This book starts with the basic ideas in uncertainty propagation using Monte Carlo methods and the generation of random variables and stochastic processes for some common distributions encountered in engineering applications. It then introduces a class of powerful simulation techniques called Markov Chain Monte Carlo method (MCMC), an important machinery behind Subset Simulation that allows one to generate samples for investigating rare scenarios in a probabilistically consistent manner. The theory of Subset Simulation is then presented, addressing related practical issues encountered in the actual implementation. The book also introduces the reader to probabilistic failure analysis and reliability-based sensitivity analysis, which are laid out in a context that can be efficiently tackled with Subset Simulation or Monte Carlo simulation in general. The book is supplemented with an Excel VBA code that provides a user-friendly tool for the reader to gain hands-on experience with Monte Carlo simulation.
* Presents a powerful simulation method called Subset Simulation for efficient engineering risk assessment and failure and sensitivity analysis
* Illustrates examples with MS Excel spreadsheets, allowing readers to gain hands-on experience with Monte Carlo simulation
* Covers theoretical fundamentals as well as advanced implementation issues
* A companion website is available to include the developments of the software ideas
This book is essential reading for graduate students, researchers and engineers interested in applying Monte Carlo methods for risk assessment and reliability based design in various fields such as civil engineering, mechanical engineering, aerospace engineering, electrical engineering and nuclear engineering. Project managers, risk managers and financial engineers dealing with uncertainty effects may also find it useful.
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ISBN-13
978-1-118-39806-7 (9781118398067)
Schweitzer Klassifikation
1 - ENGINEERING RISK ASSESSMENT WITH SUBSET SIMULATION [Seite 3]
2 - Contents [Seite 9]
3 - About the Authors [Seite 15]
4 - Preface [Seite 17]
5 - Acknowledgements [Seite 19]
6 - Nomenclature [Seite 21]
7 - 1 Introduction [Seite 23]
7.1 - 1.1 Formulation [Seite 24]
7.2 - 1.2 Context [Seite 27]
7.3 - 1.3 Extreme Value Theory [Seite 27]
7.4 - 1.4 Exclusion [Seite 28]
7.5 - 1.5 Organization of this Book [Seite 29]
7.6 - 1.6 Remarks on the Use of Risk Analysis [Seite 29]
7.7 - 1.7 Conventions [Seite 30]
7.8 - References [Seite 30]
8 - 2 A Line of Thought [Seite 31]
8.1 - 2.1 Numerical Integration [Seite 32]
8.2 - 2.2 Perturbation [Seite 32]
8.3 - 2.3 Gaussian Approximation [Seite 34]
8.3.1 - 2.3.1 Single Design Point [Seite 34]
8.3.2 - 2.3.2 Multiple Design Points [Seite 36]
8.4 - 2.4 First/Second-Order Reliability Method [Seite 36]
8.4.1 - 2.4.1 Context [Seite 37]
8.4.2 - 2.4.2 Design Point [Seite 38]
8.4.3 - 2.4.3 FORM [Seite 39]
8.4.4 - 2.4.4 SORM [Seite 40]
8.4.5 - 2.4.5 Connection with Gaussian Approximation [Seite 44]
8.5 - 2.5 Direct Monte Carlo [Seite 46]
8.5.1 - 2.5.1 Unbiasedness [Seite 47]
8.5.2 - 2.5.2 Mean-Square Convergence [Seite 47]
8.5.3 - 2.5.3 Asymptotic Distribution (Central Limit Theorem) [Seite 50]
8.5.4 - 2.5.4 Almost Sure Convergence (Strong Law of Large Numbers) [Seite 53]
8.5.5 - 2.5.5 Failure Probability Estimation [Seite 54]
8.5.6 - 2.5.6 CCDF Perspective [Seite 56]
8.5.7 - 2.5.7 Rare Event Problems [Seite 60]
8.5.8 - 2.5.8 Variance Reduction by Conditioning [Seite 63]
8.6 - 2.6 Importance Sampling [Seite 66]
8.6.1 - 2.6.1 Optimal Sampling Density [Seite 67]
8.6.2 - 2.6.2 Failure Probability Estimation [Seite 67]
8.6.3 - 2.6.3 Shifting Distribution [Seite 68]
8.6.4 - 2.6.4 Benefits and Side-Effects [Seite 70]
8.6.5 - 2.6.5 Bias [Seite 72]
8.6.6 - 2.6.6 Curse of Dimension [Seite 75]
8.6.7 - 2.6.7 CCDF Perspective [Seite 78]
8.7 - 2.7 Subset Simulation [Seite 80]
8.8 - 2.8 Remarks on Reliability Methods [Seite 82]
8.9 - 2A.1 Appendix: Laplace Type Integrals [Seite 83]
8.10 - References [Seite 84]
9 - 3 Simulation of Standard Random Variable and Process [Seite 87]
9.1 - 3.1 Pseudo-Random Number [Seite 87]
9.2 - 3.2 Inversion Principle [Seite 88]
9.2.1 - 3.2.1 Continuous Random Variable [Seite 89]
9.2.2 - 3.2.2 Discrete Random Variables [Seite 89]
9.3 - 3.3 Mixing Principle [Seite 90]
9.4 - 3.4 Rejection Principle [Seite 91]
9.4.1 - 3.4.1 Acceptance Probability [Seite 93]
9.5 - 3.5 Samples of Standard Distribution [Seite 94]
9.6 - 3.6 Dependent Gaussian Variables [Seite 100]
9.6.1 - 3.6.1 Cholesky Factorization [Seite 100]
9.6.2 - 3.6.2 Eigenvector Factorization [Seite 103]
9.7 - 3.7 Dependent Non-Gaussian Variables [Seite 105]
9.7.1 - 3.7.1 Nataf Transformation [Seite 105]
9.7.2 - 3.7.2 Copula [Seite 109]
9.8 - 3.8 Correlation through Constraint [Seite 111]
9.8.1 - 3.8.1 Uniform in Sphere [Seite 111]
9.8.2 - 3.8.2 Gaussian on Hyper-plane [Seite 114]
9.9 - 3.9 Stationary Gaussian Process [Seite 117]
9.9.1 - 3.9.1 Autocorrelation Function and Power Spectral Density [Seite 117]
9.9.2 - 3.9.2 Discrete-Time Process [Seite 121]
9.9.3 - 3.9.3 Sample Autocorrelation Function and Periodogram [Seite 122]
9.9.4 - 3.9.4 Time Domain Representation [Seite 123]
9.9.5 - 3.9.5 The ARMA Process [Seite 125]
9.9.6 - 3.9.6 Frequency Domain Representation [Seite 130]
9.9.7 - 3.9.7 Remarks [Seite 137]
9.10 - 3A.1 Appendix: Variance of Linear System Driven by White Noise [Seite 137]
9.11 - 3A.2 Appendix: Verification of Spectral Formula [Seite 139]
9.12 - References [Seite 140]
10 - 4 Markov Chain Monte Carlo [Seite 141]
10.1 - 4.1 Problem Context [Seite 141]
10.2 - 4.2 Metropolis Algorithm [Seite 144]
10.2.1 - 4.2.1 Proposal PDF [Seite 145]
10.2.2 - 4.2.2 Statistical Properties [Seite 145]
10.2.3 - 4.2.3 Detailed Balance [Seite 150]
10.2.4 - 4.2.4 Biased Rejection [Seite 154]
10.2.5 - 4.2.5 Reversible Chain [Seite 156]
10.3 - 4.3 Metropolis-Hastings Algorithm [Seite 156]
10.3.1 - 4.3.1 Detailed Balance [Seite 157]
10.3.2 - 4.3.2 Independent Proposal and Importance Sampling [Seite 157]
10.4 - 4.4 Statistical Estimation [Seite 159]
10.4.1 - 4.4.1 Properties of Estimator [Seite 159]
10.4.2 - 4.4.2 Chain Correlation [Seite 161]
10.4.3 - 4.4.3 Ergodicity [Seite 165]
10.5 - 4.5 Generation of Conditional Samples [Seite 170]
10.5.1 - 4.5.1 Curse of Dimension [Seite 171]
10.5.2 - 4.5.2 Independent Component MCMC [Seite 174]
10.6 - References [Seite 177]
11 - 5 Subset Simulation [Seite 179]
11.1 - 5.1 Standard Algorithm [Seite 179]
11.1.1 - 5.1.1 Simulation Level 0 (Direct Monte Carlo) [Seite 180]
11.1.2 - 5.1.2 Simulation Level (MCMC) [Seite 181]
11.2 - 5.2 Understanding the Algorithm [Seite 182]
11.2.1 - 5.2.1 Direct Monte Carlo Indispensible [Seite 182]
11.2.2 - 5.2.2 Rare Regime Explored by MCMC [Seite 183]
11.2.3 - 5.2.3 Stationary Markov Chain from the Start [Seite 183]
11.2.4 - 5.2.4 Multiple Chains [Seite 183]
11.2.5 - 5.2.5 Seeds Discarded [Seite 184]
11.2.6 - 5.2.6 CCDF Perspective [Seite 184]
11.2.7 - 5.2.7 Repeated Samples [Seite 184]
11.2.8 - 5.2.8 Uniform Conditional Probabilities [Seite 185]
11.3 - 5.3 Error Assessment in a Single Run [Seite 188]
11.3.1 - 5.3.1 Heuristic Argument [Seite 189]
11.3.2 - 5.3.2 Efficiency Over Direct Monte Carlo [Seite 191]
11.4 - 5.4 Implementation Issues [Seite 195]
11.4.1 - 5.4.1 Proposal Distribution [Seite 195]
11.4.2 - 5.4.2 Ergodicity [Seite 195]
11.4.3 - 5.4.3 Generalizations [Seite 196]
11.4.4 - 5.4.4 Level Probability [Seite 197]
11.5 - 5.5 Analysis of Statistical Properties [Seite 201]
11.5.1 - 5.5.1 Random Intervals [Seite 202]
11.5.2 - 5.5.2 Random CCDF Values [Seite 203]
11.5.3 - 5.5.3 Summary of Results [Seite 204]
11.5.4 - 5.5.4 Expectation [Seite 205]
11.5.5 - 5.5.5 Variance [Seite 207]
11.6 - 5.6 Auxiliary Response [Seite 212]
11.6.1 - 5.6.1 Statistical Properties [Seite 214]
11.6.2 - 5.6.2 Design of Driving Response [Seite 216]
11.7 - 5.7 Black Swan Events [Seite 217]
11.7.1 - 5.7.1 Diagnosis [Seite 219]
11.8 - 5.8 Applications [Seite 221]
11.9 - 5.9 Variants [Seite 223]
11.10 - References [Seite 224]
12 - 6 Analysis Using Conditional Failure Samples [Seite 227]
12.1 - 6.1 Probabilistic Failure Analysis [Seite 228]
12.2 - 6.2 Uncertain Parameter Sensitivity [Seite 229]
12.3 - 6.3 Conditional Samples from Direct Monte Carlo [Seite 230]
12.3.1 - 6.3.1 Conditional Expectation [Seite 230]
12.3.2 - 6.3.2 Parameter Sensitivity [Seite 232]
12.4 - 6.4 Conditional Samples from Subset Simulation [Seite 238]
12.4.1 - 6.4.1 Sample Partitioning [Seite 239]
12.4.2 - 6.4.2 Conditioning Structure [Seite 241]
12.4.3 - 6.4.3 Conditional Expectation [Seite 242]
12.4.4 - 6.4.4 Parameter Sensitivity [Seite 246]
12.5 - References [Seite 253]
13 - 7 Spreadsheet Implementation [Seite 255]
13.1 - 7.1 Microsoft Excel and VBA [Seite 255]
13.1.1 - 7.1.1 Excel Spreadsheet [Seite 256]
13.1.2 - 7.1.2 Illustrative Example - Polynomial Function [Seite 258]
13.1.3 - 7.1.3 Visual Basic for Applications (VBA) [Seite 264]
13.1.4 - 7.1.4 VBA User-Defined Functions [Seite 267]
13.1.5 - 7.1.5 VBA Subroutines [Seite 269]
13.1.6 - 7.1.6 Macro Recorder [Seite 273]
13.2 - 7.2 Software Package UPSS [Seite 277]
13.2.1 - 7.2.1 Installation in Excel 2003 [Seite 277]
13.2.2 - 7.2.2 Installation in Excel 2010 [Seite 280]
13.2.3 - 7.2.3 Software Context [Seite 282]
13.2.4 - 7.2.4 Deterministic System Modeling [Seite 283]
13.2.5 - 7.2.5 Uncertainty Modeling [Seite 284]
13.2.6 - 7.2.6 Uncertainty Propagation [Seite 284]
13.2.7 - 7.2.7 Pre-Processing Tools [Seite 287]
13.2.8 - 7.2.8 Post-Processing Tools [Seite 290]
13.3 - 7.3 Tutorial Example - Polynomial Function [Seite 291]
13.3.1 - 7.3.1 Deterministic System Modeling [Seite 292]
13.3.2 - 7.3.2 Uncertainty Modeling [Seite 292]
13.3.3 - 7.3.3 Uncertainty Propagation [Seite 294]
13.3.4 - 7.3.4 Direct Monte Carlo [Seite 296]
13.3.5 - 7.3.5 Subset Simulation [Seite 297]
13.4 - 7.4 Tutorial Example - Slope Stability [Seite 300]
13.4.1 - 7.4.1 Problem Context [Seite 300]
13.4.2 - 7.4.2 Deterministic System Modeling [Seite 301]
13.4.3 - 7.4.3 Uncertainty Modeling [Seite 301]
13.4.4 - 7.4.4 Histogram Tool [Seite 303]
13.4.5 - 7.4.5 Uncertainty Propagation [Seite 304]
13.4.6 - 7.4.6 CCDF of Driving Variable [Seite 308]
13.4.7 - 7.4.7 Auxiliary Variable [Seite 308]
13.5 - 7.5 Tutorial Example - Portal Frame [Seite 310]
13.5.1 - 7.5.1 Problem Context [Seite 311]
13.5.2 - 7.5.2 Deterministic System Modeling [Seite 312]
13.5.3 - 7.5.3 Uncertainty Modeling [Seite 313]
13.5.4 - 7.5.4 Uncertainty Propagation [Seite 316]
13.5.5 - 7.5.5 Transforming Standard Normal Random Variables [Seite 317]
13.5.6 - 7.5.6 Introducing Correlation [Seite 321]
13.6 - References [Seite 324]
14 - Appendix A: Mathematical Tools [Seite 325]
14.1 - A.1 Calculus [Seite 325]
14.1.1 - A.1.1 Lagrange Multiplier Method [Seite 325]
14.1.2 - A.1.2 Asymptotics [Seite 326]
14.2 - A.2 Linear Algebra [Seite 326]
14.2.1 - A.2.1 Linear Independence, Span, Basis [Seite 326]
14.2.2 - A.2.2 Orthogonality and Norm [Seite 327]
14.2.3 - A.2.3 Gram-Schmidt Procedure [Seite 328]
14.2.4 - A.2.4 Eigenvalue Problem [Seite 329]
14.2.5 - A.2.5 Real Symmetric Matrices [Seite 329]
14.2.6 - A.2.6 Function of Real Symmetric Matrices [Seite 330]
14.3 - A.3 Probability Theory [Seite 331]
14.3.1 - A.3.1 Conditional Expectation [Seite 331]
14.3.2 - A.3.2 Conditional Variance Formula [Seite 332]
14.3.3 - A.3.3 Chebyshevs Inequality [Seite 332]
14.3.4 - A.3.4 Jensens Inequality [Seite 332]
14.3.5 - A.3.5 Modes of Stochastic Convergence [Seite 333]
15 - Index [Seite 335]
