ENGINEERING RISK ASSESSMENT WITH SUBSET SIMULATION -- Contents -- About the Authors -- Preface -- Acknowledgements -- Nomenclature -- 1 Introduction -- 1.1 Formulation -- 1.2 Context -- 1.3 Extreme Value Theory -- 1.4 Exclusion -- 1.5 Organization of this Book -- 1.6 Remarks on the Use of Risk Analysis -- 1.7 Conventions -- References -- 2 A Line of Thought -- 2.1 Numerical Integration -- 2.2 Perturbation -- 2.3 Gaussian Approximation -- 2.3.1 Single Design Point -- 2.3.2 Multiple Design Points -- 2.4 First/Second-Order Reliability Method -- 2.4.1 Context -- 2.4.2 Design Point -- 2.4.3 FORM -- 2.4.4 SORM -- 2.4.5 Connection with Gaussian Approximation -- 2.5 Direct Monte Carlo -- 2.5.1 Unbiasedness -- 2.5.2 Mean-Square Convergence -- 2.5.3 Asymptotic Distribution (Central Limit Theorem) -- 2.5.4 Almost Sure Convergence (Strong Law of Large Numbers) -- 2.5.5 Failure Probability Estimation -- 2.5.6 CCDF Perspective -- 2.5.7 Rare Event Problems -- 2.5.8 Variance Reduction by Conditioning -- 2.6 Importance Sampling -- 2.6.1 Optimal Sampling Density -- 2.6.2 Failure Probability Estimation -- 2.6.3 Shifting Distribution -- 2.6.4 Benefits and Side-Effects -- 2.6.5 Bias -- 2.6.6 Curse of Dimension -- 2.6.7 CCDF Perspective -- 2.7 Subset Simulation -- 2.8 Remarks on Reliability Methods -- 2A.1 Appendix: Laplace Type Integrals -- References -- 3 Simulation of Standard Random Variable and Process -- 3.1 Pseudo-Random Number -- 3.2 Inversion Principle -- 3.2.1 Continuous Random Variable -- 3.2.2 Discrete Random Variables -- 3.3 Mixing Principle -- 3.4 Rejection Principle -- 3.4.1 Acceptance Probability -- 3.5 Samples of Standard Distribution -- 3.6 Dependent Gaussian Variables -- 3.6.1 Cholesky Factorization -- 3.6.2 Eigenvector Factorization -- 3.7 Dependent Non-Gaussian Variables -- 3.7.1 Nataf Transformation -- 3.7.2 Copula.

3.8 Correlation through Constraint -- 3.8.1 Uniform in Sphere -- 3.8.2 Gaussian on Hyper-plane -- 3.9 Stationary Gaussian Process -- 3.9.1 Autocorrelation Function and Power Spectral Density -- 3.9.2 Discrete-Time Process -- 3.9.3 Sample Autocorrelation Function and Periodogram -- 3.9.4 Time Domain Representation -- 3.9.5 The ARMA Process -- 3.9.6 Frequency Domain Representation -- 3.9.7 Remarks -- 3A.1 Appendix: Variance of Linear System Driven by White Noise -- 3A.2 Appendix: Verification of Spectral Formula -- References -- 4 Markov Chain Monte Carlo -- 4.1 Problem Context -- 4.2 Metropolis Algorithm -- 4.2.1 Proposal PDF -- 4.2.2 Statistical Properties -- 4.2.3 Detailed Balance -- 4.2.4 Biased Rejection -- 4.2.5 Reversible Chain -- 4.3 Metropolis-Hastings Algorithm -- 4.3.1 Detailed Balance -- 4.3.2 Independent Proposal and Importance Sampling -- 4.4 Statistical Estimation -- 4.4.1 Properties of Estimator -- 4.4.2 Chain Correlation -- 4.4.3 Ergodicity -- 4.5 Generation of Conditional Samples -- 4.5.1 Curse of Dimension -- 4.5.2 Independent Component MCMC -- References -- 5 Subset Simulation -- 5.1 Standard Algorithm -- 5.1.1 Simulation Level 0 (Direct Monte Carlo) -- 5.1.2 Simulation Level (MCMC) -- 5.2 Understanding the Algorithm -- 5.2.1 Direct Monte Carlo Indispensible -- 5.2.2 Rare Regime Explored by MCMC -- 5.2.3 Stationary Markov Chain from the Start -- 5.2.4 Multiple Chains -- 5.2.5 Seeds Discarded -- 5.2.6 CCDF Perspective -- 5.2.7 Repeated Samples -- 5.2.8 Uniform Conditional Probabilities -- 5.3 Error Assessment in a Single Run -- 5.3.1 Heuristic Argument -- 5.3.2 Efficiency Over Direct Monte Carlo -- 5.4 Implementation Issues -- 5.4.1 Proposal Distribution -- 5.4.2 Ergodicity -- 5.4.3 Generalizations -- 5.4.4 Level Probability -- 5.5 Analysis of Statistical Properties -- 5.5.1 Random Intervals -- 5.5.2 Random CCDF Values.

5.5.3 Summary of Results -- 5.5.4 Expectation -- 5.5.5 Variance -- 5.6 Auxiliary Response -- 5.6.1 Statistical Properties -- 5.6.2 Design of Driving Response -- 5.7 Black Swan Events -- 5.7.1 Diagnosis -- 5.8 Applications -- 5.9 Variants -- References -- 6 Analysis Using Conditional Failure Samples -- 6.1 Probabilistic Failure Analysis -- 6.2 Uncertain Parameter Sensitivity -- 6.3 Conditional Samples from Direct Monte Carlo -- 6.3.1 Conditional Expectation -- 6.3.2 Parameter Sensitivity -- 6.4 Conditional Samples from Subset Simulation -- 6.4.1 Sample Partitioning -- 6.4.2 Conditioning Structure -- 6.4.3 Conditional Expectation -- 6.4.4 Parameter Sensitivity -- References -- 7 Spreadsheet Implementation -- 7.1 Microsoft Excel and VBA -- 7.1.1 Excel Spreadsheet -- 7.1.2 Illustrative Example - Polynomial Function -- 7.1.3 Visual Basic for Applications (VBA) -- 7.1.4 VBA User-Defined Functions -- 7.1.5 VBA Subroutines -- 7.1.6 Macro Recorder -- 7.2 Software Package UPSS -- 7.2.1 Installation in Excel 2003 -- 7.2.2 Installation in Excel 2010 -- 7.2.3 Software Context -- 7.2.4 Deterministic System Modeling -- 7.2.5 Uncertainty Modeling -- 7.2.6 Uncertainty Propagation -- 7.2.7 Pre-Processing Tools -- 7.2.8 Post-Processing Tools -- 7.3 Tutorial Example - Polynomial Function -- 7.3.1 Deterministic System Modeling -- 7.3.2 Uncertainty Modeling -- 7.3.3 Uncertainty Propagation -- 7.3.4 Direct Monte Carlo -- 7.3.5 Subset Simulation -- 7.4 Tutorial Example - Slope Stability -- 7.4.1 Problem Context -- 7.4.2 Deterministic System Modeling -- 7.4.3 Uncertainty Modeling -- 7.4.4 Histogram Tool -- 7.4.5 Uncertainty Propagation -- 7.4.6 CCDF of Driving Variable -- 7.4.7 Auxiliary Variable -- 7.5 Tutorial Example - Portal Frame -- 7.5.1 Problem Context -- 7.5.2 Deterministic System Modeling -- 7.5.3 Uncertainty Modeling -- 7.5.4 Uncertainty Propagation.

7.5.5 Transforming Standard Normal Random Variables -- 7.5.6 Introducing Correlation -- References -- Appendix A: Mathematical Tools -- A.1 Calculus -- A.1.1 Lagrange Multiplier Method -- A.1.2 Asymptotics -- A.2 Linear Algebra -- A.2.1 Linear Independence, Span, Basis -- A.2.2 Orthogonality and Norm -- A.2.3 Gram-Schmidt Procedure -- A.2.4 Eigenvalue Problem -- A.2.5 Real Symmetric Matrices -- A.2.6 Function of Real Symmetric Matrices -- A.3 Probability Theory -- A.3.1 Conditional Expectation -- A.3.2 Conditional Variance Formula -- A.3.3 Chebyshevs Inequality -- A.3.4 Jensens Inequality -- A.3.5 Modes of Stochastic Convergence -- Index.