Monte Carlo and Quasi-Monte Carlo Methods - MCQMC 2018, Rennes, France, July 1–6

Preface......Page 6
Contents......Page 9
Part I Invited Talks......Page 12
1 Introduction......Page 13
2 Mathematical Framework......Page 14
3.1 Naive Monte Carlo......Page 17
4 Large-Sample Properties of Quantile Estimators......Page 19
5.1 Control Variates......Page 21
5.2 Stratified Sampling......Page 24
5.3 Latin Hypercube Sampling......Page 26
5.4 Importance Sampling......Page 28
5.5 Conditional Monte Carlo......Page 30
5.7 Numerical Results of Point Estimators for Quantiles......Page 32
6.1 Small-Sample CIs......Page 33
6.2 Consistent Estimation of Asymptotic Variance......Page 34
6.3 Batching, Sectioning, and Other Methods......Page 35
6.4 Numerical Results of CIs for Quantiles......Page 36
7 Summary and Concluding Remarks......Page 37
References......Page 38
1 Introduction......Page 41
2.1 Model Advection-Diffusion-Reaction Problem in D......Page 44
2.2 Uncertainty Parametrization......Page 47
3 Finite Element Discretization......Page 49
3.1 Numerical Integration......Page 50
3.2 Finite Element Approximation of the Parametric Solution......Page 52
4.1 Higher Order QMC Integration......Page 54
4.2 Parametric Regularity......Page 55
4.3 Multilevel QMC Error Estimates......Page 57
4.4 Error Versus Work Analysis......Page 58
5 Bayesian Inverse UQ......Page 60
5.1 Formulation of the Bayesian Inverse Problem......Page 61
5.2 Multilevel HoQMC-FE Discretization......Page 62
6 Conclusions......Page 63
7.1 Numerical Integration......Page 64
7.2 Approximation of Functions with Point Singularities......Page 73
References......Page 75
1 Introduction......Page 78
2 R&S Basics......Page 80
3 R&S Computation......Page 82
4 Thinking Differently......Page 85
References......Page 87
 Discrepancy of Digital Sequences: New Results on a Classical QMC Topic......Page 89
1 Introduction......Page 90
2 Known Facts About the Lp Discrepancy......Page 91
3 Discrepancy of Digital Sequences......Page 94
3.2 Digital (t,s)-Sequences......Page 95
3.3 Digital (0,1)-Sequences over mathbbF2......Page 97
3.4 Digital Sequences with Optimal Order of Lp Discrepancy......Page 99
3.5 Intermediate Norms of the Local Discrepancy Function......Page 101
4 Discussion of the Asymptotic Discrepancy Estimates......Page 102
5 Weighted Discrepancy of Digital Sequences......Page 105
6 Summary......Page 107
References......Page 108
Part II Regular Talks......Page 112
1 Introduction......Page 113
2.1 The Data Set and Sliding Window Processing......Page 115
2.2 The Stochastic Block Model......Page 116
2.3 The Collapsed Posterior......Page 118
2.4 Sampling the Parameters from the Posterior......Page 119
2.5 Posterior Predictive Discrepancy......Page 121
2.6 Cumulative Discrepancy Energy via Sliding Window......Page 122
3.2 Effect of Changing the Distance Between Change Points......Page 123
3.3 Effect of Changing the Window Size......Page 125
4 Discussion......Page 126
5 Conclusion......Page 127
References......Page 128
1 Introduction......Page 130
2 Probabilistic Tools for Solving PDEs......Page 131
2.1 Pointwise Evaluations of the Solution......Page 132
2.2 A Sequential Control Variates Algorithm......Page 134
3.1 Sparse Interpolation......Page 137
3.2 Adaptive Algorithm for Sparse Interpolation......Page 138
4.1 Perturbed Version of Algorithm2......Page 139
4.3 Numerical Results......Page 140
5 Conclusion......Page 144
References......Page 145
1 Introduction......Page 147
2.3 Binary Trees......Page 149
2.4 k-ary Trees......Page 150
2.6 The Alias Method......Page 151
3 An Efficient Inverse Mapping......Page 154
3.1 Massively Parallel Construction of Radix Trees......Page 156
3.2 Massively Parallel Construction of Radix Tree Forests......Page 157
4 Results......Page 159
5 Discussion......Page 162
References......Page 163
 An Adaptive Algorithm Employing Continuous Linear Functionals......Page 164
1.1 Input and Output Spaces......Page 165
1.2 Solvability......Page 166
1.3 Computational Cost of the Algorithm and Complexity  of the Problem......Page 167
1.4 The Case for Adaptive Algorithms......Page 170
2 Assuming a Steady Decay of the Series Coefficients of the Solution......Page 171
3 Adaptive Algorithm......Page 172
4 Essential Optimality of the Adaptive Algorithm......Page 177
5 Numerical Example......Page 182
References......Page 184
 Constructing QMC Finite Element Methods for Elliptic PDEs with Random Coefficients by a Reduced CBC Construction......Page 185
1 Introduction and Problem Setting......Page 186
1.1 Existence of Solutions of the Variational Problem......Page 187
1.2 Parametric Regularity......Page 188
2 Quasi-Monte Carlo Finite Element Error......Page 192
2.1 Finite Element Approximation......Page 193
2.2 Dimension Truncation......Page 194
2.3 Quasi-Monte Carlo Integration......Page 195
2.4 Implementation of the Reduced CBC Algorithm......Page 197
3 QMC Finite Element Error Analysis......Page 199
3.1 Derivative Bounds of POD Form......Page 201
4 Combined Error Bound......Page 204
5 Derivation of the Fast Reduced CBC Algorithm......Page 205
References......Page 207
 Sudoku Latin Square Sampling for Markov Chain Simulation......Page 208
1 Introduction......Page 209
2 Monte Carlo Simulations of Markov Chains......Page 211
2.2 Simple Stratified Sampling......Page 213
2.3 Sudoku Latin Square Sampling......Page 214
3 Convergence Analysis......Page 215
3.1 Standard Monte Carlo......Page 216
3.2 Simple Stratified Sampling......Page 218
3.3 Sudoku Latin Square Sampling......Page 219
4.1 A Geo/Geo/1 Queue......Page 220
4.3 Diffusion......Page 222
5.1 Proof of Lemma 2......Page 223
5.2 Proof of Lemma 3......Page 224
6 Conclusion......Page 230
References......Page 231
 Avoiding the Sign Problem in Lattice Field Theory......Page 232
1 Introduction......Page 233
2 Integration in Lattice QCD......Page 234
3.1 Avoiding the Sign Problem in Physical One-Dimensional Systems......Page 236
3.2 Reducing High-Dimensional Integrals to Nested One-dimensional Integrals......Page 242
3.3 Avoiding the Sign Problem in High-Dimensional Integrals......Page 246
4 Conclusion......Page 248
References......Page 249
1 Introduction and Preliminaries......Page 251
2 Halton-Type Hammersley Point Sets, Polynomial Lattice Point Sets, and Results on the Star Discrepancy of Their Hybrid Point Sets......Page 254
3 Auxiliary Results......Page 257
4 Proof of Theorem 1......Page 259
5 Proof of Theorem 2......Page 266
6 Open Problems......Page 267
References......Page 268
1 Introduction......Page 270
2 Psi-Estimators......Page 273
3 Computation......Page 279
3.2 An Exact Method......Page 280
4.1 Timings......Page 281
5 Conclusion......Page 282
References......Page 283
 Infinite Swapping Algorithm for Training Restricted Boltzmann Machines......Page 284
1 Introduction......Page 285
2 Restricted Boltzmann Machines......Page 287
3 Parallel Tempering and Infinite Swapping for Gibbs Samplers......Page 288
4.1 Prediction Accuracy......Page 291
4.2 Parameters and Settings......Page 292
4.3 Data Sets......Page 294
4.4 Evaluating the INS Algorithm for Small Toy Data Sets......Page 295
5 Conclusions......Page 300
References......Page 304
1 Introduction......Page 307
2 Theoretical Results......Page 309
2.1 Gaussian Case: K=2, Uncorrelated ε1 and ε2......Page 310
2.2 General Gaussian Case......Page 311
2.3 General Case......Page 314
3 Numerical Examples......Page 315
3.1 Example 1......Page 316
3.2 Example 2......Page 319
4 Proof of the Result (14)......Page 320
5 Conclusion......Page 325
References......Page 326
1 Introduction......Page 327
2 Lower Bound on Lp(XNC)......Page 332
3 The Proofs......Page 333
References......Page 342
1 Introduction......Page 343
2 Mixed-Integer Two-Stage Stochastic Programs......Page 345
3 ANOVA Decomposition and Effective Dimension......Page 346
4 ANOVA Terms of Mixed-Integer Two-Stage Integrands......Page 348
5 Error Analysis of Randomly Shifted Lattice Rules......Page 350
6 Application to Electricity Optimization Under Uncertainty......Page 353
References......Page 359
 Approximating Gaussian Process Emulators with Linear Inequality Constraints and Noisy Observations via MC and MCMC......Page 361
1 Introduction......Page 362
2 Gaussian Process Emulators with Linear Inequality Constraints and Noisy Observations......Page 363
2.2 Imposing Linear Inequality Constraints......Page 364
2.3 Maximum a Posteriori Estimate via Quadratic Programming......Page 366
3 Numerical Experiments......Page 367
3.1 1D Toy Example Under Boundedness Constraints......Page 368
3.2 1D Toy Example Under Multiple Constraints......Page 371
3.3 Coastal Flooding Applications......Page 372
4 Conclusions......Page 377
References......Page 378
1 Introduction......Page 380
2.1 Model Equation in the Diffusive Limit......Page 382
2.2 Asymptotic-Preserving Monte Carlo Scheme......Page 383
3.1 Method and Notation......Page 384
3.2 Correlating Asymptotic-Preserving Monte Carlo Simulations......Page 386
4.1 Model Correlation Behavior......Page 390
4.2 Comparison with Classical Monte Carlo......Page 393
5 Conclusion......Page 397
References......Page 398
1 Introduction......Page 400
2 Classical Sobol' Sensitivity Analysis......Page 401
3.1 Motivation: Short Rate Models......Page 405
3.2 Temperature Derivatives......Page 408
4 Model Robustness......Page 411
4.1 Robustness of Interest Rate and Temperature Models......Page 414
5 Conclusions......Page 416
References......Page 417
1 Introduction......Page 419
2 Stochastic Partial Differential Equations and Their Approximations......Page 421
3 Covariance-Based Sampling in a Monte Carlo Setting......Page 425
4 Covariance-Based Sampling in a Multilevel Monte Carlo Setting......Page 428
5 Implementation......Page 433
6 Simulation......Page 435
References......Page 438
1 Introduction......Page 440
2 Parabolic Problems with Random Discontinuous Coefficients......Page 441
3.1 Approximated Diffusion Coefficients......Page 446
3.2 Semi-discretization by Adaptive Finite Elements......Page 448
3.3 Fully Discrete Pathwise Approximation......Page 452
4 Estimation of Moments by Multilevel Monte Carlo Methods......Page 453
5 Numerical Results......Page 456
References......Page 460
1 Introduction......Page 462
2 Preliminaries......Page 466
3 Main Results......Page 467
3.1 Proof of Theorem 1......Page 468
3.2 Proof of Theorem 2......Page 475
References......Page 478
1 Introduction......Page 480
2.1 Rank-1 Lattice Point Sets and the Associated Anti-aliasing Sets......Page 482
2.3 Fourier Pseudo-Spectral Methods Using Rank-1 Lattices......Page 483
2.4 Higher-Order Exponential Splitting......Page 486
2.5 Higher-Order Exponential Splitting on Rank-1 Lattices......Page 487
3 Numerical Results......Page 491
3.1 Convergence with Respect to Time Step Size......Page 492
3.3 Eighth-Order Splitting......Page 493
4 Conclusion......Page 495
References......Page 496
 An Analysis of the Milstein Scheme for SPDEs Without a Commutative Noise Condition......Page 498
1 Introduction......Page 499
2.1 Setting and Assumptions......Page 500
2.2 The Milstein Scheme for Non-commutative SPDEs......Page 502
2.3 Comparison of Computational Cost......Page 506
2.4 Example......Page 512
References......Page 515
1 Introduction......Page 517
2 QMC Sampling......Page 518
2.2 Mapping of QMC Points......Page 519
2.3 Error Analysis......Page 522
3 Sorting the Dataset via the Hilbert Space-Filling Curve......Page 525
4 Numerical Tests......Page 529
6 Appendix......Page 532
References......Page 533