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The proper generalized decomposition for advanced numerical simulations. A primer. (English) Zbl 1287.65001
SpringerBriefs in Applied Sciences and Technology. Cham: Springer (ISBN 978-3-319-02864-4/pbk; 978-3-319-02865-1/ebook). xiii, 117 p. (2014).
Principally speaking, the method of proper generalized decomposition (PGD), as it is called by the authors, is the technique of separation of variables combined with the alternating direction implicit (ADI) method, where the number of trial functions is sucsessively increased until a required termination criterium is satisfied. In each successive step the resulting, in general, one-dimensional problems are solved using finite elements or finite differences. When dealing with the diffusion equation also the time dependent part of the unknown solution is discretized on the time interval as a whole, thus not using step by step time incremental methods. Data in the given equations, e.g., material parameters, initial and boundary conditions, can be considered as additional variables and be included in the trial function ansatz.
The book aims “to provide a readable and practical introduction to the PGD, which will allow the reader to quickly grasp the main features of the method”. It is divided into six chapters: Introduction, PGD solution of the Poisson equation, PGD versus singular value decompostion (SVD), the transient diffusion equation, parametric models and advanced topics. All chapters include numerical examples obtained for standard equations in separable domains. References are given at the end of each chapter, unfortunately not in alphabetic order but in the order of their first appearance in the text.
In the opinion of the referee it would be interesting to know how the PGD method competes with respect to computer time consumption when compared with the standard ADI method for separable problems set up with finite elements or finite differences, which can use the available sophisticated solution algorithms for the resulting algebraic systems.

MSC:
65-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65N06 Finite difference methods for boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35K05 Heat equation
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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