Scientific Computation

Nonlinear Least Squares for Inverse Problems

Theoretical Foundations and Step-by-Step Guide for Applications

Authors: Chavent, Guy

  • Step-by-step guide to solving nonlinear inverse problems with ‘least square’ methods
  • Contains a geometric theory to analyze wellposedness and optimizability
  • Detailed analysis of practical issues when solving nonlinear least square problems
  • Self-contained presentation of strictly quasi-convex sets
  • Each chapter inlcudes an overview of the presented concepts and results
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eBook $119.00
price for USA (gross)
  • ISBN 978-90-481-2785-6
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $159.00
price for USA
  • ISBN 978-90-481-2784-9
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $159.00
price for USA
  • ISBN 978-94-007-3060-1
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
About this book

This book provides an introduction into the least squares resolution of nonlinear inverse problems. The first goal is to develop a geometrical theory to analyze nonlinear least square (NLS) problems with respect to their quadratic wellposedness, i.e. both wellposedness and optimizability. Using the results, the applicability of various regularization techniques can be checked. The second objective of the book is to present frequent practical issues when solving NLS problems. Application oriented readers will find a detailed analysis of problems on the reduction to finite dimensions, the algebraic determination of derivatives (sensitivity functions versus adjoint method), the determination of the number of retrievable parameters, the choice of parametrization (multiscale, adaptive) and the optimization step, and the general organization of the inversion code. Special attention is paid to parasitic local minima, which can stop the optimizer far from the global minimum: multiscale parametrization is shown to be an efficient remedy in many cases, and a new condition is given to check both wellposedness and the absence of parasitic local minima.

For readers that are interested in projection on non-convex sets, Part II of this book presents the geometric theory of quasi-convex and strictly quasi-convex (s.q.c.) sets. S.q.c. sets can be recognized by their finite curvature and limited deflection and possess a neighborhood where the projection is well-behaved.

Throughout the book, each chapter starts with an overview of the presented concepts and results.

About the authors

Background: Ecole Polytechnique (Paris, 1965),

Ecole Nationale Supérieure des Télécommunications (Paris,1968),

Paris-6 University (Ph. D., 1971).

Professor Chavent joined the Faculty of Paris 9-Dauphine in 1971. He is now an emeritus professor from this university. During the same span of time, he ran a research project at INRIA (Institut National de Recherche en Informatique et en Automatique), focused on industrial inverse problems (oil production and exploration, nuclear reactors, ground water management…).

Reviews

From the reviews:

“This comprehensive treatise on the nonlinear inverse problem, written by a mathematician with extensive experience in exploration geophysics, deals primarily with the nonlinear least squares (NLS) methods to solve such problems. … Chavent has authored a book with appeal to both the practitioner of the arcane art of NLS inversion as well as to the theorist seeking a rigorous and formal development of what is currently known about this subject.” (Sven Treitel, The Leading Edge, April, 2010)

“The book is organized so that readers interested in the more practical aspects can easily dip into the appropriate chapters of the book without having to work through the more theoretical details. … is recommended for readers who are interested in applying the OLS approach to nonlinear inverse problems. … This material is relatively accessible even to readers without a very strong background in analysis. The book will also be of interest to readers who want to learn more about … quasi-convex sets and Q-wellposedness.” (Brain Borchers, The Mathematical Association of America, July, 2010)


Table of contents (8 chapters)

  • Nonlinear Inverse Problems: Examples and Difficulties

    Chavent, Guy

    Pages 5-28

  • Computing Derivatives

    Chavent, Guy

    Pages 29-78

  • Choosing a Parameterization

    Chavent, Guy

    Pages 79-159

  • Output Least Squares Identifiability and Quadratically Wellposed NLS Problems

    Chavent, Guy

    Pages 161-207

  • Regularization of Nonlinear Least Squares Problems

    Chavent, Guy

    Pages 209-270

Buy this book

eBook $119.00
price for USA (gross)
  • ISBN 978-90-481-2785-6
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $159.00
price for USA
  • ISBN 978-90-481-2784-9
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $159.00
price for USA
  • ISBN 978-94-007-3060-1
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Nonlinear Least Squares for Inverse Problems
Book Subtitle
Theoretical Foundations and Step-by-Step Guide for Applications
Authors
Series Title
Scientific Computation
Copyright
2010
Publisher
Springer Netherlands
Copyright Holder
Springer Science+Business Media B.V.
eBook ISBN
978-90-481-2785-6
DOI
10.1007/978-90-481-2785-6
Hardcover ISBN
978-90-481-2784-9
Softcover ISBN
978-94-007-3060-1
Series ISSN
1434-8322
Edition Number
1
Number of Pages
XIV, 360
Topics