Random fields estimation / Alexander G. Ramm.

Format:

Book

Author/Creator:

Ramm, A. G. (Alexander G.)

Contributor:

Ramm, A. G. (Alexander G.).

Language:

English

Subjects (All):

Random fields.

Estimation theory.

Physical Description:

xiii, 373 pages ; 24 cm

Place of Publication:

Hackensack, NJ : World Scientific, [2006]

Summary:

This book contains a novel theory of random fields estimation of Wiener type, developed originally by the author and presented here. No assumption about the Gaussian or Markovian nature of the fields are made. The theory, constructed entirely within the framework of covariancetheory, is based on a detailed analytical study of a new class of multidimensional integral equations basic in estimation theory.

This book is suitable for graduate courses in random fields estimation. It can also be used in courses in functional analysis, numerical analysis, integral equations, and scattering theory. Book jacket.

Contents:

2 Formulation of Basic Results 9

2.1 Statement of the problem 9

2.2 Formulation of the results (multidimensional case) 14

2.2.1 Basic results 14

2.2.2 Generalizations 17

2.3 Formulation of the results (one-dimensional case) 18

2.3.1 Basic results for the scalar equation 19

2.3.2 Vector equations 22

2.4 Examples of kernels of class R and solutions to the basic equation 25

2.5 Formula for the error of the optimal estimate 29

3 Numerical Solution of the Basic Integral Equation in Distributions 33

3.2 Theoretical approaches 37

3.3 Multidimensional equation 43

3.4 Numerical solution based on the approximation of the kernel 46

3.5 Asymptotic behavior of the optimal filter as the white noise component goes to zero 54

3.6 A general approach 57

4 Proofs 65

4.1 Proof of Theorem 2.1 65

4.2 Proof of Theorem 2.2 73

4.3 Proof of Theorems 2.4 and 2.5 79

4.4 Another approach 84

5 Singular Perturbation Theory for a Class of Fredholm Integral Equations Arising in Random Fields Estimation Theory 87

5.2 Auxiliary results 90

5.3 Asymptotics in the case n = 1 93

5.4 Examples of asymptotical solutions: case n = 1 98

5.5 Asymptotics in the case n > 1 103

5.6 Examples of asymptotical solutions: case n > 1 105

6 Estimation and Scattering Theory 111

6.1 The direct scattering problem 111

6.1.1 The direct scattering problem 111

6.1.2 Properties of the scattering solution 114

6.1.3 Properties of the scattering amplitude 120

6.1.4 Analyticity in k of the scattering solution 121

6.1.5 High-frequency behavior of the scattering solutions 123

6.1.6 Fundamental relation between u[superscript +] and u[superscript -] 127

6.1.7 Formula for det S(k) and state the Levinson Theorem 128

6.1.8 Completeness properties of the scattering solutions 131

6.2 Inverse scattering problems 134

6.2.1 Inverse scattering problems 134

6.2.2 Uniqueness theorem for the inverse scattering problem 134

6.2.3 Necessary conditions for a function to be a scattering amplitude 135

6.2.4 A Marchenko equation (M equation) 136

6.2.5 Characterization of the scattering data in the 3D inverse scattering problem 138

6.2.6 The Born inversion 141

6.3 Estimation theory and inverse scattering in R[superscript 3] 150

7 Applications 159

7.1 What is the optimal size of the domain on which the data are to be collected? 159

7.2 Discrimination of random fields against noisy background 161

7.3 Quasioptimal estimates of derivatives of random functions 169

7.3.2 Estimates of the derivatives 170

7.3.3 Derivatives of random functions 172

7.3.4 Finding critical points 180

7.3.5 Derivatives of random fields 181

7.4 Stable summation of orthogonal series and integrals with randomly perturbed coefficients 182

7.4.2 Stable summation of series 184

7.4.3 Method of multipliers 185

7.5 Resolution ability of linear systems 185

7.5.2 Resolution ability of linear systems 187

7.5.3 Optimization of resolution ability 191

7.5.4 A general definition of resolution ability 196

7.6 Ill-posed problems and estimation theory 198

7.6.2 Stable solution of ill-posed problems 205

7.6.3 Equations with random noise 216

7.7 A remark on nonlinear (polynomial) estimates 230

8 Auxiliary Results 233

8.1 Sobolev spaces and distributions 233

8.1.1 A general imbedding theorem 233

8.1.2 Sobolev spaces with negative indices 236

8.2 Eigenfunction expansions for elliptic selfadjoint operators 241

8.2.1 Resoluion of the identity and integral representation of selfadjoint operators 241

8.2.2 Differentiation of operator measures 242

8.2.3 Carleman operators 246

8.2.4 Elements of the spectral theory of elliptic operators in L[superscript 2](R[superscript r]) 249

8.3 Asymptotics of the spectrum of linear operators 260

8.3.1 Compact operators 260

8.3.1.2 Minimax principles and estimates of eigenvalues and singular values 262

8.3.2 Perturbations preserving asymptotics of the spectrum of compact operators 265

8.3.2.1 Statement of the problem 265

8.3.2.2 A characterization of the class of linear compact operators 266

8.3.2.3 Asymptotic equivalence of s-values of two operators 268

8.3.2.4 Estimate of the remainder 270

8.3.2.5 Unbounded operators 274

8.3.2.6 Asymptotics of eigenvalues 275

8.3.2.7 Asymptotics of eigenvalues (continuation) 283

8.3.2.8 Asymptotics of s-values 284

8.3.2.9 Asymptotics of the spectrum for quadratic forms 287

8.3.2.10 Proof of Theorem 2.3 293

8.3.3 Trace class and Hilbert-Schmidt operators 297

8.3.3.1 Trace class operators 297

8.3.3.2 Hilbert-Schmidt operators 298

8.3.3.3 Determinants of operators 299

8.4 Elements of probability theory 300

8.4.1 The probability space and basic definitions 300

8.4.2 Hilbert space theory 306

8.4.3 Estimation in Hilbert space L[superscript 2]([Omega], U, P) 310

8.4.4 Homogeneous and isotropic random fields 312

8.4.5 Estimation of parameters 315

8.4.6 Discrimination between hypotheses 317

8.4.7 Generalized random fields 319

8.4.8 Kalman filters 320

Appendix A Analytical Solution of the Basic Integral Equation for a Class of One-Dimensional Problems 325

A.2 Proofs 329

Appendix B Integral Operators Basic in Random Fields Estimation Theory 337

B.2 Reduction of the basic integral equation to a boundary-value problem 341

B.3 Isomorphism property 349

B.4 Auxiliary material 354.

Notes:

Based partly on the author's earlier book: Random fields estimation theory. Harlow, Essex, England : Longman Scientific & Technical ; New York : Wiley, 1990.

Includes bibliographical references (pages 363-369) and index.

Related Work:

Ramm, A. G. (Alexander G.). Random fields estimation theory.

ISBN:

9812565361

OCLC:

187707403

Publisher Number:

9789812565365