Reduced Basis Methods for Partial Differential Equations: An Introduction

Reduced Basis Methods for Partial Differential Equations provides a basic introduction to reduced basis (RB) methods for problems involving the repeated solution of partial differential equations (PDEs), which arise in engineering and applied sciences. 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 mathematical presentation is made more stimulating with representative examples of applicative interest in the context of both linear and nonlinear PDEs. Moreover, the inclusion of many pseudocodes enables the reader to easily implement the algorithms illustrated throughout. This book is ideal for upper division undergraduate students and, more generally, individuals interested in scientific computing.

A set of MATLAB code files for reduced-order modeling of parametrized PDEs is available for download.