Inverse problem theory and methods for model parameter estimation / Albert Tarantola.

Format:

Book

Author/Creator:

Tarantola, Albert.

Contributor:

Class of 1924 Book Fund.

Language:

English

Subjects (All):

Inverse problems (Differential equations).

Physical Description:

xii, 342 pages : illustrations (some color) ; 26 cm

Place of Publication:

Philadelphia, PA : Society for Industrial and Applied Mathematics, [2005]

Summary:

While the prediction of observations is a forward problem, the use of actual observations to infer the properties of a model is an inverse problem. Inverse problems are difficult because they may not have a unique solution. The description of uncertainties plays a central role in the theory, which is based on probability theory. This book proposes a general approach that is valid for linear as well as for nonlinear problems. The philosophy is essentially probabilistic and allows the reader to understand the basic difficulties appearing in the resolution of inverse problems. The book attempts to explain how a method of acquisition of information can be applied to actual real-world problems, and many of the arguments are heuristic.

Prompted by recent developments in inverse theory, Inverse Problem Theory and Methods for Model Parameter Estimation is a completely rewritten version of a 1987 book by the same author. In this version there are many algorithmic details for Monte Carlo methods, least-squares discrete problems, and least-squares problems involving functions. In addition, some notions are clarified, the role of optimization techniques is underplayed, and Monte Carlo methods are taken much more seriously. The first part of the book deals exclusively with discrete inverse problems with a finite number of parameters, while the second part of the book deals with general inverse problems.

The book is directed to all scientists, including applied mathematicians, facing the problem of quantitative interpretation of experimental data in fields such as physics, chemistry, biology, image processing, and information sciences. Considerable effort has been made so that this book can serve either as a reference manual for researchers or as a textbook in a course for undergraduate or graduate students.

Contents:

1 The General Discrete Inverse Problem 1

1.1 Model Space and Data Space 1

1.2 States of Information 6

1.3 Forward Problem 20

1.4 Measurements and A Priori Information 24

1.5 Defining the Solution of the Inverse Problem 32

1.6 Using the Solution of the Inverse Problem 37

2 Monte Carlo Methods 41

2.2 The Movie Strategy for Inverse Problems 44

2.3 Sampling Methods 48

2.4 Monte Carlo Solution to Inverse Problems 51

2.5 Simulated Annealing 54

3 The Least-Squares Criterion 57

3.1 Preamble: The Mathematics of Linear Spaces 57

3.2 The Least-Squares Problem 62

3.3 Estimating Posterior Uncertainties 70

3.4 Least-Squares Gradient and Hessian 75

4 Least-Absolute-Values Criterion and Minimax Criterion 81

4.2 Preamble: e[subscript p]-Norms 82

4.3 The e[subscript p]-Norm Problem 86

4.4 The e[subscript 1]-Norm Criterion for Inverse Problems 89

4.5 The e[subscript infinity]-Norm Criterion for Inverse Problems 96

5 Functional Inverse Problems 101

5.1 Random Functions 101

5.2 Solution of General Inverse Problems 108

5.3 Introduction to Functional Least Squares 108

5.4 Derivative and Transpose Operators in Functional Spaces 119

5.5 General Least-Squares Inversion 133

5.6 Example: X-Ray Tomography as an Inverse Problem 140

5.7 Example: Travel-Time Tomography 143

5.8 Example: Nonlinear Inversion of Elastic Waveforms 144

6.1 Volumetric Probability and Probability Density 159

6.2 Homogeneous Probability Distributions 160

6.3 Homogeneous Distribution for Elastic Parameters 164

6.4 Homogeneous Distribution for Second-Rank Tensors 170

6.5 Central Estimators and Estimators of Dispersion 170

6.6 Generalized Gaussian 174

6.7 Log-Normal Probability Density 175

6.8 Chi-Squared Probability Density 177

6.9 Monte Carlo Method of Numerical Integration 179

6.10 Sequential Random Realization 181

6.11 Cascaded Metropolis Algorithm 182

6.12 Distance and Norm 183

6.13 The Different Meanings of the Word Kernel 183

6.14 Transpose and Adjoint of a Differential Operator 184

6.15 The Bayesian Viewpoint of Backus (1970) 190

6.16 The Method of Backus and Gilbert 191

6.17 Disjunction and Conjunction of Probabilities 195

6.18 Partition of Data into Subsets 197

6.19 Marginalizing in Linear Least Squares 200

6.20 Relative Information of Two Gaussians 201

6.21 Convolution of Two Gaussians 202

6.22 Gradient-Based Optimization Algorithms 203

6.23 Elements of Linear Programming 223

6.24 Spaces and Operators 230

6.25 Usual Functional Spaces 242

6.26 Maximum Entropy Probability Density 245

6.27 Two Properties of e[subscript p]-Norms 246

6.28 Discrete Derivative Operator 247

6.29 Lagrange Parameters 249

6.30 Matrix Identities 249

6.31 Inverse of a Partitioned Matrix 250

6.32 Norm of the Generalized Gaussian 250

7 Problems 253

7.1 Estimation of the Epicentral Coordinates of a Seismic Event 253

7.2 Measuring the Acceleration of Gravity 256

7.3 Elementary Approach to Tomography 259

7.4 Linear Regression with Rounding Errors 266

7.5 Usual Least-Squares Regression 269

7.6 Least-Squares Regression with Uncertainties in Both Axes 273

7.7 Linear Regression with an Outlier 275

7.8 Condition Number and A Posteriori Uncertainties 279

7.9 Conjunction of Two Probability Distributions 285

7.10 Adjoint of a Covariance Operator 288

7.11 Problem 7.1 Revisited 289

7.12 Problem 7.3 Revisited 289

7.13 An Example of Partial Derivatives 290

7.14 Shapes of the e[subscript p]-Norm Misfit Functions 290

7.15 Using the Simplex Method 293

7.16 Problem 7.7 Revisited 295

7.17 Geodetic Adjustment with Outliers 296

7.18 Inversion of Acoustic Waveforms 297

7.19 Using the Backus and Gilbert Method 304

7.20 The Coefficients in the Backus and Gilbert Method 308

7.21 The Norm Associated with the 1D Exponential Covariance 308

7.22 The Norm Associated with the 1D Random Walk 311

7.23 The Norm Associated with the 3D Exponential Covariance 313.

Notes:

Includes bibliographical references (pages 317-332) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Class of 1924 Book Fund.

ISBN:

0898715725

OCLC:

56672375