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Inverse and Ill-Posed Problems

E-BookPDFDRM AdobeE-Book
580 Seiten
Englisch
Elsevier Science & Techn.erschienen am10.05.2014
Inverse and Ill-Posed Problemsmehr

Produkt

KlappentextInverse and Ill-Posed Problems
Details
Weitere ISBN/GTIN9781483272658
ProduktartE-Book
EinbandartE-Book
FormatPDF
Format HinweisDRM Adobe
Erscheinungsjahr2014
Erscheinungsdatum10.05.2014
Seiten580 Seiten
SpracheEnglisch
Artikel-Nr.3165136
Rubriken
Genre9200

Inhalt/Kritik

Inhaltsverzeichnis
1;Front Cover;1
2;Inverse and Ill-Posed Problems;4
3;Copyright Page;5
4;Table of Contents;6
5;Contributors;10
6;Preface;14
7;CHAPTER 1. A FEW GEOMETRICAL FEATURES OF INVERSE AND ILL-POSED PROBLEMS;18
7.1;I. INTRODUCTION;18
7.2;II. WELL-POSED QUESTIONS FOR (STRONGLY) ILL-POSED PROBLEMS;20
7.3;III. JOINT GENERALIZED COORDINATES;27
7.4;IV. JOINT TRAJECTORIES;32
7.5;REFERENCES;34
8;CHAPTER 2. THE INVERSE PROBLEM OF AQUIFER TRANSMISSIVITY IDENTIFICATION;36
8.1;ABSTRACT;36
8.2;1. INTRODUCTION;37
8.3;2. COMPUTATIONAL METHODS FOR TRANSMISSIVITY IDENTIFICATION;38
8.4;3. THE LINEAR FUNCTIONAL STRATEGY;40
8.5;ACKNOWLEDGEMENT;45
8.6;REFERENCES;45
9;CHAPTER 3. RELIABILITY OF INFORMATION OBTAINED FROM APPROXIMATELY-SOLVED PROBLEMS;46
9.1;I. ANALYSIS OF APPROXIMATE SOLUTIONS;46
9.2;II. APPROXIMATE SOLUTION SETS;48
9.3;III. PRACTICAL IMPLICATIONS;50
9.4;REFERENCES;52
10;CHAPTER 4. THREE TOPICS IN ILL-POSED PROBLEMS;54
10.1;I. INTRODUCTION;54
10.2;II. PARTIAL SPLINE METHODS FOR INCLUDING DISCONTINUITIES IN OTHERWISE SMOOTH REGULARIZED SOLUTIONS OF ILL POSED PROBLEMS WITH NOISY DATA;56
10.3;III. COMPUTATIONAL PROBLEMS COMMON TO PARTIAL SPLINE MODELS;62
10.4;IV.THE USE OF GCV AS A STOPPING RULE IN THE ITERATIVE SOLUTION OF LARGE LINEAR SYSTEMS.;64
10.5;V. GCV AND CONSTRAINED REGULARIZATION FOR THE PARAMETER ESTIMATION PROBLEM;65
10.6;REFERENCES;66
11;CHAPTER 5. A NEW APPROACH TO CLASSIFICATION AND REGULARIZATION OF ILL-POSED OPERATOR EQUATIONS;70
11.1;1. INTRODUCTION;70
11.2;2. ON THE ROLE OF OUTER INVERSES IN "SOLVABILITY" AND "REGULAJRIZATION" OF ILL-POSED PROBLEMS;73
11.3;3. REGULARIZERS OF TYPES I AND II. APPROXIMATE OUTER AND APPROXIMATE RIGHT INVERSES;79
11.4;4. CHARACTERIZATIONS OF ILL-POSED PROBLEMS OF TYPE I OR II;82
11.5;5. REMARKS;87
11.6;REFERENCES;90
12;CHAPTER 6. ON THE OPTIMALITY OF REGULARIZATION METHODS;94
12.1;I. INTRODUCTION;94
12.2;II . OPTIMALITY DEFINITIONS: POLLUTED RIGHT-HAND TERM;95
12.3;III. OPTIMALITY DEFINITIONS: OPERATOR POLLUTED ALSO;96
12.4;IV. CONSTRUCTION OP OL-OPTIMAL METHODS;97
12.5;V. QUASIOPTIMALITY CONDITIONS;98
12.6;VI. OPTIMALITY OP TIKHONOV METHOD;99
12.7;VII. QUASIOPTIMAL CHOICES OF PARAMETER IN TIKHONOV METHOD;102
12.8;VIII. OPTIMALITY ON THE SOURCE SETS;103
12.9;IX. OPTIMALITY OP LAVRENTIEV, TIKHONOV AND ITERATION METHODS ON SOURCE SETS;105
12.10;X. DISCREPANCY PRINCIPLE AND QUASIOPTIMALITY ON SOURCE SETS;109
12.11;REFERENCES;111
13;CHAPTER 7. OPTIMAL PARAMETER CHOICE FOR ORDINARY AND ITERATED TIKHONOV REGULAR!ZATION;114
13.1;ABSTRACT;114
13.2;I. INTRODUCTION;115
13.3;II. PARAMETER CHOICE FOR ITERATED TIKHONOV REGULARIZATION;119
13.4;III. A VARIANT OF MARTI'S METHOD;129
13.5;REFERENCES;141
14;CHAPTER 8. PARAMETER CHOICE FOR TIKHONOV REGULARIZATION OF ILL-POSED PROBLEMS;144
14.1;ABSTRACT;144
14.2;1. INTRODUCTION;145
14.3;2. OPTIMAL CHOICE OF THE REGULARIZATION PARAMETER FOR ITERATED TIKHONOV REGULARIZATION;147
14.4;3. FINITE DIMENSIONAL APPROXIMATIONS;151
14.5;REFERENCES;165
15;CHAPTER 9. FREDHOLM INTEGRAL EQUATIONS OF FIRST KIND AND THE METHOD OF CORRELOGRAM;168
15.1;0. INTRODUCTION;168
15.2;I. ASYMPTOTIC CONVERGENCE OF EIGENFUNCTION EXPANSIONS;168
15.3;II. PROBABILISTIC METHODS;174
15.4;III. THE METHOD OF CORRELOGRAM;178
15.5;REFERENCES;181
16;CHAPTER 10. ON ILL-POSED PROBLEMS AND THE METHOD OF CONJUGATE GRADIENTS;182
16.1;I. INTRODUCTION;182
16.2;II. THE v-METHOD;185
16.3;III. THE CONVERGENCE RATE OF THE cg-METHOD;187
16.4;IV. INEXACT DATA;189
16.5;V. SOME CONCLUDING REMARKS;191
16.6;REFERENCES;192
17;CHAPTER 11. CONVERGENCE OF THE CONJUGATE GRADIENT METHOD FOR COMPACT OPERATORS;194
17.1;I. INTRODUCTION;194
17.2;II.PRELIMINARIES ON CG AND ILL-POSED PROBLEMS;195
17.3;III. THE ORDER OF CONVERGENCE OF CG;197
17.4;REMARK :;200
17.5;REFERENCES;200
18;CHAPTER 12. COMPARISON PRINCIPLES FOR ITERATIVE METHODS;202
18.1;I. THE COMPARISON PRINCIPLE;203
18.2;II. THE LANDWEBER METHOD;205
18.3;III. AN IMPLICITE ITERATIVE METHOD;207
18.4;REFERENCES;210
19;CHAPTER 13. COMPUTATION OF ROUGH SOLUTIONS OF ABEL INTEGRAL EQUATIONS;212
19.1;I. INTRODUCTION: INTEGRAL EQUATION;212
19.2;II. DISCRETIZATION;215
19.3;III. PROOF OF CONVERGENCE;219
19.4;IV. ANUMERICAL CASE STUDY;224
19.5;ACKNOWLEDGEMENTS;226
19.6;REFERENCES;227
20;CHAPTER 14. ITERATIVE METHODS FOR THE APPROXIMATE SOLUTION OF ILL-POSED PROBLEMS WITH A PRIORI INFORMATION AND THEIR APPLICATIONS;228
20.1;I. INTRODUCTION;229
20.2;II. WEAK CONVERGENCE OF ITERATES;231
20.3;III. QUASI-CONTRACTIONS FOR CONVEX CONSTRAINTS;234
20.4;IV. APPLICATION TO LINEAR EQUATIONS;236
20.5;V. APPLICATION TO NONLINEAR EQUATIONS;238
20.6;VI. NUMERICAL EXPERIMENTS;240
20.7;REFERENCES;245
21;CHAPTER 15. An Overview of Numerical Methods for Nonlinear Ill-Posed Problems;248
21.1;I. INTRODUCTION;248
21.2;II. APPROXIMATE SOLUTION METHODS;250
21.3;III. THE LEVENBERG-MARQUARDT METHOD;252
21.4;IV. THE PENALIZED LEAST SQUARES METHOD;255
21.5;V. THE CONSTRAINED LEAST SQUARES METHOD;258
21.6;ACKNOWLEDGEMENTS;260
21.7;REFERENCES;261
22;CHAPTER 16. SEVERELY ILL-POSED RADON PROBLEMS;264
22.1;ABSTRACT;264
22.2;1. INTRODUCTION;264
22.3;2. THE RADON INVERSION FORMULA;267
22.4;3. PROJECTION COMPLETION METHODS;269
22.5;4. RECONSTRUCTION ALGORITHMS FOR INCOMPLETE DATA PROBLEMS;271
22.6;5. THE INTERIOR PROBLEM;273
22.7;REFERENCES;275
23;CHAPTER 17. PROJECTION THEOREMS FOR FAR FIELD PATTERNS AND THE INVERSE SCATTERING PROBLEM;278
23.1;I. INTRODUCTION;278
23.2;II. THE INVERSE SCATTERING PROBLEM FOR ACOUSTIC WAVES;280
23.3;III. A PROJECTION THEOREM FOR FAR FIELD PATTERNS;282
23.4;IV. GENERALIZED HERGLOTZ DOMAINS AND THE RAYLEIGH HYPOTHESIS;285
23.5;V. OPTIMAL SOLUTIONS OF THE INVERSE SCATTERING PROBLEM;288
23.6;REFERENCES;292
24;CHAPTER 18. A NUMERICAL METHOD FOR AN INVERSE SCATTERING PROBLEM;296
24.1;1. Introduction;296
24.2;2. The Method;298
24.3;3. A Comparison between our method and the one by Colton and Monk;299
24.4;4. Choice of the Regularizing Norm;300
24.5;5. Numerical Examples;302
24.6;6. References;303
25;CHAPTER 19. APPLIED INVERSE PROBLEMS IN OPTICS;308
25.1;I. INTRODUCTION;309
25.2;II. EXPERIMENTAL DATA INVERSION;310
25.3;III. THE CONFOCAL SCANNING MICROSCOPE;317
25.4;IV. PARTICLE SIZING AND VELOCIMETRY;322
25.5;REFERENCES;329
26;CHAPTER 20. SOME REMARKS ON LOCATING RADIATION SOURCES;332
26.1;ABSTRACT;332
26.2;REFERENCES AND RELATED LITERATURE;340
27;CHAPTER 21. ON THE APPROXIMATE SOLUTION OF A TWO-DIMENSIONAL INVERSE HEAT CONDUCTION PROBLEM;342
27.1;1. INTRODUCTION;342
27.2;2. STATEMENT OF THE PROBLEM;343
27.3;3. SEQUENTIAL REGULARIZATION FOR A ONE-DIMENSIONAL PROBLEM;346
27.4;4. APPROXIMATE SOLUTION OF ONE-DIMENSIONAL IHCP's;347
27.5;5. AN APPROXIMATION METHOD FOR THE SOLUTION OF TWO-DIMENSIONAL IHCP's;350
27.6;6. COMPUTATIONAL RESULTS;355
27.7;REFERENCES;361
27.8;ACKNOWLEDGEMENTS;361
28;CHAPTER 22. MODIFIED EQUATIONS FOR APPROXIMATING THE SOLUTION OF A CAUCHY PROBLEM FOR THE HEAT EQUATION;362
28.1;1. INTRODUCTION;362
28.2;2. DERIVATION OF THE MODIFIED EQUATIONS;363
28.3;3. ERROR ESTIMATES FOR THE MODIFIED EQUATIONS;365
28.4;REFERENCES;367
29;CHAPTER 23. STABILITY ESTIMATES FOR ILL-POSED CAUCHY PROBLEMS FOR PARABOLIC EQUATIONS;368
29.1;1. INTRODUCTION;368
29.2;2.RECONSTRUCTION FROM INTERIOR OBSERVATION FOR THE HEAT EQUATION IN THE QUARTER PLANE;369
29.3;3. THE HEAT EQUATION WITH NONCHARACTERISTIC CHAUCHY DATA ON THE (HALF) LINE;374
29.4;4. LINEAR PARABOLIC EQUATIONS WITH NONCHARACTERISTIC CAUCHY DATA: SMOOTH COEFFICIENTS;377
29.5;5. LINEAR PARABOLIC EQUATIONS WITH NONCHARACTERISTIC CAUCHY DATA: IRREGULAR COEFFICIENTS;380
29.6;REFERENCES;385
30;CHAPTER 24. A BOUNDARY ELEMENT COLLOCATION METHOD FOR THE NEUMANN PROBLEM OF THE HEAT EQUATION;386
30.1;I. INTRODUCTION;386
30.2;II. NEUMANN PROBLEM;387
30.3;III. BOUNDARY INTEGRAL EQUATION;389
30.4;IV. SOLUTION OF THE INTEGRAL EQUATION;394
30.5;V. APPROXIMATION;399
30.6;ACKNOWLEDGEMENTS;401
30.7;REFERENCES;401
31;CHAPTER 25. SUFFICIENT CONDITIONS FOR THE SOLUTION OF THE INVERSE VIBRATING BEAM PROBLEM;402
31.1;1. PRELIMINARIES;402
31.2;2. NECESSARY CONDITIONS;403
31.3;3. DISCRETE APPROXIMATIONS;408
31.4;4. THE CONTINUUM LIMIT;410
31.5;ACKNOWLEDGEMENTS;413
31.6;REFERENCES;414
32;CHAPTER 26. ON STABILIZING ILL-POSED PROBLEMS AGAINST ERRORS IN GEOMETRY AND MODELING;416
32.1;I. INTRODUCTION;416
32.2;II. CONTINUOUS DEPENDENCE ON GEOMETRY;419
32.3;III. CONTINUOUS DEPENDENCE ON MODELING;426
32.4;REFERENCES;433
33;CHAPTER 27. ON AN ILL-POSED PROBLEM FOR CONSTANT ALPHA FORCE-FREE FIELDS;434
33.1;1. INTRODUCTION;434
33.2;2. THE INITIAL BOUNDARY VALUE PROBLEM;435
33.3;3. LEAST SQUARES SOLUTIONS;437
33.4;ACKNOWLEDGEMENT;439
33.5;REFERENCES;440
34;CHAPTER 28. SOME INVERSE AND ILL-POSED PROBLEMS IN COMPUTATIONAL FLUID DYNAMICS;442
34.1;1. Abstract;442
34.2;2. The First Biharmonic Equation;442
34.3;3. Double Splitting ADI for the First Biharmonic;444
34.4;4. Separation and Reattachment of Viscous Flows;451
34.5;5. The Arieli-Murphy Algorithm;455
34.6;REFERENCES;457
35;CHAPTER 29. IMPROVED CONTINUOUS DEPENDENCE RESULTS FOR A CLASS OF EVOLUTIONARY EQUATIONS;460
35.1;I. INTRODUCTION;460
35.2;II. PROBLEM;461
35.3;III. SPECIAL CASE;463
35.4;IV. GENERAL CASE;466
35.5;REFERENCES;467
36;CHAPTER 30. SOME BOUNDARY VALUE PROBLEMS FOR THE WAVE EQUATION;468
36.1;References;476
37;CHAPTER 31. ON THE LOW FREQUENCY ASYMPTOTICS OF THE EXTERIOR 2-D DIRICHLET PROBLEM IN DYNAMIC ELASTICITY;478
37.1;ABSTRACT;478
37.2;I. INTRODUCTION;479
37.3;II. BOUNDARY INTEGRAL EQUATIONS;482
37.4;III. LOW FREQUENCY ASYMPTOTICS;487
37.5;IV. SPLINE BOUNDARY ELEMENT APPROXIMATIONS AND ILL POSEDNESS;492
37.6;REFERENCES;498
38;CHAPTER 32. INVERSE AND ILL-POSED PROBLEMS IN RESERVOIR SIMULATION;500
38.1;ABSTRACT;500
38.2;I. INTRODUCTION;501
38.3;II. ILL-POSEDNESS OF CERTAIN MODEL PROBLEMS;502
38.4;III. PARAMETER ESTIMATION PROBLEMS;505
38.5;REFERENCES;512
39;CHAPTER 33. RATE OF CONVERGENCE FOR THE ESTIMATION OF A COEFFICIENT IN A TWO POINT BOUNDARY VALUE PROBLEM;516
39.1;1. INTRODUCTION;516
39.2;2. ESTIMATE OF THE RATE OF CONVERGENCE;517
39.3;REFERENCES;526
40;CHAPTER 34. IDENTIFIABILITY OF DISTRIBUTED PARAMETERS;530
40.1;REFERENCES;538
41;CHAPTER 35. ON THE REGULARIZATION OF LINEAR DIFFERENTIAL-ALGEBRAIC EQUATIONS;540
41.1;References;556
42;CHAPTER 36. LIMITS OF ABSTRACT SPLINES;558
42.1;0. Introduction;558
42.2;1. SPLINES AND PSEUDO-INVERSES;558
42.3;2. CONVERGENCE OF SPLINE PROJECTORS;561
42.4;3. CARDINAL SPLINES;563
42.5;4. TENSOR PRODUCT SPLINES;567
42.6;5. BLENDING SPLINES;570
42.7;REFERENCES;573
43;LIST OF PARTICIPANTS;576
44;NOTES AND REPORTS IN MATHEMATICS IN SCIENCE AND ENGINEERING;585mehr