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This book provides a basic introduction to reduced basis (RB)
methods for problems involving the repeated solution of partial
differential equations (PDEs) arising from engineering and applied
sciences, such as PDEs depending on several parameters and
PDE-constrained optimization. The book presents a general
mathematical formulation of RB methods, analyzes their fundamental
theoretical properties, discusses the related algorithmic and
implementation aspects, and highlights their built-in algebraic and
geometric structures. More specifically, the authors discuss
alternative strategies for constructing accurate RB spaces using
greedy algorithms and proper orthogonal decomposition techniques,
investigate their approximation properties and analyze
offline-online decomposition strategies aimed at the reduction of
computational complexity. Furthermore, they carry out both a priori
and a posteriori error analysis. The whole mathematical
presentation is made more stimulating by the use of representative
examples of applicative interest in the context of both linear and
nonlinear PDEs. Moreover, the inclusion of many pseudocodes allows
the reader to easily implement the algorithms illustrated
throughout the text. The book will be ideal for upper undergraduate
students and, more generally, people interested in scientific
computing. All these pseudocodes are in fact implemented in a
MATLAB package that is freely available at
https://github.com/redbkit
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