
Parameter Estimation and Inverse Problems
Description
Key Features
- Includes appendices for review of needed concepts in linear, statistics, and vector calculus.
- Accessible to students and professionals without a highly specialized mathematical background.
Readership
The book is primarily used as a textbook for graduate and advanced undergraduate students taking courses in geophysical inverse problems. It is also used as a reference for geoscientists and researchers in academe and industry.
Table of Contents
Preface
Chapter One. Introduction
1.1. Classification of Parameter Estimation and Inverse Problems
1.2. Examples of Parameter Estimation Problems
1.3. Examples of Inverse Problems
1.4. Discretizing Integral Equations
1.5. Why Inverse Problems Are Difficult
1.6. Exercises
1.7. Notes and Further Reading
Chapter Two. Linear Regression
2.1. Introduction to Linear Regression
2.2. Statistical Aspects of Least Squares
2.3. An Alternative View of the 95% Confidence Ellipsoid
2.4. Unknown Measurement Standard Deviations
2.5. L1 Regression
2.6. Monte Carlo Error Propagation
2.7. Exercises
2.8. Notes and Further Reading
Chapter Three. Rank Deficiency and Ill-Conditioning
3.1. The SVD and the Generalized Inverse
3.2. Covariance and Resolution of the Generalized Inverse Solution
3.3. Instability of the Generalized Inverse Solution
3.4. A Rank Deficient Tomography Problem
3.5. Discrete Ill-Posed Problems
3.6. Exercises
3.7. Notes and Further Reading
Chapter Four. Tikhonov Regularization
4.1. Selecting Good Solutions to Ill-Posed Problems
4.2. SVD Implementation of Tikhonov Regularization
4.3. Resolution, Bias, and Uncertainty in the Tikhonov Solution
4.4. Higher-Order Tikhonov Regularization
4.5. Resolution in Higher-order Tikhonov Regularization
4.6. The TGSVD Method
4.7. Generalized Cross-Validation
4.8. Error Bounds
4.9. Exercises
4.10. Notes and Further Reading
Chapter Five. Discretizing Problems Using Basis Functions
5.1. Discretization by Expansion of the Model
5.2. Using Representers as Basis Functions
5.3. The Method of Backus and Gilbert
5.4. Exercises
5.5. Notes and Further Reading
Chapter Six. Iterative Methods
6.1. Introduction
6.2. Iterative Methods for Tomography Problems
6.3. The Conjugate Gradient Method
6.4. The CGLS Method
6.5. Resolution Analysis for Iterative Methods
6.6. Exercises
6.7. Notes and Further Reading
Chapter Seven. Additional Regularization Techniques
7.1. Using Bounds as Constraints
7.2. Sparsity Regularization
7.3. Using IRLS to Solve L1 Regularized Problems
7.4. Total Variation
7.5. Exercises
7.6. Notes and Further Reading
Chapter Eight. Fourier Techniques
8.1. Linear Systems in the Time and Frequency Domains
8.2. Linear Systems in Discrete Time
8.3. Water Level Regularization
8.4. Tikhonov Regularization in the Frequency Domain
8.5. Exercises
8.6. Notes and Further Reading
Chapter Nine. Nonlinear Regression
9.1. Introduction to Nonlinear Regression
9.2. Newton's Method for Solving Nonlinear Equations
9.3. The Gauss-Newton and Levenberg-Marquardt Methods for Solving Nonlinear Least Squares Problems
9.4. Statistical Aspects of Nonlinear Least Squares
9.5. Implementation Issues
9.6. Exercises
9.7. Notes and Further Reading
Chapter Ten. Nonlinear Inverse Problems
10.1. Regularizing Nonlinear Least Squares Problems
10.2. Occam's Inversion
10.3. Model Resolution in Nonlinear Inverse Problems
10.4. Exercises
10.5. Notes and Further Reading
Chapter Eleven. Bayesian Methods
11.1. Review of the Classical Approach
11.2. The Bayesian Approach
11.3. The Multivariate Normal Case
11.4. The Markov Chain Monte Carlo Method
11.5. Analyzing MCMC Output
11.6. Exercises
11.7. Notes and Further Reading
Chapter Twelve. Epilogue
Appendix A. Review of Linear Algebra
Appendix B. Review of Probability and Statistics
Appendix C. Review of Vector Calculus
Appendix D. Glossary of Notation
Bibliography
Index
Product details
- No. of pages: 376
- Language: English
- Copyright: © Academic Press 2012
- Published: December 10, 2011
- Imprint: Academic Press
- eBook ISBN: 9780123850492
About the Authors
Richard C. Aster
Affiliations and Expertise
Brian Borchers

Affiliations and Expertise
Clifford H. Thurber

Affiliations and Expertise
Ratings and Reviews
Latest reviews
(Total rating for all reviews)
Andri H. Thu Dec 28 2017
Parameter estimation and inverse problems
I enjoy reading this book. Well explained.