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Reduced basis methods for partial differential equations. An introduction. (English) Zbl 1337.65113

Unitext 92. La Matematica per il 3+2. Cham: Springer (ISBN 978-3-319-15430-5/pbk; 978-3-319-15431-2/ebook). xi, 296 p. (2016).
This new book, being one of the first on solving partial differential equations (PDEs) by reduced basis methods, gives an excellent introduction to the subject and its numerical approach. The basic idea is that of carrying out a projection of the problem on a low-dimensional subspace of the solution vector space spanned by specifically selected basis functions, being provided with a set of more exact solutions that belong to certain parameters. The book contains a nice basic mathematical introduction and several general formulations of the subject and problem, it gives algorithms with error estimates and a priori and a posteriori accuracy estimates, it considers implementations in detail, and examples of linear, nonlinear and nonaffine type. There is also a useful appendix on basic theory (suitable spaces, maps, operators, interpolation, orthogonal polynomial systems). Concretely, the first three chapters of the volume provide an introduction, examples and the basic methods and properties (e.g. complexity reduction, a posteriori error estimates, bounding of errors in practice, examples). Then the algebraic and geometric view of the methods is given, including the theory in the fifth chapter with \(n\)-widths, and the dimension of the set of solutions. The following part of the book begins with computations and examples, the construction of the spaces for the reduced bases by singular value decomposition and greedy methods (in the seventh chapter, including a priori and a posteriori accuracy estimates), and finally examples and computations and extensions (nonlinear and nonaffine problems, further useful applications for PDE-constrained optimisation problems for instance, and examples of heat-transfer and mass-transfer type). In each chapter of the book, there are at the end exercises included for the reader, with respect to the content of that part of the work.

MSC:

65Mxx Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
65Nxx Numerical methods for partial differential equations, boundary value problems
65-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis
65Y20 Complexity and performance of numerical algorithms
80A20 Heat and mass transfer, heat flow (MSC2010)
80Mxx Basic methods in thermodynamics and heat transfer
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