Nonlinear Least Squares for Inverse Problems: Theoretical Foundations and Step-by-Step Guide for Applications

ISBN-13: 9789048127849 / Angielski / Twarda / 2009 / 360 str.

Guy Chavent

Opis

The domain of inverse problems has experienced a rapid expansion, driven by the increase in computing power and the progress in numerical modeling. When I started working on this domain years ago, I became somehow fr- tratedtoseethatmyfriendsworkingonmodelingwhereproducingexistence, uniqueness, and stability results for the solution of their equations, but that I was most of the time limited, because of the nonlinearity of the problem, to provethatmyleastsquaresobjectivefunctionwasdi?erentiable....Butwith my experience growing, I became convinced that, after the inverse problem has been properly trimmed, the ?nal least squares problem, the one solved on the computer, should be Quadratically (Q)-wellposed, thatis, both we- posed and optimizable: optimizability ensures that a global minimizer of the least squares function can actually be found using e?cient local optimization algorithms, and wellposedness that this minimizer is stable with respect to perturbation of the data. But the vast majority of inverse problems are nonlinear, and the clas- cal mathematical tools available for their analysis fail to bring answers to these crucial questions: for example, compactness will ensure existence, but provides no uniqueness results, and brings no information on the presence or absenceofparasiticlocalminimaorstationarypoints..

Recenzje

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)

Spis treści

Nonlinear Least Squares.- Nonlinear Inverse Problems: Examples and Difficulties.- Computing Derivatives.- Choosing a Parameterization.- Output Least Squares Identifiability and Quadratically Wellposed NLS Problems.- Regularization of Nonlinear Least Squares Problems.- A generalization of convex sets.- Quasi-Convex Sets.- Strictly Quasi-Convex Sets.- Deflection Conditions for the Strict Quasi-convexity of Sets.

O autorze

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…).

Blurb

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.