Structural Reliability: Statistical Learning Perspectives

Abstract

1 A Discussion on Structural Reliability Methods.- 1.1 Performance and Limit State Functions.- 1.2 Methods Based on the Limit State Function.- 1.3 Transformation of Basic Variables.- 1.3.1 Normal Variables.- 1.3.2 Normal Translation.- 1.3.3 Rosenblatt Transformation.- 1.3.4 Nataf Transformation.- 1.3.5 Polynomial Chaoses.- 1.4 FORM and SORM.- 1.4.1 Basic Equations.- 1.4.2 Discussion.- 1.5 Monte Carlo Methods.- 1.5.1 Importance Sampling.- 1.5.2 Directional Simulation.- 1.5.3 General Characteristics of Simulation Methods.- 1.6 Solver Surrogate Methods.- 1.6.1 Response Surface Method.- 1.6.2 Neural Networks and Support Vector Machines.- 1.6.3 Characteristics of the Response Surface Method.- 1.7 Regression and Classification.- 1.8 FORM and SORM Approximations with Statistical Learning Devices.- 1.9 Methods Based on the Performance Function.- 1.10 Summary.- 2 Fundamental Concepts of Statistical Learning.- 2.1 Introduction.- 2.2 The Basic Learning Problem.- 2.3 Cost and Risk Functions.- 2.4 The Regularization Principle.- 2.5 Complexity and Vapnik-Chervonenkis Dimension.- 2.6 Error Bounds and Structured Risk Minimization.- 2.7 Risk Bounds for Regression.- 2.8 Stringent and Adaptive Models.- 2.9 The Curse of Dimensionality.- 2.10 Dimensionality Increase.- 2.11 Sample Complexity.- 2.12 Selecting a Learning Method in Reliability Analysis.- 2.12.1 Classification Techniques.- 2.12.2 Remarks on Probability Density Estimation.- 2.12.3 Characteristics of Samples in Structural Reliability.- 2.12.4 A Look from the Viewpoint of Information Theory.- 2.12.5 Recommended Methods.- 3 Dimension Reduction and Data Compression.- 3.1 Introduction.- 3.2 Principal Component Analysis.- 3.3 Kernel PCA.- 3.3.1 Basic Equations.- 3.3.2 Kernel Properties and Construction.- 3.3.3 Example 1: Structure of a Monte Carlo Cloud..- 3.3.4 Example 2: Transformation of Reliability Problems.- 3.4 Karhunen-Loeve Expansion.- 3.5 Discrete Wavelet Transform..- 3.6 Data Compression Techniques..- 3.6.1 Vector Quantization.- 3.6.2 Expectation-Maximization..- 4 Classification Methods I - Neural Networks.- 4.1 Introduction.- 4.2 Probabilistic and Euclidean methods.- 4.2.1 Bayesian Classification.- 4.2.2 Classification Trees.- 4.2.3 Concluding Remarks.- 4.3 Multi-Layer Perceptrons..- 4.3.1 Hyperplane Discrimination.- 4.3.2 Polyhedral Discrimination.- 4.4 General Nonlinear Two-Layer Perceptrons.- 4.4.1 Training Algorithms.- 4.4.2 Example.- 4.4.3 Complexity and Dimensionality Issues.- 4.5 Radial Basis Function Networks.- 4.5.1 Approximation Properties.- 4.5.2 A First Comparison of MLP and RBFN.- 4.6 Elements of a General Training Algorithm.- 5 Classification Methods II -1 A Discussion on Structural Reliability Methods.- 1.1 Performance and Limit State Functions.- 1.2 Methods Based on the Limit State Function.- 1.3 Transformation of Basic Variables.- 1.3.1 Normal Variables.- 1.3.2 Normal Translation.- 1.3.3 Rosenblatt Transformation.- 1.3.4 Nataf Transformation.- 1.3.5 Polynomial Chaoses.- 1.4 FORM and SORM.- 1.4.1 Basic Equations.- 1.4.2 Discussion.- 1.5 Monte Carlo Methods.- 1.5.1 Importance Sampling.- 1.5.2 Directional Simulation.- 1.5.3 General Characteristics of Simulation Methods.- 1.6 Solver Surrogate Methods.- 1.6.1 Response Surface Method.- 1.6.2 Neural Networks and Support Vector Machines.- 1.6.3 Characteristics of the Response Surface Method.- 1.7 Regression and Classification.- 1.8 FORM and SORM Approximations with Statistical Learning Devices.- 1.9 Methods Based on the Performance Function.- 1.10 Summary.- 2 Fundamental Concepts of Statistical Learning.- 2.1 Introduction.- 2.2 The Basic Learning Problem.- 2.3 Cost and Risk Functions.- 2.4 The Regularization Principle.- 2.5 Complexity and Vapnik-Chervonenkis Dimension.- 2.6 Error Bounds and Structured Risk Minimization.- 2.7 Risk Bounds for Regression.- 2.8 Stringent and Adaptive Models.- 2.9 The Curse of Dimensionality.- 2.10 Dimensionality Increase.- 2.11 Sample Complexity.- 2.12 Selecting a Learning Method in Reliability Analysis.- 2.12.1 Classification Techniques.- 2.12.2 Remarks on Probability Density Estimation.- 2.12.3 Characteristics of Samples in Structural Reliability.- 2.12.4 A Look from the Viewpoint of Information Theory.- 2.12.5 Recommended Methods.- 3 Dimension Reduction and Data Compression.- 3.1 Introduction.- 3.2 Principal Component Analysis.- 3.3 Kernel PCA.- 3.3.1 Basic Equations.- 3.3.2 Kernel Properties and Construction.- 3.3.3 Example 1: Structure of a Monte Carlo Cloud..- 3.3.4 Example 2: Transformation of Reliability Problems.- 3.4 Karhunen-Loeve Expansion.- 3.5 Discrete Wavelet Transform..- 3.6 Data Compression Techniques..- 3.6.1 Vector Quantization.- 3.6.2 Expectation-Maximization..- 4 Classification Methods I - Neural Networks.- 4.1 Introduction.- 4.2 Probabilistic and Euclidean methods.- 4.2.1 Bayesian Classification.- 4.2.2 Classification Trees.- 4.2.3 Concluding Remarks.- 4.3 Multi-Layer Perceptrons..- 4.3.1 Hyperplane Discrimination.- 4.3.2 Polyhedral Discrimination.- 4.4 General Nonlinear Two-Layer Perceptrons.- 4.4.1 Training Algorithms.- 4.4.2 Example.- 4.4.3 Complexity and Dimensionality Issues.- 4.5 Radial Basis Function Networks.- 4.5.1 Approximation Properties.- 4.5.2 A First Comparison of MLP and RBFN.- 4.6 Elements of a General Training Algorithm.- 5 Classification Methods II - Support Vector Machines.- 5.1 Introduction.- 5.2 Support Vector Machines.- 5.2.1 Linearly Separable Classes..- 5.2.2 Nonlinear Separation.- 5.2.3 Solution of the Optimization Problem..- 5.3 A Remark on Polynomial Chaoses.- 5.4 Genetic Algorithm..- 5.4.1 General Considerations..- 5.4.2 Algorithm.- 5.5 Active Learning Algorithms.- 5.5.1 Algorithm Based on Margin Shrinking.- 5.5.2 Algorithm Based on Version Space Shrinking.- 5.6 A Comparison with Neural Classifiers.- 5.7 Complexity, Dimensionality and Induction of SV Machines.- 5.8 Application Examples.- 5.8.1 Parabolic Limit State Function.- 5.8.2 A Linear Limit State Function with Nonlinear Performance Function.- 5.8.3 Two- and Twenty-Dimensional SORM Functions.- 5.8.4 Ten Dimensional Problem.- 5.8.5 An Application of the Version Space Algorithm.- 5.8.6 Bound of the VC Dimension of the SORM Function.- 5.9 An Application to Stochastic Stability.- 5.9.1 Asymptotic Moment Stability.- 5.9.2 Numerical Example.- 5.10 Other Kernel Classification Algorithms.- 6 Regression Methods.- 6.1 Introduction.- 6.2 The Response Surface Method Revisited.- 6.2.1 Dimensionality Problems.- 6.2.2 Performance Function Approximation.- 6.2.3 Naive Inductive Principle.- 6.3 Neural Networks.- 6.3.1 Boosting.- 6.3.2 A Second Comparison of MLP and RBFN.- 6.3.3 Example: Full Probabilistic Analysis with Stochastic Finite Elements.- 6.4 Support Vector Regression.- 6.4.1 Support Vector Approach to Non-Separable Classes.- 6.4.2 Extension to Function Approximation..- 6.4.3 Example: Random Eigenvalues of a Frame.- 6.5 Time-Dependent MLP for Random Vibrations.- 7 Classification Approaches to Reliability Indexation.- 7.1 Introduction.- 7.2 A Discussion on Reliability Indices.- 7.3 A Comparison of Hyperplane Approximations.- 7.4 Secant Hyperplane Reliability Index.- 7.4.1 Index Properties.- 7.5 Volumetric Reliability Index.- 7.5.1 Derivation of the Index.- 7.5.2 Index Properties.- References.- Essential Symbols.

Cite this work

No public copy is stored for this record, so only the metadata is cited.

Stored only when both papers are in the corpus — 71 citations exist upstream.