Chavent Nonlinear Least Squares for Inverse Problems

Theoretical Foundations and Step-by-Step Guide for Applications
1. Auflage 2010
ISBN: 978-90-481-2785-6
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 360 Seiten

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....

Autoren/Hrsg.

Chavent, Guy

Inhaltsverzeichnis

1;Preface;7
2;Contents;10
3;I Nonlinear Least Squares;14
3.1;Nonlinear Inverse Problems: Examples and Difficulties;17
3.1.1;Example 1: Inversion of Knott--Zoeppritz Equations;18
3.1.2;An Abstract NLS Inverse Problem;21
3.1.3;Analysis of NLS Problems;22
3.1.3.1;Wellposedness;22
3.1.3.2;Optimizability;24
3.1.3.3;Output Least Squares Identifiability and Quadratically Wellposed Problems;24
3.1.3.4;Regularization;26
3.1.3.5;Derivation;32
3.1.4;Example 2: 1D Elliptic Parameter Estimation Problem;33
3.1.5;Example 3: 2D Elliptic Nonlinear Source Estimation Problem;36
3.1.6;Example 4: 2D Elliptic Parameter Estimation Problem;38
3.2;Computing Derivatives;41
3.2.1;Setting the Scene;42
3.2.2;The Sensitivity Functions Approach;45
3.2.3;The Adjoint Approach;45
3.2.4;Implementation of the Adjoint Approach;50
3.2.5;=3pt plus2pt minus2ptExample 1: The Adjoint Knott--Zoeppritz Equations;53
3.2.6;Examples 3 and 4: Discrete Adjoint Equations;58
3.2.6.1;Discretization Step 1: Choice of a DiscretizedForward Map;59
3.2.6.2;Discretization Step 2: Choice of a DiscretizedObjective Function;64
3.2.6.3;Derivation Step 0: Forward Map and Objective Function;64
3.2.6.4;Derivation Step 1: State-Space Decomposition;65
3.2.6.5;Derivation Step 2: Lagrangian;66
3.2.6.6;Derivation Step 3: Adjoint Equation;68
3.2.6.7;Derivation Step 4: Gradient Equation;70
3.2.7;Examples 3 and 4: Continuous Adjoint Equations;71
3.2.8;Example 5: Differential Equations, Discretized VersusDiscrete Gradient;77
3.2.8.1;Implementing the Discretized Gradient;80
3.2.8.2;Implementing the Discrete Gradient;80
3.2.9;Example 6: Discrete Marching Problems;85
3.3;Choosing a Parameterization;91
3.3.1;Calibration;92
3.3.1.1;On the Parameter Side;92
3.3.1.2;On the Data Side;95
3.3.1.3;Conclusion;96
3.3.2;How Many Parameters Can be Retrieved from the Data?;96
3.3.3;Simulation Versus Optimization Parameters;100
3.3.4;Parameterization by a Closed Form Formula;102
3.3.5;Decomposition on the Singular Basis;103
3.3.6;Multiscale Parameterization;105
3.3.6.1;Simulation Parameters for a Distributed Parameter;105
3.3.6.2;Optimization Parameters at Scale k;106
3.3.6.3;Scale-By-Scale Optimization;107
3.3.6.4;Examples of Multiscale Bases;117
3.3.6.5;Summary for Multiscale Parameterization;120
3.3.7;Adaptive Parameterization: Refinement Indicators;120
3.3.7.1;Definition of Refinement Indicators;121
3.3.7.2;Multiscale Refinement Indicators;128
3.3.7.3;Application to Image Segmentation;133
3.3.7.4;Coarsening Indicators;134
3.3.7.5;A Refinement/Coarsening Indicators Algorithm;136
3.3.8;Implementation of the Inversion;138
3.3.8.1;Constraints and Optimization Parameters;138
3.3.8.2;Gradient with Respect to OptimizationParameters;141
3.3.9;Maximum Projected Curvature: A Descent Step for Nonlinear Least Squares;147
3.3.9.1;Descent Algorithms;147
3.3.9.2;Maximum Projected Curvature (MPC) Step;149
3.3.9.3;Convergence Properties for the TheoreticalMPC Step;155
3.3.9.4;Implementation of the MPC Step;156
3.3.9.5;Performance of the MPC Step;160
3.4;Output Least Squares Identifiability and QuadraticallyWellposed NLS Problems;172
3.4.1;The Linear Case;174
3.4.2;Finite Curvature/Limited Deflection Problems;176
3.4.3;Identifiability and Stability of the Linearized Problems;185
3.4.4;A Sufficient Condition for OLS-Identifiability;187
3.4.5;The Case of Finite Dimensional Parameters;190
3.4.6;Four Questions to Q-Wellposedness;193
3.4.6.1;Case of Finite Dimensional Parameters;194
3.4.6.2;Case of Infinite Dimensional Parameters;195
3.4.7;Answering the Four Questions;195
3.4.8;Application to Example 2: 1D Parameter Estimation with H1 Observation;202
3.4.8.1;Linear Stability;204
3.4.8.2;Deflection Estimate;209
3.4.8.3;Curvature Estimate;210
3.4.8.4;Conclusion: OLS-Identifiability;211
3.4.9;Application to Example 4: 2D Parameter Estimation,with H1 Observation;211
3.5;Regularization of Nonlinear Least Squares Problems;219
3.5.1;Levenberg--Marquardt--Tychonov (LMT) Regularization;219
3.5.1.1;Linear Problems;221
3.5.1.2;Finite Curvature/Limited Deflection(FC/LD) Problems;229
3.5.1.3;General Nonlinear Problems;241
3.5.2;Application to the Nonlinear 2D Source Problem;247
3.5.3;State-Space Regularization;256
3.5.3.1;Dense Observation: Geometric Approach;258
3.5.3.2;Incomplete Observation: Soft Analysis;266
3.5.4;Adapted Regularization for Example 4: 2D ParameterEstimation with H1 Observation;269
3.5.4.1;Which Part of a is Constrained by the Data?;270
3.5.4.2;How to Control the Unconstrained Part?;272
3.5.4.3;The Adapted-Regularized Problem;274
3.5.4.4;Infinite Dimensional Linear Stabilityand Deflection Estimates;275
3.5.4.5;Finite Curvature Estimate;277
3.5.4.6;OLS-Identifiability for the Adapted RegularizedProblem;278
4;II A Generalization of Convex Sets;281
4.1;Quasi-Convex Sets;283
4.1.1;Equipping the Set D with Paths;285
4.1.2;Definition and Main Properties of q.c. Sets;289
4.2;Strictly Quasi-Convex Sets;306
4.2.1;Definition and Main Properties of s.q.c. Sets;307
4.2.2;Characterization by the Global Radius of Curvature;311
4.2.3;Formula for the Global Radius of Curvature;323
4.3;Deflection Conditions for the Strict Quasi-convexityof Sets;328
4.3.1;The General Case: D F;334
4.3.2;The Case of an Attainable Set D = (C);344
5;Bibliography;351
6;Index;351