A parallel nearly implicit time-stepping scheme

Across-the-space parallelism still remains the most mature, convenient and natural way to parallelize large scale problems. One of the major problems here is that implicit time stepping is often difficult to parallelize due to the structure of the system. Approximate implicit schemes have been suggested to circumvent the problem. These schemes have attractive stability properties and they are also very well parallelizable.\ud The purpose of this article is to give an overall assessment of the parallelism of the method

Computational science curriculum in Utrecht

In 1993 Utrecht University has started a curriculum in Computational Science, starting at the undergraduate level and leading to the Dutch `Doctorandus' degree (wich is more or less comparable to the Master's degree). The curriculum has been st up as a joint collaboration between the Departments of Mathematics & Computer Science, and Physics. It aims at a complete and self-contained educational program that should fulll society's growing demand for scientic computing, and it does so by trying to make students familiar with computational models (physics), applied mathematics (with emphasis on numerical analysis), and computer possibilities (computer science). In our presentation we will discuss the ideas behind this new study, the perspectives for students with respect to carreer, and we will report on our experiences during the rst two years of existence of the new curriculum

Parallel Iterative Solution Methods for Linear Systems arising from Discretized PDE's

In these notes we will present an overview of a number of related iterative methods for the solution of linear systems of equations. These methods are so-called Krylov projection type methods and the include popular methods as Conjugate Gradients, Bi-Conjugate Gradients, CGST Bi-CGSTAB, QMR, LSQR and GMRES. We will show how these methods can be derived from simple basic iteration formulas. We will not give convergence proofs, but we will refer for these, as far as available, to litterature. Iterative methods are often used in combination with so-called preconditioning operators (approximations for the inverses of the operator of the system to be solved). Since these preconditions are not essential in the derivation of the iterative methods, we will not give much attention to them in these notes. However, in most of the actual iteration schemes, we have included them in order to facilitate the use of these schemes in actual computations. For the application of the iterative schemes one usually thinks of linear sparse systems, e.g., like those arising in the finite element or finite difference approximatious of (systems of) partial differential equations. However, the structure of the operators plays no explicit role in any of these schemes, and these schemes might also successfully be used to solve certain large dense linear systems. Depending on the situation that might be attractive in terms of numbers of floating point operations. It will turn out that all of the iterative are parallelizable in a straight forward manner. However, especially for computers with a memory hierarchy (i.e. like cache or vector registers), and for distributed memory computers, the performance can often be improved significantly through rescheduling of the operations. We will discuss parallel implementations, and occasionally we will report on experimental findings

Gedachten over de mathematisering van de samenleving gezien 'vanuit de Wiskunde'

Dit essay is geschreven, vanuit een wiskundige invalshoek, op verzoek van de OCV. Het moet dienen als voorbereiding op een advies aan de Minister van OCenW over hoe in de wetenschap kan worden ingegaan op de mathematisering van de samenleving. De OCV gaf hiervoor een aantal opties aan: 1. Het vergroten van de toepassingsgerichtheid van de opleiding. 2. Het vergroten van de toepassingsgerichtheid van het onderzoek aan wiskunde-faculteiten 3. Stimulering van wiskundig onderzoek in de betreende "afnemende" vakgebieden" 4. Stimulering van samenwerkingsprojecten tussen onderzoeksgroepen in de wiskunde en andere vakgebieden Ik heb geprobeerd om in een drietal paragrafen een visie te geven op plaats en toekomstperspectief van de wiskunde in het algemeen en in Nederland in het bijzonder. Op grond van de beschreven overwegingen worden, aan het eind van de paragrafen 2 en 3, pre-adviezen geformuleerd. Bij deze overwegingen en pre-adviezen hebben de door de OCV genoemde opties als leidraad gefungeerd, maar ik heb me er niet toe beperkt, met name omdat ik het nodig vond de wiskunde als ondeelbaar geheel te blijven beschouwen en daarvan niet een enkel, zij het maatschappelijk belangrijk, aspect nadrukkelijk te isoleren. De adviezen strekken zich dus uit over de beoefening van de wiskunde als geheel maar beogen wel om de link met de samenleving en het bedrijfsleven te versterken. In hoeverre deze adviezen budget-neutraal kunnen worden uitgevoerd weet ik niet, dat hangt mede af van de grootte van het beoogde eect

An iterative solution method for solving f(A)x = b, using Krylov subspace information obtained for the symmetric positive definite matrix A

AbstractThe conjugate gradients method generates successive approximations xi for the solution of the linear system Ax = b, where A is symmetric positive definite and usually sparse. It will be shown how intermediate information obtained by the conjugate gradients (cg) algorithm (or by the closely related Lanczos algorithm) can be used to solve f(A)x = b iteratively in an efficient way, for suitable functions f. The special case f(A) = A2 is discussed in particular. We also consider the problem of solving Ax = b for different right-hand sides b. A variant on a well-known algorithm for that problem is proposed, which does not seem to suffer from the usual loss of orthogonality in the standard cg and Lanczos algorithms

Closer to the solutions: iterative linear solvers

The solution of dense linear systems received much attention after the second world war, and by the end of the sixties, most of the problems associated with it had been solved. For a long time, Wilkinson's \The Algebraic Eigenvalue Problem" [107], other than the title suggests, became also the standard textbook for the solution of linear systems. When it became clear that partial dierential equations could be solved numerically, to a level of accuracy that was of interest for application areas (such as reservoir engineering, and reactor diusion modeling), there was a strong need for the fast solution of the discretized systems, and iterative methods became popular for these problems

Parallel preconditioning for sparse linear equations

A popular class of preconditioners is known as incomplete factorizations. They can be thought of as approximating the exact LU factorization of a given matrix A (e.g. computed via Gaussian elimination) by disallowing certain ll-ins. As opposed to other PDE-based preconditioners such asmultigrid and domain decomposition, this class of preconditioners are primarily algebraic in nature and can in principle be applied to any sparse matrices. In this paper we will discuss some new viewpoints for the construction of eective preconditioners. In particular, we will discuss parallelization aspects, including re-ordering, series expansion and domain decomposition techniques. Generally, this class of preconditioner does not possess a high degree of parallelism in its original form. Re-ordering and approximations by truncating certain series expansion will increase the parallelism, but usually with a deterioration in convergence rate. Domain decomposition oers a compromise

Approximate and Incomplete Factorizations

In this chapter, we give a brief overview of a particular class of preconditioners known as incomplete factorizations. They can be thought of as approximating the exact LU factorization of a given matrix A (e.g. computed via Gaussian elimination) by disallowing certain ll-ins. As opposed to other PDE-based preconditioners such as multigrid and domain decomposition, this class of preconditioners are primarily algebraic in nature and can in principle be applied to any sparse matrices. When applied to PDE problems, they are usually not optimal in the sense that the condition number of the preconditioned system will grow as the mesh size h is reduced, although usually at a slower rate than for the unpreconditioned system. On the other hand, they are often quite robust with respect to other more algebraic features of the problem such as rough and anisotropic coecients and strong convection terms. We will describe the basic ILU and (modied) MILU preconditioners. Then we will review brie y several variants: more lls, relaxed ILU, shifted ILU, ILQ, as well as block and multilevel variants. We will also touch on a related class of approximate factorization methods which arise more directly from approximating a partial dierential operator by a product of simpler operators. Finally, we will discuss parallelization aspects, including re-ordering, series expansion and domain decomposition techniques. Generally, this class of preconditioner does not possess a high degree of parallelism in its original form. Re-ordering and approximations by truncating certain series expansion will increase the parallelism, but usually with a deterioration in convergence rate. Domain decomposition oers a compromise

Alternatives to the Rayleigh quotient for the quadratic eigenvalue problem

We consider the quadratic eigenvalue problem a²Ax + aBx + Cx = 0. Suppose that u is an approximation to an eigenvector x (for instance obtained by a subspace method), and that we want to determine an approximation to the corresponding eigenvalue a. The usual approach is to impose the Galerkin condition r(ø, u) = (ø²A + øB + C)u | u from which it follows that ø must be one of the two solutions to the quadratic equation (u*Au)ø² + (u*Bu)ø + (u*Cu) = 0. An unnatural aspect is that if u = x, the second solution has in general no meaning. When u is not very accurate, it may not be clear which solution is the best. Moreover, when the discriminant of the equation is small, the solutions may be very sensitive to perturbations in u. In this paper we therefore examine alternative approximations to a. We compare the approaches theoretically and by numerical experiments. The methods are extended to approximations from subspaces and to the polynomial eigenvalue problem

Differences in the effects of rouding errors in Krylov solvers for symmetric indefinite linear systems

The three­term Lanczos process for a symmetric matrix leads to bases for Krylov subspaces of increasing dimension. The Lanczos basis, together with the recurrence coe#cients, can be used for the solution of symmetric indefinite linear systems, by solving a reduced system in one way or another. This leads to well­known methods: MINRES (minimal residual), GMRES (generalized minimal residual), and SYMMLQ (symmetric LQ). We will discuss in what way and to what extent these approaches di#er in their sensitivity to rounding errors. In our analysis we will assume that the Lanczos basis is generated in exactly the same way for the di#erent methods, and we will not consider the errors in the Lanczos process itself. We will show that the method of solution may lead, under certain circumstances, to large additional errors, which are not corrected by continuing the iteration process. Our findings are supported and illustrated by numerical examples