دانلود کتاب Sparse grids and applications - Miami 2016

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Preface......Page 6
Contents......Page 7
Contributors......Page 9
1 Introduction......Page 11
2 Interpolation......Page 14
3 The Importance of Nesting......Page 15
3.1 PseudoGauss Nested Points......Page 16
3.2 Leja Nested Points......Page 17
4 Lebesgue Constants......Page 19
5 Comparison Between Leja Points and PseudoGauss Points in Collocation Calculations......Page 22
6 Conclusion......Page 24
References......Page 25
1 Introduction......Page 28
2 Least-Squares Regression......Page 30
3 Full Grids and Sparse Grids......Page 31
4.1 Well-Posedness and Error Decay......Page 34
4.2 Application to Sparse Grids......Page 40
5 Numerical Experiments......Page 46
5.1 Error Decay......Page 47
5.2 Balancing the Error......Page 48
6 Conclusion......Page 49
References......Page 50
1 Introduction......Page 51
2 Adaptivity with Sparse Grids......Page 53
2.1 Interpolation on Spatially-Adaptive Sparse Grids......Page 54
2.2 Interpolation with Dimension-Adaptive Sparse Grids......Page 56
3.1 Generalized Polynomial Chaos......Page 59
3.2 Multilevel Approaches for Generalized Polynomial Chaos......Page 60
3.3 Stochastic Dimensionality Reduction......Page 62
4.1 Second-Order Linear Oscillator with External Forcing......Page 65
4.2 A simple Fluid-Structure Interaction Example......Page 71
5 Conclusions and Outlook......Page 73
References......Page 74
1 Introduction......Page 77
2 Sparse Grids......Page 79
2.1 Hierarchical Ancestors and the Fundamental Property......Page 80
2.2 Interpolation on Sparse Grids......Page 81
3 Limiting Ranges of Sparse Grid Function Values......Page 82
3.1 Limitation from Above and Below......Page 83
3.2 Minimal Extension Set......Page 84
3.3 Computing Coefficients of the Extension Set......Page 85
3.4 Intersection Search......Page 87
4 Approximation of Gaussians with Extended Sparse Grids......Page 90
4.1 Intersection Search and Candidate Sets for Regular Sparse Grids......Page 91
4.2 Extension Sets and Convergence for Regular Grids......Page 93
4.3 Extension Sets for Adaptively Refined Grids......Page 96
5 Conclusions......Page 97
References......Page 98
1 Introduction......Page 100
1.1 High-Dimensional PDEs in High-Performance Computing......Page 101
2 Theory of the Classical Combination Technique......Page 102
3 The Combination Technique in Parallel......Page 105
4 Dealing with System Faults......Page 106
5.1 Method 1: Comparing Combination Solutions Pairwise via a Maximum Norm......Page 107
5.3 Cost and Parallelization......Page 111
5.4 Detection Rates......Page 112
6.2 SDC Injection......Page 113
6.3 Results: Detection Rates and Errors......Page 114
6.4 Results: Scaling......Page 116
6.5 Dealing with False Positives......Page 117
7 Extensions to Quantities of Interest......Page 119
8 Conclusion......Page 120
References......Page 121
1 Introduction and Background......Page 123
2 Transformed Quadrature Rules......Page 125
2.1 Standard One-Dimensional Quadrature Rules......Page 128
2.2 Sparse Quadrature for High Dimensional Integrals......Page 130
3 Comparison of the Transformed Sparse Grid Quadrature Method......Page 131
4 Numerical Tests of the Sparse Grid Transformed Quadrature Rules......Page 134
4.1 Comparison of Maps......Page 135
4.2 Effect of Dimension......Page 137
References......Page 138
1 Introduction......Page 141
2 Dynamic Portfolio Choice Models......Page 144
2.1 Approximation......Page 145
2.2 Optimization......Page 146
2.4 Dynamic Programming......Page 147
3.1 Hierarchical Bases and Sparse Subspace Selection......Page 149
3.2 Spatially Adaptive Refinement......Page 153
4 Spatially Adaptive Sparse Grid Dynamic Programming Scheme......Page 155
4.1 Spatially Adaptive Sparse Grid Dynamic Programming......Page 156
4.3 Choice of Basis Functions, Extrapolation, and Refinement......Page 159
5 Numerical Example......Page 162
5.1 Transaction Costs Model......Page 163
5.2 Error Measurement......Page 165
5.3 Results......Page 167
6 Conclusion......Page 174
Appendix: Euler Equation Errors......Page 175
References......Page 177
1 Introduction......Page 180
2 Multidimensional Hierarchical Interpolation Strategy......Page 182
2.1 Multidimensional Hierarchy of Nodes and Functions......Page 183
2.2 Multidimensional Interpolation......Page 187
2.3 Piece-Wise Constant Hierarchy......Page 189
3 Adaptive Interpolation......Page 192
4.1 Influence of the Type of Hierarchy......Page 196
4.2 Influence of the Refinement Method......Page 198
4.3 Application to Kermack-McKendrick SIR Model......Page 200
5 Conclusions......Page 201
References......Page 202
1 Introduction......Page 205
2 Decomposition......Page 207
3 Truncation......Page 209
3.1 Knapsack Problem......Page 213
4.1 Finite-Dimensional Case......Page 215
4.2 Infinite-Dimensional Case......Page 217
5.1 High-Dimensional Interpolation and Integration......Page 219
5.2 Monte Carlo Path Simulation......Page 221
5.3 Multilevel Quadrature......Page 223
5.4 Partial Differential Equations......Page 224
5.5 Uncertainty Quantification......Page 225
Appendix......Page 228
References......Page 231
1 Introduction......Page 233
2.1 Definition and Properties......Page 235
2.2 Hierarchization with B-Splines......Page 238
3 Fundamental Property......Page 239
4.1 Hierarchical Fundamental Transformation......Page 241
4.3 Translation-Invariant Fundamental Transformation......Page 243
5.1 Definition and Properties......Page 244
5.2 Modified Fundamental Splines......Page 246
6.1 Generation of the Spatially Adaptive Grid......Page 248
6.3 Test Functions and Results......Page 250
6.4 Comparison of Runtime and Memory Consumption......Page 253
References......Page 254
Editorial Policy......Page 256
Lecture Notesin Computational Scienceand Engineering......Page 258
Texts in Computational Scienceand Engineering......Page 264