Numerical Integration: On Advanced Computer Systems

Front Cover
Springer Science & Business Media, Sep 28, 1994 - Computers - 341 pages
This monograph is a comprehensive treatment of the theoretical and computational aspects of numerical integration.
The authors give a unique overview of the topic by bringing into line many recent research results not yet presented coherently; the extensive bibliography lists 268 items. Particular emphasis is given to the potential parallelism of numerical integration problems and to utilizing it by means of dynamic load distribution techniques. The book discusses the basics and provides methodologies for producing efficient and reliable software for numerical integration on advanced computer systems. The book addresses researchers, graduate students, and computational scientists.
 

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Contents

Applications of Numerical Integration
5
11 Computational Statistics
6
113 Bayesian Statistics
8
12 Integral Transforms
9
122 Application to Differential Equations
10
123 Finite Transforms and Fourier Series
11
124 Computational Aspects
12
13 Finite Element Methods
13
61 Error Estimation
166
612 Use of Two Integration Formulas with Different Accuracy
167
613 Use of Null Rules
169
614 Use of the Same Formula on Different Subdivisions
173
615 Use of Randomized Formulas
174
62 Sampling Strategy
176
621 Efficiency Assessment
177
622 NonAdaptive Algorithms
179

132 Galerkin Method
15
133 Fundamentals of Finite Element Methods Basic Principles
16
14 Boundary Integral Methods
19
142 Collocation Methods
21
Fundamentals of Numerical Integration
25
22 Integration Methods
26
23 Conditioning of Integration Problems
30
24 Inherent Uncertainty of Numerical Integration
33
Mathematical Aspects
35
Integrals
37
312 Multiple Dimensions
38
32 Improper Integrals
39
322 MultiDimensional Improper Integrals
41
33 Special Types of Integrals
42
34 Integration Regions
44
35 Weight Functions
45
36 The Preprocessing of Integrals
47
362 Decomposition of Integration Regions
54
363 Iteration of Integrals
55
Univariate Integration Formulas
57
412 Formula Construction by Approximation
60
413 Formula Construction by Acceleration
64
414 Linear Transformation of Quadrature Formulas
66
42 Simple Interpolatory Quadrature Formulas
67
421 Degree of Accuracy
68
422 Convergence
70
424 Open NewtonCotes Formulas
73
425 ClenshawCurtis Formulas
74
426 Gauss Formulas
75
427 Radau und Lobatto Formulas
78
43 Compound Formulas
79
431 Compound Trapezoidal Rule
82
432 IMT Formulas
84
44 Romberg Formulas
85
45 Nonlinear Extrapolation Methods
87
Multivariate Integration Formulas
91
51 Construction Principles
92
513 Construction by Transformation
93
514 Construction by Iteration
94
515 Construction by Decomposition
97
52 Polynomial Rules
99
522 Construction of Interpolatory Formulas Degree of Accuracy
103
523 Simple Interpolatory Formulas
113
524 Compound Formulas
117
53 LowDiscrepancy Formulas
118
532 OneDimensional LowDiscrepancy Sequences
124
533 MultiDimensional LowDiscrepancy Sequences
128
54 Pseudorandom Methods
134
542 Variance Reduction Schemes
137
543 Generation of Pseudorandom Numbers
140
544 Generation of Uniform Random Numbers Linear Congruential Methods
141
545 Generation of NonUniform Random Numbers
144
55 Lattice Rules
147
552 Basic Definitions
149
553 Practical Representation of Lattice Rules
151
554 Error Analysis of Lattice Rules
152
555 Efficiency of Lattice Rules
159
56 Various Special Methods
161
Algorithmic and Computational Aspects
163
Integration Algorithms
165
63 Reliability Enhancement
184
632 Detection of NonIntegrable Functions
185
64 Efficiency Enhancement
187
65 Integration of Multiple Integrands
190
Load Distribution
193
71 Static Load Distribution
195
72 Dynamic Load Distribution
196
721 Dynamic Load Balancing Algorithms
197
722 Asynchronous Single Task Pool Methods
198
723 Synchronous Single Task Pool Methods
200
724 Multiple Task Pool Methods
201
725 Synchronous Multiple Task Pool Methods
205
726 Asynchronous Multiple Task Pool Methods
206
727 Hierarchical Methods
209
Potential Parallelism in Integration Algorithms
211
82 Global Extrapolation Level
214
83 Subdivision Level
215
84 Subregion Level
218
85 Local Extrapolation Level
219
87 Integrand Evaluation Level
221
Parallelization Schemes for Integration Algorithms
223
91 Constituent Parts of Parallel Tasks
226
92 Performance Characteristics of Parallel Tasks
227
922 Degree of Parallelism
228
924 Predictability of Execution Times
229
Practical Parallelization of Integration Algorithms
233
1011 Pipelined Processors
234
1012 SIMD Computers
235
102 Vectorization of Integration Methods
236
1022 Integration Formula Level
237
103 Parallelization of Integration Methods
238
1032 Subregion Level
239
1033 Subdivision Level
240
1034 Subdivision and Subregion Level
246
1035 Subdivision Subregion and Integration Formula Level
251
105 Vectorization and Parallelization of Abscissa Computations
252
106 Vectorization and Parallelization of Reduction Operations
254
Parallel Data Management
257
111 Data Structures for Global Subdivision Strategies
259
1112 Other Implementations of Priority Queues
263
112 Concurrent Access to Data Structures
269
1121 Concurrent Access to Priority Queues
270
1122 New Types of Data Structures
274
IV Software Aspects
277
Assessment of Numerical Integration Software
279
1211 Reliability
280
1212 Performance
282
1213 Portability
283
122 Techniques of Assessment
285
1221 Test Batteries and Performance Profiles
286
1222 Fundamental Approaches to Assessing Numerical Integration Software
290
Architecture Adaptive Integration Algorithms
293
131 Required Information
294
132 Practical Experience
300
Glossary of Notation
303
Bibliography
309
Authors
330
Index
334
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