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Über dieses Buch

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© software used to generate these examples.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
This chapter summarizes several recurrent issues related to efficient numerical simulations of problems encountered in engineering sciences. In order to alleviate such issues, model reduction techniques constitute an appealing alternative to standard discretization techniques. First, reduction techniques based on Proper Orthogonal Decompositions are revisited. Their use is illustrated and discussed, suggesting the interest of a priori separated representations which are at the heart of the Proper Generalized Decomposition (PGD). The main ideas behind the PGD are described, underlying its potential for addressing standard computational mechanics models in a non-standard way, within a new computational engineering paradigm. The chapter ends with a brief overview of some recent PGD applications in different areas, proving the potentiality of this novel technique.
Francisco Chinesta, Roland Keunings, Adrien Leygue

Chapter 2. PGD Solution of the Poisson Equation

Abstract
This chapter describes the main features of the PGD technique, in particular the one related to the construction of a separated representation of the unknown field involved in a partial differential equation. For this purpose, we consider the solution of the two-dimensional Poisson equation in a square domain. The solution is sought as a finite sum of terms, each one involving the product of functions of each coordinate. The solution is then calculated by means of a sequence of one-dimensional problems. The chapter starts with the simplest case, that is later extended to cover more complex problems: non-constant source terms, non-homogeneous Dirichlet and Neumann boundary conditions, and high-dimensional problems. Carefully solved numerical examples are discussed to illustrate the theoretical developments.
Francisco Chinesta, Roland Keunings, Adrien Leygue

Chapter 3. PGD Versus SVD

Abstract
The issue of separability is of major importance when using the Proper Generalized Decomposition. Efficient computer implementations require the separated representation of model parameters, boundary conditions and/or source terms. This can be performed by applying the Singular Value Decomposition (SVD) or its multi-dimensional counterpart, the so-called High Order Singular Value Decomposition. This chapter revisits these concepts. We then point out and discuss the subtle connections between SVD and PGD. This allows us to illustrate how the PGD solver can compress separated representations, and also to justify the fact that in certain circumstances the PGD solution procedure can be viewed as the calculation of an on-the-fly compressed representation.
Francisco Chinesta, Roland Keunings, Adrien Leygue

Chapter 4. The Transient Diffusion Equation

Abstract
This chapter addresses the efficient solution of transient problems by considering space-time separated representations. In this case, the transient solution is calculated from a sequence of space and time problems. Such non-incremental solution procedure can lead to impressive computing-time savings, as discussed in detail. Finally, numerical examples are considered that demonstrate the efficiency of the proposed strategy.
Francisco Chinesta, Roland Keunings, Adrien Leygue

Chapter 5. Parametric Models

Abstract
Since separated representations allow one to circumvent the curse of dimensionality, one can consider model parameters, boundary conditions, initial conditions or geometrical parameters defining the computational domain, as extra-coordinates of the problem. Thus, standard models become multi-dimensional, but by solving them only once and offline using the PGD, the solution of the model is available for any choice of the parameters considered as extra-coordinates. This parametric solution can then be used online for different purposes, such as real time simulation, efficient optimization or inverse analysis, or simulation-based control. In this chapter, we illustrate the procedures for considering (a) model parameters, (b) constant and non-constant Dirichlet and Neumann boundary conditions, (c) initial conditions and (d) geometrical parameters, as extra-coordinates of a resulting multi-dimensional model.
Francisco Chinesta, Roland Keunings, Adrien Leygue

Chapter 6. Advanced Topics

Abstract
This chapter addresses some advanced topics whose development remains work in progress. The first topic concerns the efficient treatment of non-linear models where standard strategies can fail for high-dimensional problems. The second topic concerns the use of advective stabilization when the involved fields are approximated in a separated form. Finally, we introduce a discrete form of the PGD solver, the one that we consider in computer implementations, that is then extended for considering a separated representation constructor based on residual minimization. Residual minimization is particularly suitable for addressing non-symmetric differential operators, for which the standard procedure described in the previous chapters can be inefficient (slow convergence and non-optimal representations).
Francisco Chinesta, Roland Keunings, Adrien Leygue

Backmatter

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