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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
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.
Content Level »Research
Keywords »analysis of NLS problems - analysis of nonlinear least square problems - choice of parametrization - inverse problem - inversion code - inversion method - inversion methods - least square method - least square methods - least square problem - least square problems;
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.