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© 2016

Reduced Basis Methods for Partial Differential Equations

An Introduction

Benefits

  • The first textbook on reduced basis methods, an approximation technique that has become very popular and successful in engineering and applied sciences

  • Offers a balanced presentation of theoretical and computational aspects supported by a broad variety of relevant problems

  • A first mathematically-driven foray into the field of reduced order modeling

Textbook

Part of the UNITEXT book series (UNITEXT, volume 92)

Also part of the La Matematica per il 3+2 book sub series (UNITEXTMAT, volume 92)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Alfio Quarteroni, Andrea Manzoni, Federico Negri
    Pages 1-10
  3. Alfio Quarteroni, Andrea Manzoni, Federico Negri
    Pages 11-38
  4. Alfio Quarteroni, Andrea Manzoni, Federico Negri
    Pages 39-72
  5. Alfio Quarteroni, Andrea Manzoni, Federico Negri
    Pages 73-86
  6. Alfio Quarteroni, Andrea Manzoni, Federico Negri
    Pages 87-113
  7. Alfio Quarteroni, Andrea Manzoni, Federico Negri
    Pages 115-140
  8. Alfio Quarteroni, Andrea Manzoni, Federico Negri
    Pages 141-154
  9. Alfio Quarteroni, Andrea Manzoni, Federico Negri
    Pages 155-180
  10. Alfio Quarteroni, Andrea Manzoni, Federico Negri
    Pages 181-192
  11. Alfio Quarteroni, Andrea Manzoni, Federico Negri
    Pages 193-214
  12. Alfio Quarteroni, Andrea Manzoni, Federico Negri
    Pages 215-243
  13. Alfio Quarteroni, Andrea Manzoni, Federico Negri
    Pages 245-263
  14. Back Matter
    Pages 265-296

About this book

Introduction

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.

Keywords

Computational complexity reduction Finite element method Parametrized differential problems Reduced basis methods Reduced order modeling

Authors and affiliations

  1. 1.Ecole Polytechnique Fédérale de LausanneLausanneSwitzerland

About the authors

Prof. Alfio Quarteroni, Dr. Andrea Manzoni and Federico Negri - Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland.

Bibliographic information

Industry Sectors
Aerospace
Energy, Utilities & Environment
Finance, Business & Banking

Reviews

“This textbook provides a basic introduction to reduced basis methods for partial differential equation. Written for undergrad and grad students in this fast growing field of numerical analysis and scientific computing, the methods described allow the authors to solve in real time or in a many-query context a parametric system or equation or to calculate a related output of interest. … MATLAB software is provided as a companion to the book for a selected class of problem.” (Gianluigi Rozza, Mathematical Reviews, October, 2016)