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The problem of solving large, sparse, linear systems of algebraic
equations is vital in scientific computing, even for applications
originating from quite different fields. A Survey of Preconditioned
Iterative Methods presents an up to date overview of iterative
methods for numerical solution of such systems. Typically, the
methods considered are well suited for the kind of systems arising
from the discretization of partial differential equations. The
focus of this presentation is on the family of Krylov subspace
solvers, of which the Conjugate Gradient algorithm is a typical
example. In addition to an introduction to the basic principles of
such methods, a large number of specific algorithms for symmetric
and nonsymmetric problems are discussed. When solving linear
systems by iteration, a preconditioner is usually introduced in
order to speed up convergence. In many cases, the selection of a
proper preconditioner is crucial to the resulting computational
performance. For this reason, this book pays special attention to
different preconditioning strategies. Although aimed at a wide
audience, the presentation assumes that the reader has basic
knowledge of linear algebra, and to some extent, of partial
differential equations. The comprehensive bibliography in this
survey is provides an entry point to the enormous amount of
published research in the field of iterative methods.
When researchers gather around lunch tables, at conferences, or in
bars, there are some topics that are more or less compulsory. The
discussions are about the ho- less management of the university or
the lab where they are working, the lack of funding for important
research, politicians' inability to grasp the potential of a p-
ticularly promising ?eld, and the endless series of committees that
seem to produce very little progress. It is common to meet
excellent researchers claiming that they have almost no time to do
research because writing applications, lecturing, and - tending to
committee work seem to take most of their time. Very few ever come
into a position to do something about it. With Simula we have this
chance. We were handed a considerable annual grant and more or less
left to ourselves to do whatever we thought would produce the best
possible results. We wanted to create a place where researchers
could have the time and conditions necessary to re?ect over
dif?cult problems, uninterrupted by mundane dif?culties; where
doctoral students could be properly supervised and learn the craft
of research in a well-organized and professional manner; and where
entrepreneurs could ?nd professional support in developing their
research-based - plications and innovations.
Since the dawn of computing, the quest for a better understanding
of Nature has been a driving force for technological development.
Groundbreaking achievements by great scientists have paved the way
from the abacus to the supercomputing power of today. When trying
to replicate Nature in the computer's silicon test tube, there is
need for precise and computable process descriptions. The scienti?c
?elds of Ma- ematics and Physics provide a powerful vehicle for
such descriptions in terms of Partial Differential Equations
(PDEs). Formulated as such equations, physical laws can become
subject to computational and analytical studies. In the
computational setting, the equations can be discreti ed for
ef?cient solution on a computer, leading to valuable tools for
simulation of natural and man-made processes. Numerical so- tion of
PDE-based mathematical models has been an important research topic
over centuries, and will remain so for centuries to come. In the
context of computer-based simulations, the quality of the computed
results is directly connected to the model's complexity and the
number of data points used for the computations. Therefore,
computational scientists tend to ?ll even the largest and most
powerful computers they can get access to, either by increasing the
si e of the data sets, or by introducing new model terms that make
the simulations more realistic, or a combination of both. Today,
many important simulation problems can not be solved by one single
computer, but calls for parallel computing.
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