Engineering Risk Assessment With Subset Simulation

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

Siu-Kui Au, University of Liverpool, UK

Yu Wang, City University of Hong Kong, China

About the Authors xiii

Preface xv

Acknowledgements xvii

Nomenclature xix

1 Introduction 1

1.1 Formulation 2

1.2 Context 5

1.3 Extreme Value Theory 5

1.4 Exclusion 6

1.5 Organization of this Book 7

1.6 Remarks on the Use of Risk Analysis 7

1.7 Conventions 8

References 8

2 A Line of Thought 9

2.1 Numerical Integration 10

2.2 Perturbation 10

2.3 Gaussian Approximation 12

2.3.1 Single Design Point 12

2.3.2 Multiple Design Points 14

2.4 First/Second-Order Reliability Method 14

2.4.1 Context 15

2.4.2 Design Point 16

2.4.3 FORM 17

2.4.4 SORM 18

2.4.5 Connection with Gaussian Approximation 22

2.5 Direct Monte Carlo 24

2.5.1 Unbiasedness 25

2.5.2 Mean-Square Convergence 25

2.5.3 Asymptotic Distribution (Central Limit Theorem) 28

2.5.4 Almost Sure Convergence (Strong Law of Large Numbers) 31

2.5.5 Failure Probability Estimation 32

2.5.6 CCDF Perspective 34

2.5.7 Rare Event Problems 38

2.5.8 Variance Reduction by Conditioning 41

2.6 Importance Sampling 44

2.6.1 Optimal Sampling Density 45

2.6.2 Failure Probability Estimation 45

2.6.3 Shifting Distribution 46

2.6.4 Benefits and Side-Effects 48

2.6.5 Bias 50

2.6.6 Curse of Dimension 53

2.6.7 CCDF Perspective 56

2.7 Subset Simulation 58

2.8 Remarks on Reliability Methods 60

2A.1 Appendix: Laplace Type Integrals 61

References 62

3 Simulation of Standard Random Variable and Process 65

3.1 Pseudo-Random Number 65

3.2 Inversion Principle 66

3.2.1 Continuous Random Variable 67

3.2.2 Discrete Random Variables 67

3.3 Mixing Principle 68

3.4 Rejection Principle 69

3.4.1 Acceptance Probability 71

3.5 Samples of Standard Distribution 72

3.6 Dependent Gaussian Variables 78

3.6.1 Cholesky Factorization 78

3.6.2 Eigenvector Factorization 81

3.7 Dependent Non-Gaussian Variables 83

3.7.1 Nataf Transformation 83

3.7.2 Copula 87

3.8 Correlation through Constraint 89

3.8.1 Uniform in Sphere 89

3.8.2 Gaussian on Hyper-plane 92

3.9 Stationary Gaussian Process 95

3.9.1 Autocorrelation Function and Power Spectral Density 95

3.9.2 Discrete-Time Process 99

3.9.3 Sample Autocorrelation Function and Periodogram 100

3.9.4 Time Domain Representation 101

3.9.5 The ARMA Process 103

3.9.6 Frequency Domain Representation 108

3.9.7 Remarks 115

3A.1 Appendix: Variance of Linear System Driven by White Noise 115

3A.2 Appendix: Verification of Spectral Formula 117

References 118

4 Markov Chain Monte Carlo 119

4.1 Problem Context 119

4.2 Metropolis Algorithm 122

4.2.1 Proposal PDF 123

4.2.2 Statistical Properties 123

4.2.3 Detailed Balance 128

4.2.4 Biased Rejection 132

4.2.5 Reversible Chain 134

4.3 Metropolis–Hastings Algorithm 134

4.3.1 Detailed Balance 135

4.3.2 Independent Proposal and Importance Sampling 135

4.4 Statistical Estimation 137

4.4.1 Properties of Estimator 137

4.4.2 Chain Correlation 139

4.4.3 Ergodicity 143

4.5 Generation of Conditional Samples 148

4.5.1 Curse of Dimension 149

4.5.2 Independent Component MCMC 152

References 155

5 Subset Simulation 157

5.1 Standard Algorithm 157

5.1.1 Simulation Level 0 (Direct Monte Carlo) 158

5.1.2 Simulation Level i = 1,…,m − 1 (MCMC) 159

5.2 Understanding the Algorithm 160

5.2.1 Direct Monte Carlo Indispensible 160

5.2.2 Rare Regime Explored by MCMC 161

5.2.3 Stationary Markov Chain from the Start 161

5.2.4 Multiple Chains 161

5.2.5 Seeds Discarded 162

5.2.6 CCDF Perspective 162

5.2.7 Repeated Samples 162

5.2.8 Uniform Conditional Probabilities 163

5.3 Error Assessment in a Single Run 166

5.3.1 Heuristic Argument 167

5.3.2 Efficiency Over Direct Monte Carlo 169

5.4 Implementation Issues 173

5.4.1 Proposal Distribution 173

5.4.2 Ergodicity 173

5.4.3 Generalizations 174

5.4.4 Level Probability 175

5.5 Analysis of Statistical Properties 179

5.5.1 Random Intervals 180

5.5.2 Random CCDF Values 181

5.5.3 Summary of Results 182

5.5.4 Expectation 183

5.5.5 Variance 185

5.6 Auxiliary Response 190

5.6.1 Statistical Properties 192

5.6.2 Design of Driving Response 194

5.7 Black Swan Events 195

5.7.1 Diagnosis 197

5.8 Applications 199

5.9 Variants 201

References 202

6 Analysis Using Conditional Failure Samples 205

6.1 Probabilistic Failure Analysis 206

6.2 Uncertain Parameter Sensitivity 207

6.3 Conditional Samples from Direct Monte Carlo 208

6.3.1 Conditional Expectation 208

6.3.2 Parameter Sensitivity 210

6.4 Conditional Samples from Subset Simulation 216

6.4.1 Sample Partitioning 217

6.4.2 Conditioning Structure 219

6.4.3 Conditional Expectation 220

6.4.4 Parameter Sensitivity 224

References 231

7 Spreadsheet Implementation 233

7.1 Microsoft Excel and VBA 233

7.1.1 Excel Spreadsheet 234

7.1.2 Illustrative Example – Polynomial Function 236

7.1.3 Visual Basic for Applications (VBA) 242

7.1.4 VBA User-Defined Functions 245

7.1.5 VBA Subroutines 247

7.1.6 Macro Recorder 251

7.2 Software Package UPSS 255

7.2.1 Installation in Excel 2003 255

7.2.2 Installation in Excel 2010 258

7.2.3 Software Context 260

7.2.4 Deterministic System Modeling 261

7.2.5 Uncertainty Modeling 262

7.2.6 Uncertainty Propagation 262

7.2.7 Pre-Processing Tools 265

7.2.8 Post-Processing Tools 268

7.3 Tutorial Example – Polynomial Function 269

7.3.1 Deterministic System Modeling 270

7.3.2 Uncertainty Modeling 270

7.3.3 Uncertainty Propagation 272

7.3.4 Direct Monte Carlo 274

7.3.5 Subset Simulation 275

7.4 Tutorial Example – Slope Stability 278

7.4.1 Problem Context* 278

7.4.2 Deterministic System Modeling 279

7.4.3 Uncertainty Modeling 279

7.4.4 Histogram Tool 281

7.4.5 Uncertainty Propagation 282

7.4.6 CCDF of Driving Variable 286

7.4.7 Auxiliary Variable 286

7.5 Tutorial Example – Portal Frame 288

7.5.1 Problem Context* 289

7.5.2 Deterministic System Modeling 290

7.5.3 Uncertainty Modeling 291

7.5.4 Uncertainty Propagation 294

7.5.5 Transforming Standard Normal Random Variables 295

7.5.6 Introducing Correlation 299

References 302

A Appendix: Mathematical Tools 303

A.1 Calculus 303

A.1.1 Lagrange Multiplier Method 303

A.1.2 Asymptotics 304

A.2 Linear Algebra 304

A.2.1 Linear Independence, Span, Basis 304

A.2.2 Orthogonality and Norm 305

A.2.3 Gram–Schmidt Procedure 306

A.2.4 Eigenvalue Problem 307

A.2.5 Real Symmetric Matrices 307

A.2.6 Function of Real Symmetric Matrices 308

A.3 Probability Theory 309

A.3.1 Conditional Expectation 309

A.3.2 Conditional Variance Formula 310

A.3.3 Chebyshev’s Inequality 310

A.3.4 Jensen’s Inequality 310

A.3.5 Modes of Stochastic Convergence 311

Index 313

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