-plus-diagonal matrix, which arises from the Sinc-Galerkin discretization of differential equations. Our main contribution is to construct effective preconditioners for this structured coefficient matrix and to derive tight bounds for eigenvalues of the preconditioned matrices. Moreover, we use numerical examples to show that the new preconditioners, when applied to the preconditioned GMRES method, are efficient for solving the system of linear equations. Mathematics Subject Classification (2000):65F10, 65F15, 65T10Research subsidized by The Special Funds for Major State Basic Research Projects G1999032803Research supported in part by RGC Grant Nos. 7132/00P and 7130/02P, and HKU CRCG Grant Nos 10203501, 10203907 and 10203408     

A robust structured preconditioner for time-harmonic parabolic optimal control problems

We consider the iterative solution of optimal control problems constrained by the time-harmonic parabolic equations. Due to the time-harmonic property of the control equations, a suitable discretization of the corresponding optimality systems leads to a large complex linear system with special two-by-two block matrix of saddle point form. For this algebraic system, an efficient preconditioner is constructed, which results in a fast Krylov subspace solver, that is robust with respect to the mesh size, frequency, and regularization parameters. Furthermore, the implementation is straightforward and the computational complexity is of optimal order, linear in the number of degrees of freedom. We show that the eigenvalue distribution of the corresponding preconditioned matrix leads to a condition number bounded above by 2. Numerical experiments confirming the theoretical derivations are presented, including comparisons with some other existing preconditioners.     

GENUINE-OPTIMAL CIRCULANT PRECONDITIONERS FOR WIENER-HOPF EQUATIONS

1. IntroductionWienerHopf equations are integral equations defined on the haif line:where rr > 0, a(.) C L1(ro and g(.) E L2(at). Here R = (--oo,oo) and ty [0,oo). Inou-r discussions, we assume that a(.) is colljugate symmetric, i.e. a(--t) = a(t). WienerHop f equations arise in a variety of practical aPplicatiolls in mathematics and ellgineering, forinstance, in the linear prediction problems fOr stationary stochastic processes [8, pp.145--146],diffuSion problems and scattering problems […     

Schwarz preconditioned CG algorithm for the mortar finite element

We propose a simple and effective hybrid (multiplicative) Schwarz precondtioner for solving systems of algebraic equations resulting from the mortar finite element discretization of second order elliptic problems on nonmatching meshes. The preconditioner is embedded in a variant of the classical preconditioned conjugate gradient (PCG) for an effective implementation reducing the cost of computing the matrix-vector multiplication in each iteration of the PCG. In fact, it serves as a framework for effective implementation of a class of hybrid Schwarz preconditioners. The preconditioners of this class are based on solving a sequence of non-overlapping local subproblems exactly, and the coarse problems either exactly or inexactly (approximately). The classical PCG algorithm is reformulated in order to make reuse of the results of matrix-vector multiplications that are already available from the preconditioning step resulting in an algorithm which is cost effective. An analysis of the proposed preconditioner, with numerical results, showing scalability with respect to the number of subdomains, and a convergence which is independent of the jumps of the coefficients are given.     

Block two-stage preconditioners

Linear systems of the form Ax = b, where the matrix A is symmetric and positive definite, often arise from the discretization of elliptic partial differential equations. A very successful method for solving these linear systems is the preconditioned conjugate gradient method. In this paper, we study parallel preconditioners for the conjugate gradient method based on the block two-stage iterative methods. Sufficient conditions for the validity of these preconditioners are given. Computational results of these preconditioned conjugate gradient methods on two parallel computing systems are presented.     

Equivalent operator preconditioning for elliptic problems with nonhomogeneous mixed boundary conditions

The numerical solution of linear elliptic partial differential equations often involves finite element discretization, where the discretized system is usually solved by some conjugate gradient method. The crucial point in the solution of the obtained discretized system is a reliable preconditioning, that is to keep the condition number of the systems under control, no matter how the mesh parameter is chosen. The PCG method is applied to solving convection-diffusion equations with nonhomogeneous mixed boundary conditions. Using the approach of equivalent and compact-equivalent operators in Hilbert space, it is shown that for a wide class of elliptic problems the superlinear convergence of the obtained preconditioned CGM is mesh independent under FEM discretization.     

Circulant integral operators as preconditioners for Wiener-Hopf equations

In this paper, we study the solutions of finite-section Wiener-Hopf equations by the preconditioned conjugate gradient method. Our main aim is to give an easy and general scheme of constructing good circulant integral operators as preconditioners for such equations. The circulant integral operators are constructed from sequences of conjugate symmetric functions {C }. Letk(t) denote the kernel function of the Wiener-Hopf equation and be its Fourier transform. We prove that for sufficiently large if {C } is uniformly bounded on the real lineR and the convolution product of the Fourier transform ofC with uniformly onR, then the circulant preconditioned Wiener-Hopf operator will have a clustered spectrum. It follows that the conjugate gradient method, when applied to solving the preconditioned operator equation, converges superlinearly. Several circulant integral operators possessing the clustering and fast convergence properties are constructed explicitly. Numerical examples are also given to demonstrate the performance of different circulant integral operators as preconditioners for Wiener-Hopf operators.Research supported in part by HKRGC grant no. 221600070.     

A variant of block incomplete factorization preconditioners for a symmetricH-matrix

We propose a variant of parallel block incomplete factorization preconditioners for a symmetric block-tridiagonalH-matrix. Theoretical properties of these block preconditioners are compared with those of block incomplete factorization preconditioners for the corresponding comparison matrix. Numerical results of the preconditioned CG(PCG) method using these block preconditioners are compared with those of PCG using other types of block incomplete factorization preconditioners. Lastly, parallel computations of the block incomplete factorization preconditioners are carried out on the Cray C90.     

COMBINATIVE PRECONDITIONERS OF MODIFIED INCOMPLETE CHOLESKY FACTORIZATION AND SHERMAN-MORRISON-WOODBURY UPDATE FOR SELF-ADJOINT ELLIPTIC DIRICHLET-PERIODIC BOUNDARY VALUE PROBLEMS

For the system of linear equations arising from discretization of the second-order self-adjoint elliptic Dirichlet-periodic boundary value problems,by making use of the specialstructure of the coefficient matrix we present a class of combinative preconditioners whichare technical combinations of modified incomplete Cholesky factorizations and Sherman-Morrison-Woodbury update.Theoretical analyses show that the condition numbers of thepreconditioned matrices can be reduced to(?)(h~(-1)),one order smaller than the conditionnumber(?)(h~(-2))of the original matrix.Numerical implementations show that the resultingpreconditioned conjugate gradient methods are feasible,robust and efficient for solving thisclass of linear systems.     

Indefinitely preconditioned conjugate gradient method for large sparse equality and inequality constrained quadratic problems

This paper is concerned with the numerical solution of a symmetric indefinite system which is a generalization of the Karush–Kuhn–Tucker system. Following the recent approach of Luk?an and Vl?ek, we propose to solve this system by a preconditioned conjugate gradient (PCG) algorithm and we devise two indefinite preconditioners with good theoretical properties. In particular, for one of these preconditioners, the finite termination property of the PCG method is stated. The PCG method combined with a parallel version of these preconditioners is used as inner solver within an inexact Interior‐Point (IP) method for the solution of large and sparse quadratic programs. The numerical results obtained by a parallel code implementing the IP method on distributed memory multiprocessor systems enable us to confirm the effectiveness of the proposed approach for problems with special structure in the constraint matrix and in the objective function. Copyright © 2002 John Wiley & Sons, Ltd.     

BANDED TOEPLITZ PRECONDITIONERS FOR TOEPLITZ MATRICES FROM SINC METHODS

We give general expressions, analyze algebraic properties and derive eigenvalue bounds for a sequence of Toeplitz matrices associated with the sinc discretizations of various orders of differential operators. We demonstrate that these Toeplitz matrices can be satisfactorily preconditioned by certain banded Toeplitz matrices through showing that the spectra of the preconditioned matrices are uniformly bounded. In particular, we also derive eigenvalue bounds for the banded Toeplitz preconditioners. These results are elementary in constructing high-quality structured preconditioners for the systems of linear equations arising from the sinc discretizations of ordinary and partial differential equations, and are useful in analyzing algebraic properties and deriving eigenvalue bounds for the corresponding preconditioned matrices. Numerical examples are given to show effectiveness of the banded Toeplitz preconditioners.     

Modified block preconditioners for the discretized time-harmonic Maxwell equations in mixed form

In this paper, building on the previous work by Greif and Schötzau [Preconditioners for the discretized time-harmonic Maxwell equations in mixed form, Numer. Linear Algebra Appl. 14 (2007) 281–297] and Benzi and Olshanskii [An augmented lagrangian-based approach to the Oseen problem, SIAM J. Sci. Comput. 28 (2006) 2095–2113], we present the improved preconditioning techniques for the iterative solution of the saddle point linear systems, which arise from the finite element discretization of the mixed formulation of the time-harmonic Maxwell equations. The modified block diagonal and triangular preconditioners considered are based on augmentation with using the symmetric nonsingular weighted matrix. We discuss the spectral properties of the preconditioned matrix in detail and generalize the results of the above-mentioned paper by Greif and Schötzau. Numerical experiments are given to demonstrate the efficiency of the presented preconditioners.     

BCCB preconditioners for systems of BVM‐based numerical integrators

Boundary value methods (BVMs) for 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 block‐circulant preconditioner with circulant blocks (BCCB preconditioner) is proposed to speed up the convergence rate of the GMRES method. The BCCB preconditioner is shown to be invertible when the BVM is A_k1,k2‐stable. The spectrum of the preconditioned matrix is clustered and therefore, the preconditioned GMRES method converges fast. Moreover, the operation cost in each iteration of the preconditioned GMRES method by using our BCCB preconditioner is less than that required by using block‐circulant preconditioners proposed earlier. In numerical experiments, we compare the number of iterations of various preconditioners. Copyright © 2003 John Wiley & Sons, Ltd.     

On the solution of indefinite systems arising in nonlinear programming problems

This work is concerned with the convergence properties and the numerical analysis of the preconditioned conjugate gradient (PCG) method applied to the solution of indefinite linear systems arising in nonlinear optimization. Our approach is based on the choice of quasidefinite preconditioners and of a suitable factorization routine. Some theoretical and numerical results about these preconditioners are obtained. Furthermore, we show the behaviour of the PCG method for different formulations of the indefinite system and we compare the effectiveness of the proposed variants. Copyright © 2007 John Wiley & Sons, Ltd.     

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.

     

Applying approximate LU-factorizations as preconditioners in eight iterative methods for solving systems of linear algebraic equations

Many problems arising in different fields of science and engineering can be reduced, by applying some appropriate discretization, either to a system of linear algebraic equations or to a sequence of such systems. The solution of a system of linear algebraic equations is very often the most time-consuming part of the computational process during the treatment of the original problem, because these systems can be very large (containing up to many millions of equations). It is, therefore, important to select fast, robust and reliable methods for their solution, also in the case where fast modern computers are available. Since the coefficient matrices of the systems are normally sparse (i.e. most of their elements are zeros), the first requirement is to efficiently exploit the sparsity. However, this is normally not sufficient when the systems are very large. The computation of preconditioners based on approximate LU-factorizations and their use in the efforts to increase further the efficiency of the calculations will be discussed in this paper. Computational experiments based on comprehensive comparisons of many numerical results that are obtained by using ten well-known methods for solving systems of linear algebraic equations (the direct Gaussian elimination and nine iterative methods) will be reported. Most of the considered methods are preconditioned Krylov subspace algorithms.     

Local refinement techniques for elliptic problems on cell-centered grids; II. Optimal order two-grid iterative methods

Two preconditioning techniques for solving difference equations arising in finite difference approximation of elliptic problems on cell-centered grids are studied. It is proven that the BEPS and the FAC preconditioners are spectrally equivalent to the corresponding finite difference schemes, including a nonsymmetric one, which is of higher-order accuracy. Numerical experiments that demonstrate the fast convergence of the preconditioned iterative methods (CG and GCG-LS in the nonsymmetric case) are presented.     

Parallel block preconditioning based on SSOR and MILU

Two kinds of parallel preconditioners for the solution of large sparse linear systems which arise from the 2-D 5-point finite difference discretization of a convection-diffusion equation are introduced. The preconditioners are based on the SSOR or MILU preconditioners and can be implemented on parallel computers with distributed memories. One is the block preconditioner, in which the interface components of the coefficient matrix between blocks are ignored to attain parallelism in the forward-backward substitutions. The other is the modified block preconditioner, in which the block preconditioner is modified by taking the interface components into account. The effect of these preconditioners on the convergence of preconditioned iterative methods and timing results on the parallel computer (Cenju) are presented.