| Description |
1 online resource (377 pages) |
| 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 |
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2.5 L1 Regression2.6 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 |
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4.3 Resolution, Bias, and Uncertainty in the Tikhonov Solution4.4 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 |
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6.3 The Conjugate Gradient Method6.4 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 |
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8.6 Notes and Further Reading9 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 |
| Summary |
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 fundamental and challenging problems, introducing students and professionals to the broad range of approaches that lie in the realm of inverse theory. The authors present both the underlying theory and practical algorithms for solving inverse problems. The authors' treatment is appropriate for |
| Notes |
11.1 Review of the Classical Approach |
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Print version record |
| Subject |
Inverse problems (Differential equations)
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Inversion (Geophysics)
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Mathematical models.
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Parameter estimation.
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mathematical models.
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Inverse problems (Differential equations)
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Inversion (Geophysics)
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Mathematical models
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Parameter estimation
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| Form |
Electronic book
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| Author |
Borchers, Brian
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Thurber, Clifford H
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| ISBN |
9780123850492 |
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0123850495 |
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