Adaptive methods of computing mathematics and mechanics: stochastic variant
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| Main Authors: | , , |
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| Format: | Book |
| Language: | English Russian |
| Published: |
Singapore [u.a.]
World Scientific
1999
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Aus dem Russ. übers. |
| Physical Description: | XIX, 416 S. graph. Darst. |
| ISBN: | 9810235011 |
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MARC
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| 240 | 1 | 0 | |a Adaptivnije metodij vychislitel'noj matematiki i mechaniki |
| 245 | 1 | 0 | |a Adaptive methods of computing mathematics and mechanics |b stochastic variant |c D. G. Arsenjev, V. M. Ivanov & O. Y. Kul'chitsky |
| 264 | 1 | |a Singapore [u.a.] |b World Scientific |c 1999 | |
| 300 | |a XIX, 416 S. |b graph. Darst. | ||
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| 650 | 4 | |a Mathematical models | |
| 650 | 4 | |a Mechanics, Analytic |x Data processing | |
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| adam_text | IMAGE 1
ADAPTIVE METHODS OF
COMPUTING MATHEMATICS AND MECHANICS STOCHASTIC VARIANT
D. G. ARSENJEV,
V. M. IVANOV & O. Y. KUL CHITSKY ST. PETERSBURG STATE TECHNICAL
UNIVERSITY, RUSSIA
W$ WORLD SCIENTIFIC WB SINGAPORE * NEW JERSEY * LONDON * HONG KONG
IMAGE 2
CONTENTS
FOREWORD V
PART I. EVALUATION OF INTEGRALS AND SOLUTION OF INTEGRAL EQUATIONS 1
CHAPTER 1. FUNDAMENTALS OF THE MONTE-CARLO M E T H OD 3
1.1. IDEA OF THE MONTE-CARLO METHOD 3
1.2. SIMULATION OF IMPLEMENTATION OF A SCALAR RANDOM VARIABLE 6
1.2.1. THE TRANSFORMING FUNCTIONS METHOD 6
1.2.2. THE SUPERPOSITION METHOD 9
1.2.3. THE SELECTION METHOD 11
1.3. SIMULATION OF IMPLEMENTATION OF A VECTOR RANDOM VARIABLE 14
1.4. EVALUATION OF DEFINITE INTEGRALS BY MEANS OF MONTE-CARLO METHOD 17
CHAPTER 2. EVALUATION OF INTEGRALS BY MEANS OF STATISTIC SIMULATION
EMPLOYING ADAPTATION 21
2.1. ADAPTATION IDEA IN STATISTIC METHODS OF NUMERICAL ANALYSIS, BASED
ON THE PRINCIPLES OF IMPORTANCE SAMPLING 21
2.2. ADAPTIVE ALGORITHM FOR EVALUATING ONE-DIMENSIONAL INTEGRAL 23
2.2.1. SELECTION OF PROBABILITY DENSITIES 23
2.2.2. EVALUATION PROCEDURE 27
2.2.3. RESULTS OF NUMERICAL EXPERIMENTS 28
2.2.4. REPORT ON THE RESULTS 29
2.3. ADAPTIVE ALGORITHM OF EVALUATION OF TWO-DIMENSIONAL AND
MULTI-DIMENSIONAL INTEGRALS 32
2.3.1. DESCRIPTION OF THE ALGORITHM 32
XLII
IMAGE 3
XIV CONTENTS
2.3.2. RESULTS OF NUMERICAL EXPERIMENTS 36
2.3.3. SOME COMMENTS 36
2.4. STOCHASTIC COMPUTING ALGORITHMS AS AN OBJECT OF ADAPTIVE CONTROL 37
2.4.1. INTRODUCTION 37
2.4.2. STATEMENT OF A PROBLEM OF CONTROL OVER THE PROCESS OF COMPUTATION
38
2.4.3. SYNTHESIS OF THE OPTIMAL CONTROL OVER THE PROCESS OF COMPUTATION
42
2.4.4. STRATEGY OF ADAPTIVE OPTIMIZATION OF COMPUTATION PROCESS 47
CHAPTER 3. SEMI-STATISTICAL M E T H OD OF NUMERICAL SOLVING INTEGRAL
EQUATIONS 49
3.1. INTRODUCTION 49
3.2. BASIC RELATIONS OF THE METHOD 50
3.3. RECURRENT INVERSION FORMULAE 52
3.4. CONVERGENCE OF THE METHOD 55
3.5. ADAPTIVE ABILITIES OF THE ALGORITHM 70
3.6. QUALITATIVE CONSIDERATIONS CONCERNING CONNECTIONS BETWEEN THE
SEMI-STATISTICAL AND VARIATIONAL METHODS . 72 3.7. APPLICATION OF THE
METHOD TO SINGULAR INTEGRAL EQUATIONS 72
3.7.1. DESCRIPTION AND APPLICATION OF THE METHOD 72
3.7.2. RECURRENT INVERSION FORMULAE 76
3.7.3. ANALYSIS OF THE METHOD S ERRORS 77
3.7.4. ADAPTIVE ABILITIES OF THE ALGORITHM 80
CHAPTER 4. PROJECTION-STATISTICAL M E T H OD OF NUMERICAL SOLUTION OF
INTEGRAL EQUATIONS 82
4.1. INTRODUCTION 82
4.2. BASIC RELATIONS OF THE METHOD 82
4.3. FORMULAE OF RECURRENT INVERSION 86
4.4. THE ALGORITHM CONVERGENCE 88
4.5. MERITS OF THE METHOD 99
4.6. ADAPTIVE ABILITIES 99
4.7. PECULIARITIES OF NUMERICAL IMPLEMENTATION 101
4.8. AN ALTERNATIVE COMPUTING TECHNIQUE: APPROXIMATE SOLUTIONS SHOULD BE
AVERAGED 103
4.9. NUMERICAL EXPERIMENTS 105
IMAGE 4
CONTENTS XV
4.9.1. A TEST PROBLEM 105
4.9.2. THE PROBLEM ON STEADY-STATE FORCED SMALL TRANSVERSE VIBRATION OF
PINNED STRING CAUSED BY HARMONIC FORCE 110
CHAPTER 5. T HE PROBLEM OF VIBRATION CONDUCTIVITY 116
5.1. THE BOUNDARY-VALUE PROBLEM OF VIBRATION CONDUCTIVITY 116 5.2.
INTEGRAL EQUATIONS OF VIBRATION CONDUCTIVITY 118
5.3. REGULARIZATION OF THE EQUATIONS 124
5.4. INTEGRAL EQUATIONS WITH IMPROVED ASYMPTOTIC PROPERTIES AT SMALL SS
130
5.5. NUMERICAL SOLUTION OF THE VIBRATION CONDUCTIVITY PROBLEMS 134
5.5.1. SOLUTION OF A TEST PROBLEM 134
5.5.2. RESEARCH OF THE INFMENCE OF DISTORTION OF A SPHERE AND CHARACTER
OF AN EXTERNAL LOAD ON THE RESULTS OF NUMERICAL SOLUTION 138
CHAPTER 6. T HE FIRST BASIC PROBLEM OF THE ELASTICITY T H E O RY 141
6.1. POTENTIALS AND INTEGRAL EQUATIONS OF THE FIRST BASIC PROBLEM OF THE
ELASTICITY THEORY . . . 142
6.1.1. THE FORCE AND PSEUDO-FORCE TENSORS 142
6.1.2. INTEGRAL EQUATIONS OF THE FIRST BASIC PROBLEM . . 145 6.2.
SOLUTION OF SOME SPACE PROBLEMS OF ELASTICITY THEORY USING THE METHOD OF
POTENTIALS 146
6.2.1. SOLUTION OF THE FIRST BASIC PROBLEM FOR SOME CENTRALLY
SYMMETRICAL SPATIAL AREAS . . .. 146
6.2.2. SOLUTION OF THE FIRST BASIC PROBLEM FOR A BALL . . 148
6.2.3. SOLUTION OF THE FIRST BASIC PROBLEM FOR AN UNLIMITED MEDIUM WITH
A SPHERICAL CAVITY . 149 6.2.4. SOLUTION OF THE FIRST BASIC PROBLEM FOR
A HOLLOW BALL 149
6.3. USING SEMI-STATISTICAL METHOD FOR SOLUTION OF INTEGRAL EQUATIONS OF
THE ELASTICITY THEORY 151
6.4. FORMULAE FOR OPTIMAL DENSITY 155
6.5. RESULTS OF NUMERICAL SIMULATION 157
CHAPTER 7. T HE SECOND BASIC PROBLEM OF THE ELASTICITY T H E O RY 161
7.1. FUNDAMENTAL SOLUTIONS OF THE FIRST AND SECOND KINDS 161
7.2. BOUSSINESQ POTENTIALS 165
IMAGE 5
XVI CONTENTS
7.3. WEYL TENSOR 166
7.4. WEYL FORCE TENSORS 168
7.5. ARBITRARY LIAPUNOV SURFACE 170
C H A P T ER 8. A WAY TO SOLVE NON-STATIONARY PROBLEMS 171
8.1. THE GENERAL SCHEME OF SOLUTION OF NON-STATIONARY INTEGRAL EQUATION
171
8.2. A WAY TO INTEGRATE SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS 175
8.2.1. SETTING THE PROBLEM 175
8.2.2. ESTIMATING AN ERROR OF EVALUATION OF MATRIX EXPONENTIAL 177
8.2.3. OPTIMIZATION OF THE ALGORITHM OF APPROXIMATE COMPUTATION OF
MATRIX EXPONENTIAL 181 8.2.4. ANALYSIS OF CONDITION OF OPTIMAL PARAMETER
SELECTION 188
8.2.5. COMPARISON OF THE PROPOSED WAY TO SELECT METHOD PARAMETERS WITH
KNOWN ANALOGUES . .. 189
8.2.6. ESTIMATE OF INTEGRATION METHOD ERROR OF DIFFERENTIAL EQUATIONS
LINEAR SYSTEMS WITH CONSTANT COEFFICIENT MATRIX 195
8.2.7. ALGORITHM OF RIGID LINEAR SYSTEMS NUMERICAL INTEGRATION OF
DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENT MATRIX 200
8.2.8. ADVANTAGES OF THE ALGORITHM 202
PART II. THE RANDOM WALK METHOD.
SOLUTION OF BOUNDARY-VALUE PROBLEMS 205
CHAPTER 9. INTRODUCTION TO THE RANDOM WALK M E T H OD ( R W M) 207
CHAPTER LO.NUMERICAL SOLUTION OF THE HEAT CONDUCTIVITY
PROBLEMS BY MEANS OF THE RANDOM WALK M E T H OD 209
10.1. BASIC BOUNDARY-VALUE PROBLEMS OF FLAT STATIONARY HEAT CONDUCTIVITY
THEORY 209
10.2. MAIN IDEAS OF RWM FOR SOLUTION OF THE HEAT CONDUCTIVITY PROBLEMS
211
10.3. THE PROCESS OF RANDOM WALKING OVER CIRCLES AND ITS PROPERTIES 214
10.4. RWM ALGORITHMS FOR SOLUTION OF THE DIRICHLET PROBLEM 219
IMAGE 6
CONTENTS XVII
10.4.1. RWM ALGORITHMS ON CIRCLES WITH DISPLACED CENTERS 221
10.4.2. TRADITIONAL RWM ALGORITHM 230
10.4.3. RWM ALGORITHMS WITH PARTIAL INTEGRATION . . . 231 10.5. RWM
ALGORITHMS FOR SOLUTION OF THE NEUMANN PROBLEM 242
10.5.1. INTEGRAL REPRESENTATIONS OF THE SOLUTION 242
10.5.2. RWM ALGORITHM WITH DISPLACED CENTERS 245
10.6. NUMERICAL RESULTS 250
CHAPTER 11.THE MONTE-CARLO M E T H OD APPLIED TO PROBLEMS OF PLATE
CURVING 253
11.1. STATEMENT OF THE PROBLEM 253
11.2. INTEGRAL REPRESENTATIONS 256
11.3. CURVING OF SUPPORTED PLATES WITH LINEAR PIECE-WISE BOUNDARY 263
11.3.1. BASIC RELATIONS 263
11.3.2. TRADITIONAL RWM METHOD 266
11.3.3. RWM ALGORITHM WITH DISPLACED CENTERS 268
11.3.4. RWM ALGORITHM WITH PARTIAL INTEGRATION . . .. 273
11.4. CURVING OF SUPPORTED PLATES WITH ARBITRARY CONTOUR . . 277 11.4.1.
DESIGNING TRADITIONAL ALGORITHM IN A FORMAL WAY 277
11.4.2. DESIGNING OF CONVERGING ALGORITHM (MODULATION METHOD) 279
11.4.3. CONVERGENCE CONDITIONS 285
11.4.4. JUSTIFICATION OF THE RESULTS OBTAINED 287
11.5. CURVING OF THE FASTENED PLATES 300
11.5.1. BASIC RELATIONS 300
11.5.2. ALGORITHM WITH PARTIAL INTEGRATION 301
11.5.3. SOME WAYS TO IMPROVE EFFICIENCY OF THE ALGORITHM 309
11.6. RESULTS OF NUMERICAL SIMULATION 312
C H A P T ER 12.THE MONTE-CARLO M E T H OD APPLIED TO THE FLAT PROBLEMS
OF ELASTICITY THEORY 315
12.1. STATEMENT OF THE PROBLEM 315
12.2. DESCRIPTION OF RWM ALGORITHM WITH PARTIAL INTEGRATION 316
12.3. FINDING DISPLACEMENTS BY THE RWM 317
12.4. SOLUTION OF TEST PROBLEMS 320
IMAGE 7
XVIII CONTENTS
CHAPTER 13.APPLICATION OF MONTE-CARLO M E T H OD TOWARDS FINDING OF
TENSIONS AT DANGEROUS POINTS OF COG-WHEELS 322
13.1. SELECTION OF COMPUTATION MODEIS AND SETTING OF BOUNDARY CONDITIONS
322
13.1.1. A RIGID-RIMMED COG-WHEEL 323
13.1.2. A FLEXIBLE-RIMMED COG-WHEEL 323
13.2. RESULTS OF NUMERICAL SIMULATION 324
13.2.1. RIGID-RIMMED COG-WHEELS 324
13.2.2. FLEXIBLE-RIMMED COG-WHEELS 327
13.2.3. PROBLEM OF FINDING THE BEST GEAR-CUTTING TOOL . 328 13.3. SOME
CONCLUSIONS 330
CHAPTER 14.THE SPATIAL PROBLEM OF HEAT CONDUCTIVITY 331
14.1. DEFINING RELATIONS 332
14.2. SOLUTION OF THE STATISTICAL PROBLEM OF HEAT CONDUCTIVITY WITH
BOUNDARY CONDITIONS OF THE SECOND KIND FOR A PARALLELEPIPED AND A CUBE
333
14.3. CONSTRUCTION OF STATISTICAL ESTIMATES 336
14.4. ANALYSIS OF THE ALGORITHM CONVERGENCE 339
14.5. RESULTS OF NUMERICAL SIMULATION 342
14.6. STATEMENT OF THE FINAL ALGORITHM 344
PART III. OPTIMIZATION OF AN FEM GRID 347
CHAPTER L OE . I N T R O D U C T I ON 349
CHAPTER 16.OPTIMAL DISTRIBUTION OF NODES FOR T HE PROBLEM
OF TENSION OF A BALK OF VARIABLE SECTION 352
16.1. STATEMENT OF THE PROBLEM 352
16.2. SOLUTION OF THE PROBLEM BY MEANS OF FEM 353
CHAPTER 17.OPTIMIZATION IN GENERAL CASE 362
17.1. ASYMPTOTICALLY OPTIMAL DENSITY 362
17.2. EXAMPLES 366
CHAPTER 18.NUMERICAL SIMULATION 368
18.1. STATEMENT OF THE PROBLEM 368
18.2. STEP APPROXIMATION OF DISTRIBUTION FUNCTION 368
IMAGE 8
CONTENTS
X X
CHAPTER 1 9 . B E MM OPTIMAL NODES WITH VARIABLE SECTION,
LINEAR WITH RESPECT TO THE LENGTH 373
19.1. DETERMINATE CASE 374
19.2. OPTIMAL ASYMPTOTIC DENSITY 378
19.3. CONSTRUCTION OF AN OPTIMAL DENSITY OF NODES DISTRIBUTION 380
C H A P T ER 20.CONNECTION BETWEEN OPTIMAL DETERMINATE NODES AND OPTIMAL
DENSITY 381
C H A P T ER 21.RESULTS OF NUMERICAL EXPERIMENTS 385
21.1. SIMULATION OF DETERMINATE FEM SOLUTIONS 385
21.2. SIMULATION OF STOCHASTIC FEM SOLUTIONS 388
AFTERWORD 395
BIBLIOGRAPHY 397
INDEX 413
|
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| indexdate | 2024-07-09T18:28:56Z |
| institution | BVB |
| isbn | 9810235011 |
| language | English Russian |
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| spelling | Arsen'ev, Dmitrij G. Verfasser aut Adaptivnije metodij vychislitel'noj matematiki i mechaniki Adaptive methods of computing mathematics and mechanics stochastic variant D. G. Arsenjev, V. M. Ivanov & O. Y. Kul'chitsky Singapore [u.a.] World Scientific 1999 XIX, 416 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Aus dem Russ. übers. Datenverarbeitung Mathematisches Modell Mathematical models Mechanics, Analytic Data processing Multigrid methods (Numerical analysis) Mechanik (DE-588)4038168-7 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Mechanik (DE-588)4038168-7 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Ivanov, Vladimir M. Verfasser aut Kul'chitsky, Oleg Y. Verfasser aut GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008496674&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Arsen'ev, Dmitrij G. Ivanov, Vladimir M. Kul'chitsky, Oleg Y. Adaptive methods of computing mathematics and mechanics stochastic variant Datenverarbeitung Mathematisches Modell Mathematical models Mechanics, Analytic Data processing Multigrid methods (Numerical analysis) Mechanik (DE-588)4038168-7 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| subject_GND | (DE-588)4038168-7 (DE-588)4128130-5 |
| title | Adaptive methods of computing mathematics and mechanics stochastic variant |
| title_alt | Adaptivnije metodij vychislitel'noj matematiki i mechaniki |
| title_auth | Adaptive methods of computing mathematics and mechanics stochastic variant |
| title_exact_search | Adaptive methods of computing mathematics and mechanics stochastic variant |
| title_full | Adaptive methods of computing mathematics and mechanics stochastic variant D. G. Arsenjev, V. M. Ivanov & O. Y. Kul'chitsky |
| title_fullStr | Adaptive methods of computing mathematics and mechanics stochastic variant D. G. Arsenjev, V. M. Ivanov & O. Y. Kul'chitsky |
| title_full_unstemmed | Adaptive methods of computing mathematics and mechanics stochastic variant D. G. Arsenjev, V. M. Ivanov & O. Y. Kul'chitsky |
| title_short | Adaptive methods of computing mathematics and mechanics |
| title_sort | adaptive methods of computing mathematics and mechanics stochastic variant |
| title_sub | stochastic variant |
| topic | Datenverarbeitung Mathematisches Modell Mathematical models Mechanics, Analytic Data processing Multigrid methods (Numerical analysis) Mechanik (DE-588)4038168-7 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| topic_facet | Datenverarbeitung Mathematisches Modell Mathematical models Mechanics, Analytic Data processing Multigrid methods (Numerical analysis) Mechanik Numerisches Verfahren |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008496674&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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