Parameter Estimation and Inverse Problems

by Aster; Borchers; Thurber

• ISBN13: 9780120656042
• ISBN10: 0120656043
• Edition: CD
• Format: Hardcover
• Copyright: 2004-12-28
• Publisher: Elsevier Science

Summary

Parameter Estimation and Inverse Problems primarily serves as a textbook for advanced undergraduate and introductory graduate courses. Class notes have been developed and reside on the World Wide Web for faciliting use and feedback by teaching colleagues. The authors' treatment promotes an understanding of fundamental and practical issus associated with parameter fitting and inverse problems including basic theory of inverse problems, statistical issues, computational issues, and an understanding of how to analyze the success and limitations of solutions to these probles. The text is also a practical resource for general students and professional researchers, where techniques and concepts can be readily picked up on a chapter-by-chapter basis. Parameter Estimation and Inverse Problems is structured around a course at New Mexico Tech and is designed to be accessible to typical graduate students in the physical sciences who may not have an extensive mathematical background. It is accompanied by a Web site that contains Matlab code corresponding to all examples. * Designed to be accessible to graduate students and professionals in physical sciences without an extensive mathematical background * Includes three appendices for review of linear algebra and crucial concepts in statistics * Battle-tested in courses at several universities *MATLAB exercises facilitate exploration of material

Table of Contents

Preface	xi
Introduction	1	(14)
Classification of Inverse Problems	1	(3)
Examples of Parameter Estimation Problems	4	(3)
Examples of Inverse Problems	7	(4)
Why Inverse Problems Are Hard	11	(3)
Exercises	14	(1)
Notes and Further Reading	14	(1)
Linear Regression	15	(26)
Introduction to Linear Regression	15	(2)
Statistical Aspects of Least Squares	17	(9)
Unknown Measurement Standard Deviations	26	(4)
L1 Regression	30	(5)
Monte Carlo Error Propagation	35	(1)
Exercises	36	(4)
Notes and Further Reading	40	(1)
Discretizing Continuous Inverse Problems	41	(14)
Integral Equations	41	(1)
Quadrature Methods	41	(5)
Expansion in Terms of Representers	46	(1)
Expansion in Terms of Orthonormal Basis Functions	47	(1)
The Method of Backus and Gilbert	48	(4)
Exercises	52	(2)
Notes and Further Reading	54	(1)
Rank Deficiency and Ill-Conditioning	55	(34)
The SVD and the Generalized Inverse	55	(7)
Covariance and Resolution of the Generalized Inverse Solution	62	(2)
Instability of the Generalized Inverse Solution	64	(3)
An Example of a Rank-Deficient Problem	67	(6)
Discrete Ill-Posed Problems	73	(12)
Exercises	85	(2)
Notes and Further Reading	87	(2)
Tikhonov Regularization	89	(30)
Selecting a Good Solution	89	(2)
SVD Implementation of Tikhonov Regularization	91	(4)
Resolution, Bias, and Uncertainty in the Tikhonov Solution	95	(3)
Higher-Order Tikhonov Regularization	98	(5)
Resolution in Higher-Order Tikhonov Regularization	103	(2)
The TGSVD Method	105	(1)
Generalized Cross Validation	106	(3)
Error Bounds	109	(5)
Exercises	114	(3)
Notes and Further Reading	117	(2)
Iterative Methods	119	(20)
Introduction	119	(1)
Iterative Methods for Tomography Problems	120	(6)
The Conjugate Gradient Method	126	(5)
The CGLS Method	131	(4)
Exercises	135	(1)
Notes and Further Reading	136	(3)
Additional Regularization Techniques	139	(14)
Using Bounds as Constraints	139	(4)
Maximum Entropy Regularization	143	(3)
Total Variation	146	(5)
Exercises	151	(1)
Notes and Further Reading	152	(1)
Fourier Techniques	153	(18)
Linear Systems in the Time and Frequency Domains	153	(5)
Deconvolution from a Fourier Perspective	158	(3)
Linear Systems in Discrete Time	161	(3)
Water Level Regularization	164	(4)
Exercises	168	(2)
Notes and Further Reading	170	(1)
Nonlinear Regression	171	(20)
Newton's Method	171	(3)
The Gauss-Newton and Levenberg-Marquardt Methods	174	(3)
Statistical Aspects	177	(4)
Implementation Issues	181	(5)
Exercises	186	(3)
Notes and Further Reading	189	(2)
Nonlinear Inverse Problems	191	(10)
Regularizing Nonlinear Least Squares Problems	191	(4)
Occam's Inversion	195	(4)
Exercises	199	(1)
Notes and Further Reading	199	(2)
Bayesian Methods	201	(18)
Review of the Classical Approach	201	(1)
The Bayesian Approach	202	(5)
The Multivariate Normal Case	207	(5)
Maximum Entropy Methods	212	(2)
Epilogue	214	(2)
Exercises	216	(1)
Notes and Further Reading	217	(2)
A REVIEW OF LINEAR ALGEBRA	219	(32)
A.1 Systems of Linear Equations	219	(3)
A.2 Matrix and Vector Algebra	222	(6)
A.3 Linear Independence	228	(1)
A.4 Subspaces of Rn	229	(4)
A.5 Orthogonality and the Dot Product	233	(4)
A.6 Eigenvalues and Eigenvectors	237	(3)
A.7 Vector and Matrix Norms	240	(2)
A.8 The Condition Number of a Linear System	242	(2)
A.9 The QR Factorization	244	(1)
A.10 Linear Algebra in Spaces of Functions	245	(2)
A.11 Exercises	247	(2)
A.12 Notes and Further Reading	249	(2)
B REVIEW OF PROBABILITY AND STATISTICS	251	(22)
B.1 Probability and Random Variables	251	(6)
B.2 Expected Value and Variance	257	(1)
B.3 Joint Distributions	258	(4)
B.4 Conditional Probability	262	(2)
B.5 The Multivariate Normal Distribution	264	(1)
B.6 The Central Limit Theorem	265	(1)
B.7 Testing for Normality	265	(2)
B.8 Estimating Means and Confidence Intervals	267	(2)
B.9 Hypothesis Tests	269	(2)
B.10 Exercises	271	(1)
B.11 Notes and Further Reading	272	(1)
C REVIEW OF VECTOR CALCULUS	273	(8)
C.1 The Gradient, Hessian, and Jacobian	273	(2)
C.2 Taylor's Theorem	275	(1)
C.3 Lagrange Multipliers	276	(2)
C.4 Exercises	278	(2)
C.5 Notes and Further Reading	280	(1)
D GLOSSARY OF NOTATION	281	(2)
Bibliography	283	(8)
Index	291