Many problems in scientific computing are intractable with
classical numerical techniques. These fail, for example, in the
solution of high-dimensional models due to the exponential increase
of the number of degrees of freedom.
Recently, the authors of this book and their collaborators have
developed a novel technique, called Proper Generalized
Decomposition (PGD) that has proven to be a significant step
forward. The PGD builds by means of a successive enrichment
strategy a numerical approximation of the unknown fields in a
separated form. Although first introduced and successfully
demonstrated in the context of high-dimensional problems, the PGD
allows for a completely new approach for addressing more standard
problems in science and engineering. Indeed, many challenging
problems can be efficiently cast into a multi-dimensional
framework, thus opening entirely new solution strategies in the PGD
framework. For instance, the material parameters and boundary
conditions appearing in a particular mathematical model can be
regarded as extra-coordinates of the problem in addition to the
usual coordinates such as space and time. In the PGD framework,
this enriched model is solved only once to yield a parametric
solution that includes all particular solutions for specific values
of the parameters.
The PGD has now attracted the attention of a large number of
research groups worldwide. The present text is the first available
book describing the PGD. It provides a very readable and practical
introduction that allows the reader to quickly grasp the main
features of the method.Throughout the book, the PGD is applied to
problems of increasing complexity, and the methodology is
illustrated by means of carefully selected numerical examples.
Moreover, the reader has free access to the Matlab(c) software used
to generate these examples."
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