# Reduced Basis Methods for Partial Differential Equations: An Introduction

Springer, Aug 19, 2015 - Mathematics - 296 pages

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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### Contents

 Chapter 1 Introduction 1 Analysis and HighFidelity Approximation 11 Basic Principles Basic Properties 39 Chapter 4 On the Algebraic and Geometric Structure of RB Methods 73 Chapter 5 The Theoretical Rationale Behind 87 Chapter 6 Construction of RB Spaces by SVDPOD 114 Chapter 7 Construction of RB Spaces by the Greedy Algorithm 141 Setting up the Problem 155
 Computing the Solution 181 Chapter 10 Extension to Nonaffine Problems 193 Chapter 11 Extension to Nonlinear Problems 215 Chapter 12 Reduction and Control 245 Appendix A Basic Theoretical Tools 264 References 281 Index 293 Copyright