Sparse grids and applications

Cover......Page 1
Sparse Grids and Applications......Page 4
Preface......Page 6
Contents......Page 8
Contributors......Page 10
Asymptotic Expansion Around Principal Components and the Complexity of Dimension Adaptive Algorithms......Page 12
1 Introduction and Problem Formulation......Page 13
2 Takens' Theorem and the Delay Embedding Scheme......Page 15
3 The Regression Problem and the Regularized Least Squares Approach......Page 17
3.1 Minimization for an Arbitrary Basis......Page 19
3.2 Minimization for a Kernel Basis in a Reproducing Kernel Hilbert Space......Page 20
4 Discretization via Sparse Grids......Page 21
4.1 Multilevel Hierarchical Bases and Regular Sparse Grids......Page 22
4.2 Space-Adaptive Sparse Grids......Page 25
4.3 Dimension-Adaptive Sparse Grids......Page 27
5.1 Hénon Map in 2d......Page 32
5.2 Jump Map in 5d......Page 34
5.3 Small Dataset of the ANN and CI Forecasting Competition 2006/2007......Page 35
6 Concluding Remarks......Page 39
References......Page 40
1 Introduction......Page 42
1.1 Tensorial Quantities......Page 43
2 Discretisation of Diffusion Problem with Uncertain Coefficient......Page 45
2.2 Stochastic Discretisation......Page 47
2.3 Quadrature Rules and Sparse Integration Grids......Page 51
3 The Canonical Tensor Format......Page 53
4.1 Computation of the Maximum Norm and Corresponding Index......Page 55
4.2 Computation of the Characteristic......Page 57
4.3 Computation of Level Sets, Frequency, Mean Value, and Variance......Page 59
5 Complexity Analysis......Page 61
6 Numerical Experiments......Page 62
7 Conclusion......Page 64
References......Page 65
1 Introduction......Page 68
2 Sparse Grids......Page 69
2.1 Hierarchical Subspace-Splitting......Page 70
2.2 Properties of the Hierarchical Subspaces......Page 73
2.3 Sparse Grids......Page 76
2.4 Hierarchy Using Constant Functions......Page 81
3 Sparse Grid Combination Technique......Page 84
3.1 Optimised Combination Technique......Page 88
References......Page 89
1 Introduction......Page 93
2.1 Input Data......Page 95
2.2 Delay Embedding into a Feature Space......Page 96
3 Sparse Grid Discretization......Page 98
3.1 Sparse Grid Combination Technique......Page 99
4 Numerical Results......Page 101
4.1 Experimental Data......Page 102
4.2 Quality Assessment......Page 103
4.3 Forecasting Using a Single Currency Pair......Page 104
4.4 Forecasting Using Multiple Currency Pairs......Page 108
4.5 Towards a Practical Trading Strategy......Page 110
5 Conclusions......Page 113
References......Page 114
1 Introduction......Page 117
2 Multilevel Monte Carlo Method......Page 118
3.1 Dimension-Adaptive Algorithm......Page 120
3.2 Time-Adaptive Algorithm......Page 122
4 Numerical Results......Page 124
4.1 Dimension-Adaptive Algorithm......Page 125
4.2 Time-Adaptive Algorithm......Page 126
References......Page 129
1 Introduction......Page 131
2.1 One-Dimensional Model......Page 133
2.2 Multi-dimensional Case and Dependence Modelling......Page 134
2.3 Representation of the Multi-dimensional Process as Lévy Process......Page 135
2.4 Option Pricing......Page 136
3.1 Time Discretization and Weak Formulation......Page 138
3.2 Space Discretization by a Sparse Grid Generating System......Page 140
3.3 Preconditioning......Page 144
3.4 Operator Application......Page 147
3.4.1 BottomUp Algorithm......Page 148
3.4.3 Multi-dimensional Case......Page 149
3.5 Galerkin Recurrence Formula for the Kou Model......Page 151
4.1 European Put Option......Page 153
4.2 Two-Dimensional Example......Page 154
4.3 Three-Dimensional Option......Page 157
5 Concluding Remarks......Page 158
References......Page 159
1 Tensors, Tensor Representations......Page 161
3 Function Approximations......Page 163
4 Sparse Grid Approximations......Page 164
5 From Sparse Grid to r-Term Tensor Format......Page 165
6 Conclusion......Page 168
References......Page 169
1 Introduction......Page 170
2 Sparse Grids......Page 171
4 Deterministic Reformulation......Page 173
5 Quadrature in the Stochastic Variable......Page 175
6 Finite Element Approximation in the Spatial Variable......Page 176
7 Multilevel Quadrature Method for the Expectation......Page 177
8 Multilevel Quadrature Method for the Second Moment......Page 180
9 Numerical Results......Page 182
9.1 Finite Dimensional Stochastics......Page 183
9.2 Infinite Dimensional Stochastics......Page 186
References......Page 188
1 Introduction......Page 189
2 Problem Formulation......Page 191
3 A Stochastic Collocation Method......Page 192
3.1 Sparse Grids......Page 193
3.2 Moment Estimation......Page 195
3.3 Adaptation Procedure......Page 196
4 An Existing Stochastic Collocation Alternative......Page 199
5.1 Algorithmic Comparison......Page 200
5.2 Numerical Comparison......Page 202
6.1 Model Formulation......Page 206
6.2 Results......Page 208
7 Conclusion......Page 209
References......Page 210
1 Introduction......Page 212
2 Linear Gyrokinetics......Page 214
3 The Gyrokinetic Electromagnetic Numerical Experiment GENE......Page 215
3.1 Fluxtube Domain......Page 216
3.2 Modes of Linear Computations......Page 217
4.2 Dimension Adaptive Combination Schemes......Page 218
4.3 Optimizing the Combination Coefficients......Page 220
5.1 Alignment of the Full Grids......Page 221
5.2 Classical Combination Technique......Page 222
5.3 Anisotropic Grids......Page 224
6 Summary......Page 227
References......Page 228
1 Introduction......Page 230
2 Evaluation Costs of Sparse Grid Functions......Page 232
3.1.1 Preliminaries......Page 234
3.1.2 Abstract Formulation and Truth Approximation......Page 235
3.1.3 Reduced Basis Approximation......Page 236
3.3 Construction of Reduced Basis Spaces......Page 238
3.4 Model Problem (Thermal Block)......Page 239
4 Sparse Grids in the Parameter Domain......Page 240
4.1 Greedy Sampling Strategy with Adaptive Sparse Grids......Page 241
4.2 Greedy Sampling Strategy with Sparse Grid Interpolant......Page 242
5 Interpolation of the Output Function......Page 243
5.1 Interpolation on Adaptive Sparse Grids......Page 244
5.2 The Output Function on a Logarithmic Scale......Page 245
6 Conclusion......Page 248
References......Page 249
2 Sparse Grids......Page 251
3 Adaptivity......Page 253
4 Problem-Awareness......Page 255
5 Examples and Strategies......Page 257
5.1 Straightforward Adaptive Refinement......Page 258
5.2 Criterion for Adaptive Refinement and Dimensional Adaptivity......Page 259
5.3 Strategies for Adaptive Refinement......Page 261
5.4 Choice of Basis Functions......Page 263
5.5 Weighted/Guided Refinement......Page 265
6 Conclusions......Page 267
References......Page 268
1 Introduction......Page 270
2.1 Construction......Page 272
2.2 An Example......Page 274
2.3 Principal Components......Page 275
3 PCA and Asymptotic Expansion......Page 277
4 A Simple and Some More Complex Complexity Results......Page 280
References......Page 282
Editorial Policy
......Page 284
Lecture Notesin Computational Science
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Texts in Computational Science
and Engineering......Page 290