On the eigenvalue distribution of a class of preconditioning methods

Summary A class of preconditioning methods depending on a relaxation parameter is presented for the solution of large linear systems of equationAx=b, whereA is a symmetric positive definite matrix. The methods are based on an incomplete factorization of the matrixA and include both pointwise and blockwise factorization. We study the dependence of the rate of convergence of the preconditioned conjugate gradient method on the distribution of eigenvalues ofC ^–1 A, whereC is the preconditioning matrix. We also show graphic representations of the eigenvalues and present numerical tests of the methods.     

Conjugate direction methods with variable storage

In this paper we study conjugate gradient algorithms for large optimization problems. These methods accelerate (or precondition) the conjugate gradient method by means of quasi-Newton matrices, and are designed to utilize a variable amount of storage, depending on how much information is retained in the quasi-Newton matrices. We are concerned with the behaviour of such methods on the underlying quadratic model, and in particular, with finite termination properties.     

On the preconditioning of matrices with skew-symmetric splittings

The rates of convergence of iterative methods with standard preconditioning techniques usually degrade when the skew-symmetric part S of the matrix is relatively large. In this paper, we address the issue of preconditioning matrices with such large skew-symmetric parts. The main idea of the preconditioner is to split the matrix into its symmetric and skew-symmetric parts and to invert the (shifted) skew-symmetric matrix. Successful use of the method requires the solution of a linear system with matrix I+S. An efficient method is developed using the normal equations, preconditioned by an incomplete orthogonal factorization.Numerical experiments on various systems arising in physics show that the reduction in terms of iteration count compensates for the additional work per iteration when compared to standard preconditioners.     

Chebyshev-type methods and preconditioning techniques

Recently, a Newton’s iterative method is attracting more and more attention from various fields of science and engineering. This method is generally quadratically convergent. In this paper, some Chebyshev-type methods with the third order convergence are analyzed in detail and used to compute approximate inverse preconditioners for solving the linear system Ax = b. Theoretic analysis and numerical experiments show that Chebyshev’s method is more effective than Newton’s one in the case of constructing approximate inverse preconditioners.     

On the relation between dynamic relaxation and semi-iterative matrix methods

Summary The well known relation between some matrix iterative methods for the solution of elliptic partial differential equations and time dependent parabolic equations is extended to establish a correspondence between the Chebyshev semi-iterative method and a time dependent hyperbolic equation. The method of dynamic relaxation based upon finding the transient solution to the corresponding dynamical system of equations is shown to be an exact analogue of the Chebyshev method provided the appropriate choice of inertia and damping coefficients is made. The possibility of developing further techniques based upon different physical models and leading to improved convergence is noted.     

On regularization methods based on dynamic programming techniques

In this article, we investigate the connection between regularization theory for inverse problems and dynamic programming theory. This is done by developing two new regularization methods, based on dynamic programming techniques. The aim of these methods is to obtain stable approximations to the solution of linear inverse ill-posed problems. We follow two different approaches and derive a continuous and a discrete regularization method. Regularization properties for both methods are proved as well as rates of convergence. A numerical benchmark problem concerning integral operators with convolution kernels is used to illustrate the theoretical results.     

Algebraic multilevel preconditioning methods. I

Summary A recursive way of constructing preconditioning matrices for the stiffness matrix in the discretization of selfadjoint second order elliptic boundary value problems is proposed. It is based on a sequence of nested finite element spaces with the usual nodal basis functions. Using a nodeordering corresponding to the nested meshes, the finite element stiffness matrix is recursively split up into two-level block structures and is factored approximately in such a way that any successive Schur complement is replaced (approximated) by a matrix defined recursively and thereform only implicitely given. To solve a system with this matrix we need to perform a fixed number (v) of iterations on the preceding level using as an iteration matrix the preconditioning matrix already defined on that level. It is shown that by a proper choice of iteration parameters it suffices to use \left( {1 - \gamma ^2 } \right)^{ - \tfrac{1}{2}} $$ " align="middle" border="0"> iterations for the so constructedv-foldV-cycle (wherev=2 corresponds to aW-cycle) preconditioning matrices to be spectrally equivalent to the stiffness matrix. The conditions involve only the constant in the strengthened C.-B.-S. inequality for the corresponding two-level hierarchical basis function spaces and are therefore independent of the regularity of the solution for instance. If we use successive uniform refinements of the meshes the method is of optimal order of computational complexity, if . Under reasonable assumptions of the finite element mesh, the condition numbers turn out to be so small that there are in practice few reasons to use an accelerated iterative method like the conjugate gradient method, for instance.Dedicated to the memory of Peter HenriciThe research of the second author reported here was supported in part by the Committee of Science, Bulgaria, under Grant No. 55/26.03.87     

On the convergence of stochastic dual dynamic programming and related methods

We discuss the almost-sure convergence of a broad class of sampling algorithms for multistage stochastic linear programs. We provide a convergence proof based on the finiteness of the set of distinct cut coefficients. This differs from existing published proofs in that it does not require a restrictive assumption.     

On approximated ILU and UGS preconditioning methods for linearized discretized steady incompressible Navier-Stokes equations

When the artificial compressibility method in conjunction with high-order upwind compact finite difference schemes is employed to discretize the steady-state incompressible Navier-Stokes equations, in each pseudo-time step we need to solve a structured system of linear equations approximately by, for example, a Krylov subspace method such as the preconditioned GMRES. In this paper, based on the special structure and concrete property of the linear system we construct a structured preconditioner for its coefficient matrix and estimate eigenvalue bounds of the correspondingly preconditioned matrix. Numerical examples are given to illustrate the effectiveness of the proposed preconditioning methods.     

Diagonally compensated reduction and related preconditioning methods

When solving linear algebraic equations with large and sparse coefficient matrices, arising, for instance, from the discretization of partial differential equations, it is quite common to use preconditioning to accelerate the convergence of a basic iterative scheme. Incomplete factorizations and sparse approximate inverses can provide efficient preconditioning methods but their existence and convergence theory is based mostly on M-matrices (H-matrices). In some application areas, however, the arising coefficient matrices are not H-matrices. This is the case, for instance, when higher-order finite element approximations are used, which is typical for structural mechanics problems. We show that modification of a symmetric, positive definite matrix by reduction of positive offdiagonal entries and diagonal compensation of them leads to an M-matrix. This diagonally compensated reduction can take place in the whole matrix or only at the current pivot block in a recursive incomplete factorization method. Applications for constructing preconditioning matrices for finite element matrices are described.     

A modification of conjugate direction methods

A modification of well-known conjugate direction methods is proposed and examined. The modified methods are more stable with respect to round-off error accumulation and are applicable to ill-conditioned and ill-posed problems. The advantages of the modification are illustrated by numerical experiments.     

Extrapolated alternating direction implicit iterative methods

This paper describes a new generalised (extrapolated) A.D.I. method for the solution of Laplace's equation. This method uses (i) a fixed acceleration parameter and (ii) the set of acceleration parameters of Douglas. The theory is applied to the 2-dimensional case and optimum numerical results are obtained.     

Biconjugate direction methods in Krylov subspaces

The paper addresses the orthogonal and variational properties of a family of iterative algorithms in Krylov subspaces for solving the systems of linear algebraic equations (SLAE) with sparse nonsymmetric matrices. There are proposed and studied a biconjugate residual method, squared biconjugate residual method, and stabilized conjugate residual method. Some results of numerical experiments are given for a series of model problems as well.     

On the stability of alternating‐direction explicit methods for advection‐diffusion equations

Alternating‐Direction Explicit (A.D.E.) finite‐difference methods make use of two approximations that are implemented for computations proceeding in alternating directions, e.g., from left to right and from right to left, with each approximation being explicit in its respective direction of computation. Stable A.D.E. schemes for solving the linear parabolic partial differential equations that model heat diffusion are well‐known, as are stable A.D.E. schemes for solving the first‐order equations of fluid advection. Several of these are combined here to derive A.D.E. schemes for solving time‐dependent advection‐diffusion equations, and their stability characteristics are discussed. In each case, it is found that it is the advection term that limits the stability of the scheme. The most stable of the combinations presented comprises an unconditionally stable approximation for computations carried out in the direction of advection of the system, from left to right in this case, and a conditionally stable approximation for computations proceeding in the opposite direction. To illustrate the application of the methods and verify the stability conditions, they are applied to some quasi‐linear one‐dimensional advection‐diffusion problems. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

On the shallow water equations with periodicity in one direction

The linearized 2D shallow water equations, supplemented with suitable boundary conditions in one direction and periodicity in the other direction are considered. The well-posedness of this mixed boundary value problem is proven, using the linear semi-group theory. The well-posedness of the totally periodic problem is also proven.     

A preconditioning strategy for boundary element Galerkin methods

The Dirichlet and Neumann problems for the Laplacian are reformulated in the usual way as boundary integral equations of the first kind with symmetric kernels. These integral equations are solved using Galerkin's method with piecewise-constant and piecewise-linear boundary elements, respectively. In both cases, the stiffness matrix is symmetric and positive-definite, and has a condition number of order N, the number of degrees of freedom. By contrast, the condition number of the product of the two stiffness matrices is bounded independently of N. Hence, we can use the Neumann stiffness matrix to precondition the Dirichlet stiffness matrix, and vice versa. © 1997 John Wiley & Sons, Inc.