Parameter estimation and inverse problems
Parameter Estimation and Inverse Problems, 2e provides geoscience students and professionals with answers to common questions like how one can derive a physical model from a finite set of observations containing errors, and how one may determine the quality of such a model. This book takes on these...
Saved in:
| Main Authors | , , |
|---|---|
| Format | eBook Book |
| Language | English |
| Published |
San Diego
Academic Press
2013
Elsevier Science & Technology |
| Edition | 2 |
| Series | International Geophysics |
| Subjects | |
| Online Access | Get full text |
| ISBN | 9780128100929 0123850487 0128100923 9780123850485 |
| DOI | 10.1016/C2009-0-61134-X |
Cover
Table of Contents:
- Front Cover -- Parameter Estimation and Inverse Problems -- Copyright -- Table of Contents -- Preface -- 1 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 -- 2 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 -- 3 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 -- 4 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 -- 5 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 -- 6 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 -- 7 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 -- 8 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 -- 9 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 -- 10 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 -- 11 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 -- 12 Epilogue -- A. Review of Linear Algebra -- A.1 Systems of Linear Equations -- A.2 Matrix and Vector Algebra -- A.3 Linear Independence -- A.4 Subspaces of Rn -- A.5 Orthogonality and the Dot Product -- A.6 Eigenvalues and Eigenvectors -- A.7 Vector and Matrix Norms -- A.8 The Condition Number of a Linear System -- A.9 The QR Factorization -- A.10 Complex Matrices and Vectors -- A.11 Linear Algebra in Spaces of Functions
- A.12 Exercises -- A.13 Notes and Further Reading -- B. Review of Probability and Statistics -- B.1 Probability and Random Variables -- B.2 Expected Value and Variance -- B.3 Joint Distributions -- B.4 Conditional Probability -- B.5 The Multivariate Normal Distribution -- B.6 The Central Limit Theorem -- B.7 Testing for Normality -- B.8 Estimating Means and Confidence Intervals -- B.9 Hypothesis Tests -- B.10 Exercises -- B.11 Notes and Further Reading -- C. Review of Vector Calculus -- C.1 The Gradient, Hessian, and Jacobian -- C.2 Taylor's Theorem -- C.3 Lagrange Multipliers -- C.4 Exercises -- C.5 Notes and Further Reading -- D. Glossary of Notation -- Bibliography -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- Z