H.A. van der Vorst

Numerical solution of large, sparse linear algebraic systems arising from tomographic problems

In this chapter we will digress on two classes of methods for solving the large sparse matrix systems arising from tomographic problems, viz. ART-like methods and projection methods.1 The former class has been in use for several decades, the latter is more recent.

A Petrov-Galerkin type method for solving Axk=b, where A is symmetric complex

Discretization of steady-state eddy-current equations may lead to linear system Ax=b in which the complex matrix A is not Hermitian, but may be chosen symmetric. In the positive definite Hermitian case, an iterative algorithm for solving this system can be defined. The residual vectors can be made mutually orthogonal by means of a two-term recursion relation which leads to the well-known conjugate gradient (CG) method. The proposed method is illustrated by comparing it with other methods for some eddy current examples. >

The performance of FORTRAN implementations for preconditioned conjugate gradients on vector computers

Abstract We will consider in detail the performance of FORTRAN implementations for the conjugate gradient algorithm, for the solution of large linear systems, on a number of well-known vector computers: CRAY-1, CRAY X-MP and CYBER 205. Lower bounds on the CPU-times, required for separate parts of the algorithm, are presented and these are compared to the actually observed CPU-times. It appears that these lower bounds are reasonably sharp.

The convergence behavior of ritz values in the presence of close eigenvalues

Abstract A number of results for Ritz values are presented. These are used to study the local effects in the convergence behavior of Ritz values corresponding to a pair of close eigenvalues in the spectrum. The local effects that are typical for such a situation are illustrated by numerical examples.

Conjugate gradient type methods and preconditioning

In this paper we consider various iterative methods for the numerical solution of very large, sparse linear systems of equations, which arise in the discretization of partial differential equations. As the performance of the methods generally depends on the characteristics of the problems to be solved, a judicious choice between the methods will require knowledge about the system. The aim of this paper is to review the properties of the various iteration methods, in order to assist the user in making a deliberate selection.

Solving 3D block bidiagonal linear systems on vector computers

Abstract Standard 7-point finite-difference discretization of second-order PDEs over a rectangular grid over a 3-dimensional block leads, with the usual lexicograpical ordering of the gridpoints, to block tridiagonal linear systems. In many popular iterative methods for the solution of these systems, triangular systems which have a block bidiagonal structure have to be solved. This is often recognized to be the major bottleneck, on vector computers, with respect to the computational speed, when carried out in a straightforward manner. In this paper we will discuss different techniques for the vectorization of the solution of 3D-block bidiagonal systems. We will report on actually observed performances for the ICCG algorithm, for which these bidiagonal systems have the reputation to spoil the overall performance, on some computers. The potentially most powerful of the vectorization techniques leads to long vector operations, at the cost, however, of strides and indirect addressing. Since the CYBER 205 is generally believed to stay behind in performance under such circumstances, we have chosen this machine to show in detail how these vectorization techniques can be implemented with almost equal performance as in the same contiguous vector case. Our methods are directly applicable to the ETA-10 family of supercomputers and may be adapted to other vector computers as well.

Practical aspects of parallel scientific computing

Abstract In this paper we study the parallel solution of a very simple problem, namely the solution of a lower bidiagonal system. It will appear that this problem exhibits many aspects of parallel processing and it has the advantage that the scene is not obscured by complicated algorithmic aspects. The following aspects of the parallel solution methods will be regarded: numerical stability, complexity, data organization, implementation and performance.

Parallel solution of bidiagonal systems coming from discretised PDEs

Standard algorithms for the solution of bidiagonal linear systems, which plays an important role in the numerical solution of discretized partial differential equations, are not very well suited for parallel or vector computers, and they often constitute a severe bottleneck on such machines. A number of parallel/vectorizable algorithms are considered. Performance limiting aspects of different (existing) architectures are considered, and it is shown that for many modern architectures suitable algorithms are presently available. >

Iterative Solution Methods for Large, Sparse Systems of Linear Equations Arising from Tomographic Image Reconstruction

The problem of reconstructing an object from its projections can be solved using two different approaches: analytic methods and algebraic methods. Analytic methods, based on integral transforms such as the inverse Radon transform, are considerably more efficient than algebraic methods. In many practical cases, however, such as limited angle tomography or problems where refraction plays an important role, algebraic techniques have to be used.