Parameter Estimation and Inverse Problems (International Geophysics)

✍ Scribed by Richard Aster, Brian Borchers, Clifford Thurber

• Publisher: Academic Press
• Year: 2005
• Tongue: English
• Leaves: 316
• Series: International Geophysics Series
• Category: Library

✦ Synopsis

This is one of my favorite books, it is a great reference for anyone working on signal processing algorithms for image reconstruction, & data estimation. It is well written, and very easy to understand, it has succint expressions with good intuitive explanations. It provides good examples that make it even easier to see the use and advantages of different techniques. I do recommend the book for anyone with enough math background only, it is not entry level.

✦ Table of Contents

1.1 CLASSIFICATION OF INVERSE PROBLEMS......Page 14
1.2 EXAMPLES OF PARAMETER ESTIMATION PROBLEMS......Page 17
1.3 EXAMPLES OF INVERSE PROBLEMS......Page 20
1.4 WHY INVERSE PROBLEMS ARE HARD......Page 24
1.6 NOTES AND FURTHER READING......Page 27
2.1 INTRODUCTION TO LINEAR REGRESSION......Page 28
2.2 STATISTICAL ASPECTS OF LEAST SQUARES......Page 30
2.3 UNKNOWN MEASUREMENT STANDARD DEVIATIONS......Page 39
2.4 L1 REGRESSION......Page 43
2.5 MONTE CARLO ERROR PROPAGATION......Page 48
2.6 EXERCISES......Page 49
2.7 NOTES AND FURTHER READING......Page 53
3.2 QUADRATURE METHODS......Page 54
3.3 EXPANSION IN TERMS OF REPRESENTERS......Page 59
3.4 EXPANSION IN TERMS OF ORTHONORMAL BASIS FUNCTIONS......Page 60
3.5 THE METHOD OF BACKUS AND GILBERT......Page 61
3.6 EXERCISES......Page 65
3.7 NOTES AND FURTHER READING......Page 67
4.1 THE SVD AND THE GENERALIZED INVERSE......Page 68
4.2 COVARIANCE AND RESOLUTION OF THE GENERALIZED INVERSE SOLUTION......Page 75
4.3 INSTABILITY OF THE GENERALIZED INVERSE SOLUTION......Page 77
4.4 AN EXAMPLE OF A RANK-DEFICIENT PROBLEM......Page 80
4.5 DISCRETE ILL-POSED PROBLEMS......Page 86
4.6 EXERCISES......Page 98
4.7 NOTES AND FURTHER READING......Page 100
5.1 SELECTING A GOOD SOLUTION......Page 102
5.2 SVD IMPLEMENTATION OF TIKHONOV REGULARIZATION......Page 104
5.3 RESOLUTION, BIAS, AND UNCERTAINTY IN THE TIKHONOV SOLUTION......Page 108
5.4 HIGHER-ORDER TIKHONOV REGULARIZATION......Page 111
5.5 RESOLUTION IN HIGHER-ORDER TIKHONOV REGULARIZATION......Page 116
5.6 THE TGSVD METHOD......Page 118
5.7 GENERALIZED CROSS VALIDATION......Page 119
5.8 ERROR BOUNDS......Page 122
5.9 EXERCISES......Page 127
5.10 NOTES AND FURTHER READING......Page 130
6.1 INTRODUCTION......Page 132
6.2 ITERATIVE METHODS FOR TOMOGRAPHY PROBLEMS......Page 133
6.3 THE CONJUGATE GRADIENT METHOD......Page 139
6.4 THE CGLS METHOD......Page 144
6.5 EXERCISES......Page 148
6.6 NOTES AND FURTHER READING......Page 149
7.1 USING BOUNDS AS CONSTRAINTS......Page 152
7.2 MAXIMUM ENTROPY REGULARIZATION......Page 156
7.3 TOTAL VARIATION......Page 159
7.4 EXERCISES......Page 164
7.5 NOTES AND FURTHER READING......Page 165
8.1 LINEAR SYSTEMS IN THE TIME AND FREQUENCY DOMAINS......Page 166
8.2 DECONVOLUTION FROMA FOURIER PERSPECTIVE......Page 171
8.3 LINEAR SYSTEMS IN DISCRETE TIME......Page 174
8.4 WATER LEVEL REGULARIZATION......Page 177
8.5 EXERCISES......Page 181
8.6 NOTES AND FURTHER READING......Page 183
9.1 NEWTON’S METHOD......Page 184
9.2 THE GAUSS–NEWTON AND LEVENBERG–MARQUARDT METHODS......Page 187
9.3 STATISTICAL ASPECTS......Page 190
9.4 IMPLEMENTATION ISSUES......Page 194
9.5 EXERCISES......Page 199
9.6 NOTES AND FURTHER READING......Page 202
10.1 REGULARIZING NONLINEAR LEAST SQUARES PROBLEMS......Page 204
10.2 OCCAM’S INVERSION......Page 208
10.4 NOTES AND FURTHER READING......Page 212
11.1 REVIEW OF THE CLASSICAL APPROACH......Page 214
11.2 THE BAYESIAN APPROACH......Page 215
11.3 THE MULTIVARIATE NORMAL CASE......Page 220
11.4 MAXIMUM ENTROPY METHODS......Page 225
11.5 EPILOGUE......Page 227
11.6 EXERCISES......Page 229
11.7 NOTES AND FURTHER READING......Page 230
A.1 SYSTEMS OF LINEAR EQUATIONS......Page 232
A.2 MATRIX AND VECTOR ALGEBRA......Page 235
A.3 LINEAR INDEPENDENCE......Page 241
A.4 SUBSPACES OF Rn......Page 242
A.5 ORTHOGONALITY AND THE DOT PRODUCT......Page 246
A.6 EIGENVALUES AND EIGENVECTORS......Page 250
A.7 VECTOR AND MATRIX NORMS......Page 253
A.8 THE CONDITION NUMBER OF A LINEAR SYSTEM......Page 255
A.9 THE QR FACTORIZATION......Page 257
A.10 LINEAR ALGEBRA IN SPACES OF FUNCTIONS......Page 258
A.11 EXERCISES......Page 260
A.12 NOTES AND FURTHER READING......Page 262
B.1 PROBABILITY AND RANDOM VARIABLES......Page 264
B.2 EXPECTED VALUE AND VARIANCE......Page 270
B.3 JOINT DISTRIBUTIONS......Page 271
B.4 CONDITIONAL PROBABILITY......Page 275
B.5 THE MULTIVARIATE NORMAL DISTRIBUTION......Page 277
B.7 TESTING FOR NORMALITY......Page 278
B.8 ESTIMATING MEANS AND CONFIDENCE INTERVALS......Page 280
B.9 HYPOTHESIS TESTS......Page 282
B.10 EXERCISES......Page 284
B.11 NOTES AND FURTHER READING......Page 285
C.1 THE GRADIENT, HESSIAN, AND JACOBIAN......Page 286
C.2 TAYLOR’S THEOREM......Page 288
C.3 LAGRANGE MULTIPLIERS......Page 289
C.4 EXERCISES......Page 291
C.5 NOTES AND FURTHER READING......Page 293
Appendix D GLOSSARY OF NOTATION......Page 294
BIBLIOGRAPHY......Page 296
INDEX......Page 304
International Geophysics Series......Page 310
About the CD-ROM......Page 316

✦ Subjects

Физика;Матметоды и моделирование в физике;