Parameter Estimation and Inverse Problems, Second Edition

Front Cover......Page 1
Parameter Estimation and Inverse Problems......Page 4
Copyright......Page 5
Table of Contents......Page 6
Preface......Page 10
1.1 Classification of Parameter Estimation and Inverse Problems......Page 12
1.2 Examples of Parameter Estimation Problems......Page 15
1.3 Examples of Inverse Problems......Page 19
1.4 Discretizing Integral Equations......Page 25
1.5 Why Inverse Problems Are Difficult......Page 30
1.6 Exercises......Page 33
1.7 Notes and Further Reading......Page 34
2.1 Introduction to Linear Regression......Page 36
2.2 Statistical Aspects of Least Squares......Page 38
2.3 An Alternative View of the 95% Confidence Ellipsoid......Page 48
2.4 Unknown Measurement Standard Deviations......Page 49
2.5 L1 Regression......Page 53
2.6 Monte Carlo Error Propagation......Page 58
2.7 Exercises......Page 60
2.8 Notes and Further Reading......Page 63
3.1 The SVD and the Generalized Inverse......Page 66
3.2 Covariance and Resolution of the Generalized Inverse Solution......Page 73
3.3 Instability of the Generalized Inverse Solution......Page 75
3.4 A Rank Deficient Tomography Problem......Page 79
3.5 Discrete Ill-Posed Problems......Page 85
3.6 Exercises......Page 98
3.7 Notes and Further Reading......Page 102
4.1 Selecting Good Solutions to Ill-Posed Problems......Page 104
4.2 SVD Implementation of Tikhonov Regularization......Page 106
4.3 Resolution, Bias, and Uncertainty in the Tikhonov Solution......Page 110
4.4 Higher-Order Tikhonov Regularization......Page 114
4.5 Resolution in Higher-order Tikhonov Regularization......Page 122
4.6 The TGSVD Method......Page 124
4.7 Generalized Cross-Validation......Page 126
4.8 Error Bounds......Page 130
4.9 Exercises......Page 135
4.10 Notes and Further Reading......Page 138
5.1 Discretization by Expansion of the Model......Page 140
5.2 Using Representers as Basis Functions......Page 144
5.3 The Method of Backus and Gilbert......Page 145
5.4 Exercises......Page 150
5.5 Notes and Further Reading......Page 151
6.1 Introduction......Page 152
6.2 Iterative Methods for Tomography Problems......Page 153
6.3 The Conjugate Gradient Method......Page 161
6.4 The CGLS Method......Page 166
6.5 Resolution Analysis for Iterative Methods......Page 171
6.6 Exercises......Page 177
6.7 Notes and Further Reading......Page 179
7.1 Using Bounds as Constraints......Page 180
7.2 Sparsity Regularization......Page 185
7.3 Using IRLS to Solve L1 Regularized Problems......Page 187
7.4 Total Variation......Page 197
7.5 Exercises......Page 202
7.6 Notes and Further Reading......Page 203
8.1 Linear Systems in the Time and Frequency Domains......Page 204
8.2 Linear Systems in Discrete Time......Page 210
8.3 Water Level Regularization......Page 215
8.4 Tikhonov Regularization in the Frequency Domain......Page 219
8.5 Exercises......Page 225
8.6 Notes and Further Reading......Page 226
9.2 Newton's Method for Solving Nonlinear Equations......Page 228
9.3 The Gauss-Newton and Levenberg-Marquardt Methods for Solving Nonlinear Least Squares Problems......Page 231
9.4 Statistical Aspects of Nonlinear Least Squares......Page 235
9.5 Implementation Issues......Page 239
9.6 Exercises......Page 245
9.7 Notes and Further Reading......Page 248
10.1 Regularizing Nonlinear Least Squares Problems......Page 250
10.2 Occam's Inversion......Page 255
10.3 Model Resolution in Nonlinear Inverse Problems......Page 259
10.4 Exercises......Page 262
10.5 Notes and Further Reading......Page 263
11.1 Review of the Classical Approach......Page 264
11.2 The Bayesian Approach......Page 266
11.3 The Multivariate Normal Case......Page 271
11.4 The Markov Chain Monte Carlo Method......Page 280
11.5 Analyzing MCMC Output......Page 284
11.6 Exercises......Page 289
11.7 Notes and Further Reading......Page 291
12 Epilogue......Page 292
A.1 Systems of Linear Equations......Page 294
A.2 Matrix and Vector Algebra......Page 297
A.3 Linear Independence......Page 303
A.4 Subspaces of Rn......Page 304
A.5 Orthogonality and the Dot Product......Page 309
A.6 Eigenvalues and Eigenvectors......Page 313
A.7 Vector and Matrix Norms......Page 315
A.8 The Condition Number of a Linear System......Page 317
A.9 The QR Factorization......Page 319
A.10 Complex Matrices and Vectors......Page 321
A.11 Linear Algebra in Spaces of Functions......Page 322
A.12 Exercises......Page 323
A.13 Notes and Further Reading......Page 325
B.1 Probability and Random Variables......Page 326
B.2 Expected Value and Variance......Page 332
B.3 Joint Distributions......Page 334
B.4 Conditional Probability......Page 337
B.5 The Multivariate Normal Distribution......Page 340
B.7 Testing for Normality......Page 341
B.8 Estimating Means and Confidence Intervals......Page 343
B.9 Hypothesis Tests......Page 345
B.10 Exercises......Page 347
B.11 Notes and Further Reading......Page 348
C.1 The Gradient, Hessian, and Jacobian......Page 350
C.3 Lagrange Multipliers......Page 352
C.4 Exercises......Page 355
C.5 Notes and Further Reading......Page 356
D. Glossary of Notation......Page 358
Bibliography......Page 360
C......Page 366
H......Page 367
M......Page 368
O......Page 369
S......Page 370
Z......Page 371