Nonlinear least squares for inverse problems : theoretical foundations and step-by-step guide for applications / G. Chavent.

Format:

Book

Author/Creator:

Chavent, Guy, 1943-

Series:

Scientific computation

Scientific computation, 1434-8322

Language:

English

Subjects (All):

Least squares.

Nonlinear theories.

Physical Description:

xiv, 360 pages : illustrations ; 24 cm.

Place of Publication:

Dordrecht : Springer Verlag, [2009]

Summary:

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) problem 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 function 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.

Contents:

I Nonlinear Least Squares 1

1 Nonlinear Inverse Problems: Examples and Difficulties 5

1.1 Example 1: Inversion of Knott-Zoeppritz Equations 6

1.2 An Abstract NLS Inverse Problem 9

1.3 Analysis of NLS Problems 10

1.3.1 Wellposedness 10

1.3.2 Optimizability 12

1.3.3 Output Least Squares Identifiability and Quadratically Wellposed Problems 12

1.3.4 Regularization 14

1.3.5 Derivation 20

1.4 Example 2: 1D Elliptic Parameter Estimation Problem 21

1.5 Example 3: 2D Elliptic Nonlinear Source Estimation Problem 24

1.6 Example 4: 2D Elliptic Parameter Estimation Problem 26

2 Computing Derivatives 29

2.1 Setting the Scene 30

2.2 The Sensitivity Functions Approach 33

2.3 The Adjoint Approach 33

2.4 Implementation of the Adjoint Approach 38

2.5 Example 1: The Adjoint Knott-Zoeppritz Equations 41

2.6 Examples 3 and 4: Discrete Adjoint Equations 46

2.6.1 Discretization Step 1: Choice of a Discretized Forward Map 47

2.6.2 Discretization Step 2: Choice of a Discretized Objective Function 52

2.6.3 Derivation Step 0: Forward Map and Objective Function 52

2.6.4 Derivation Step 1: State-Space Decomposition 53

2.6.5 Derivation Step 2: Lagrangian 54

2.6.6 Derivation Step 3: Adjoint Equation 56

2.6.7 Derivation Step 4: Gradient Equation 58

2.7 Examples 3 and 4: Continuous Adjoint Equations 59

2.8 Example 5: Differential Equations, Discretized Versus Discrete Gradient 65

2.8.1 Implementing the Discretized Gradient 68

2.8.2 Implementing the Discrete Gradient 68

2.9 Example 6: Discrete Marching Problems 73

3 Choosing a Parameterization 79

3.1 Calibration 80

3.1.1 On the Parameter Side 80

3.1.2 On the Data Side 83

3.1.3 Conclusion 84

3.2 How Many Parameters Can be Retrieved from the Data? 84

3.3 Simulation Versus Optimization Parameters 88

3.4 Parameterization by a Closed Form Formula 90

3.5 Decomposition on the Singular Basis 91

3.6 Multiscale Parameterization 93

3.6.1 Simulation Parameters for a Distributed Parameter 93

3.6.2 Optimization Parameters at Scale k 94

3.6.3 Scale-By-Scale Optimization 95

3.6.4 Examples of Multiscale Bases 105

3.6.5 Summary for Multiscale Parameterization 108

3.7 Adaptive Parameterization: Refinement Indicators 108

3.7.1 Definition of Refinement Indicators 109

3.7.2 Multiscale Refinement Indicators 116

3.7.3 Application to Image Segmentation 121

3.7.4 Coarsening Indicators 122

3.7.5 A Refinement/Coarsening Indicators Algorithm 124

3.8 Implementation of the Inversion 126

3.8.1 Constraints and Optimization Parameters 126

3.8.2 Gradient with Respect to Optimization Parameters 129

3.9 Maximum Projected Curvature: A Descent Step for Nonlinear Least Squares 135

3.9.1 Descent Algorithms 135

3.9.2 Maximum Projected Curvature (MPC) Step 137

3.9.3 Convergence Properties for the Theoretical MPC Step 143

3.9.4 Implementation of the MPC Step 144

3.9.5 Performance of the MPC Step 148

4 Output Least Squares Identifiability and Quadratically Wellposed NLS Problems 161

4.1 The Linear Case 163

4.2 Finite Curvature/Limited Deflection Problems 165

4.3 Identifiability and Stability of the Linearized Problems 174

4.4 A Sufficient Condition for OLS-Identifiability 176

4.5 The Case of Finite Dimensional Parameters 179

4.6 Four Questions to Q-Wellposedness 182

4.6.1 Case of Finite Dimensional Parameters 183

4.6.2 Case of Infinite Dimensional Parameters 184

4.7 Answering the Four Questions 184

4.8 Application to Example 2: ID Parameter Estimation with H₁ Observation 191

4.8.1 Linear Stability 193

4.8.2 Deflection Estimate 198

4.8.3 Curvature Estimate 199

4.8.4 Conclusion: OLS-Identifiability 200

4.9 Application to Example 4: 2D Parameter Estimation, with H₁ Observation 200

5 Regularization of Nonlinear Least Squares Problems 209

5.1 Levenberg-Marquardt-Tychonov (LMT) Regularization 209

5.1.1 Linear Problems 211

5.1.2 Finite Curvature/Limited Deflection (FC/LD) Problems 219

5.1.3 General Nonlinear Problems 231

5.2 Application to the Nonlinear 2D Source Problem 237

5.3 State-Space Regularization 246

5.3.1 Dense Observation: Geometric Approach 248

5.3.2 Incomplete Observation: Soft Analysis 256

5.4 Adapted Regularization for Example 4: 2D Parameter Estimation with H₁ Observation 259

5.4.1 Which Part of a is Constrained by the Data? 260

5.4.2 How to Control the Unconstrained Part? 262

5.4.3 The Adapted-Regularized Problem 264

5.4.4 Infinite Dimensional Linear Stability and Deflection Estimates 265

5.4.5 Finite Curvature Estimate 267

5.4.6 OLS-Identifiability for the Adapted Regularized Problem 268

II A Generalization of Convex Sets 271

6 Quasi-Convex Sets 275

6.1 Equipping the Set D with Paths 277

6.2 Definition and Main Properties of q.c. Sets 281

7 Strictly Quasi-Convex Sets 299

7.1 Definition and Main Properties of s.q.c. Sets 300

7.2 Characterization by the Global Radius of Curvature 304

7.3 Formula for the Global Radius of Curvature 316

8 Deflection Conditions for the Strict Quasi-convexity of Sets 321

8.1 The General Case: D ⊂ F 327

8.2 The Case of an Attainable Set D = φ;(C) 337.

Notes:

Includes bibliographical references (pages 345-352) and index.

ISBN:

9789048127849

904812784X

9048127858

9789048127856

OCLC:

401151181