Contents

• Preface
• List of Figures
• List of Tables
• Contributors
• Part I Fundamentals

1. Introduction to Bayesian Inverse Problems
   • Juan Chiachío-Ruano, Manuel Chiachío-Ruano and Shankar Sankararaman
   • 1.1 Introduction
   • 1.2 Sources of uncertainty
   • 1.3 Formal definition of probability
   • 1.4 Interpretations of probability
   • 1.4.1 Physical probability
   • 1.4.2 Subjective probability
   • 1.5 Probability fundamentals
   • 1.5.1 Bayes’ Theorem
   • 1.5.2 Total probability theorem
   • 1.6 The Bayesian approach to inverse problems
   • 1.6.1 The forward problem
   • 1.6.2 The inverse problem
   • 1.7 Bayesian inference of model parameters
   • 1.7.1 Markov Chain Monte Carlo methods
   • 1.7.1.1 Metropolis-Hasting algorithm
   • 1.8 Bayesian model class selection
   • 1.8.1 Computation of the evidence of a model class
   • 1.8.2 Information-theory approach to model-class selection
   • 1.9 Concluding remarks
2. Solving Inverse Problems by Approximate Bayesian Computation
   • Manuel Chiachío-Ruano, Juan Chiachío-Ruano and Maria L. Jal¢n
   • 2.1 Introduction to the ABC method
   • 2.2 Basis of ABC using Subset Simulation
   • 2.2.1 Introduction to Subset Simulation
   • 2.2.2 Subset Simulation for ABC
   • 2.3 The ABC-SubSim algorithm
   • 2.4 Summary
3. Fundamentals of Sequential System Monitoring and Prognostics Methods
   • David E. Acuña-Ureta, Juan Chíachío-Ruano, Manuel Chiachío-Ruano and Marcos E. Orchard
   • 3.1 Fundamentals
   • 3.1.1 Prognostics and SHM
   • 3.1.2 Damage response modelling
   • 3.1.3 Interpreting uncertainty for prognostics
   • 3.1.4 Prognostic performance metrics
   • 3.2 Bayesian tracking methods
   • 3.2.1 Linear Bayesian Processor: The Kalman Filter
   • 3.2.2 Unscented Transformation and Sigma Points: The Unscented Kalman Filter
   • 3.2.3 Sequential Monte Carlo methods: Particle Filters
   • 3.2.3.1 Sequential importance sampling
   • 3.2.3.2 Resampling
   • 3.3 Calculation of EOL and RUL
   • 3.3.1 The failure prognosis problem
   • 3.3.2 Future state prediction
   • 3.4 Summary
4. Parameter Identification Based on Conditional Expectation
   • Elmar Zander, Noémi Friedman and Hermann G. Matthies
   • 4.1 Introduction
   • 4.1.1 Preliminaries—basics of probability and information
   • 4.1.1.1 Random variables
   • 4.1.2 Bayes’ theorem
   • 4.1.3 Conditional expectation
   • 4.2 The Mean Square Error Estimator
   • 4.2.1 Numerical approximation of the MMSE
   • 4.2.2 Numerical examples
   • 4.3 Parameter identification using the MMSE
   • 4.3.1 The MMSE filter
   • 4.3.2 The Kalman filter
   • 4.3.3 Numerical examples
   • 4.4 Conclusion

• Part II Engineering Applications

5. Sparse Bayesian Learning and its Application in Bayesian System Identification
   • Yong Huang and James L. Beck
   • 5.1 Introduction
   • 5.2 Sparse Bayesian learning
   • 5.2.1 General formulation of sparse Bayesian learning with the ARD prior
   • 5.2.2 Bayesian Ockham's razor implementation in sparse Bayesian learning
   • 5.3 Applying sparse Bayesian learning to system identification
   • 5.3.1 Hierarchical Bayesian model class for system identification
   • 5.3.2 Fast sparse Bayesian learning algorithm
   • 5.3.2.1 Formulation
   • 5.3.2.2 Proposed fast SBL algorithm for stiffness inversion
   • 5.3.2.3 Damage assessment
   • 5.4 Case studies
   • 5.5 Concluding remarks
   • Appendices
   • Appendix A: Derivation of MAP estimation equations for α and β
6. Ultrasonic Guided-waves Based Bayesian Damage Localisation and Optimal Sensor Configuration
   • Sergio Cantero-Chinchilla, Juan Chiachío, Manuel Chiachío and Dimitrios Chronopoulos
   • 6.1 Introduction
   • 6.2 Damage localisation
   • 6.2.1 Time-frequency model selection
   • 6.2.1.1 Stochastic embedding of TF models
   • 6.2.1.2 Model parameters estimation
   • 6.2.1.3 Model class assessment
   • 6.2.2 Bayesian damage localisation
   • 6.2.2.1 Probabilistic description of ToF model
   • 6.2.2.2 Model parameter estimation
   • 6.3 Optimal sensor configuration
   • 6.3.1 Value of information for optimal design
   • 6.3.2 Expected value of information
   • 6.3.2.1 Algorithmic implementation
   • 6.4 Summary
7. Fast Bayesian Approach for Stochastic Model Updating using Modal Information from Multiple Setups
   • Wang-Ji Yan, Lambros Katafygiotis and Costas Papadimitriou
   • 7.1 Introduction
   • 7.2 Probabilistic consideration of frequency-domain responses
   • 7.2.1 PDF of multivariate FFT coefficients
   • 7.2.2 PDF of PSD matrix
   • 7.2.3 PDF of the trace of the PSD matrix
   • 7.3 A two-stage fast Bayesian operational modal analysis
   • 7.3.1 Prediction error model connecting modal responses and measurements
   • 7.3.2 Spectrum variables identification using FBSTA
   • 7.3.3 Mode shape identification using FBSDA
   • 7.3.4 Statistical modal information for model updating
   • 7.4 Bayesian model updating with modal data from multiple setups
   • 7.4.1 Structural model class
   • 7.4.2 Formulation of Bayesian model updating
   • 7.4.2.1 The introduction of instrumental variables system mode shapes
   • 7.4.2.2 Probability model connecting 'system mode shapes' and measured local mode shape
   • 7.4.2.3 Probability model for the eigenvalue equation errors
   • 7.4.2.4 Negative log-likelihood function for model updating
   • 7.4.3 Solution strategy
   • 7.5 Numerical example
   • 7.5.1 Robustness test of the probabilistic model of trace of PSD matrix
   • 7.5.2 Bayesian operational modal analysis
   • 7.5.3 Bayesian model updating
   • 7.6 Experimental study
   • 7.6.1 Bayesian operational modal analysis
   • 7.6.2 Bayesian model updating
   • 7.7 Concluding remarks
8. A Worked-out Example of Surrogate-based Bayesian Parameter and Field Identification Methods
   • Noémi Friedman, Claudia Zoccarato, Elmar Zander and Hermann G. Matthies
   • 8.1 Introduction
   • 8.2 Numerical modelling of seabed displacement
   • 8.2.1 The deterministic computation of seabed displacements
   • 8.2.2 Modified probabilistic formulation
   • 8.3 Surrogate modelling
   • 8.3.1 Computation of the surrogate by orthogonal projection
   • 8.3.2 Computation of statistics
   • 8.3.3 Validating surrogate models
   • 8.4 Efficient representation of random fields
   • 8.4.1 Karhunen-Loeve Expansion (KLE)
   • 8.4.2 Proper Orthogonal Decomposition (POD)
   • 8.5 Identification of the compressibility field
   • 8.5.1 Bayes’ Theorem
   • 8.5.2 Sampling-based procedures—the MCMC method
   • 8.5.3 The Kalman filter and its modified versions
   • 8.5.3.1 The Kalman filter
   • 8.5.3.2 The ensemble Kalman filter
   • 8.5.3.3 The PCE-based Kalman filter
   • 8.5.4 Non-linear filters
   • 8.6 Summary, conclusion, and outlook

• Appendices
  • Appendix A: FEM computation of seabed displacements
  • Appendix B: Hermite polynomials
    • B.1 Generation of Hermite Polynomials
    • B.2 Calculation of the norms
    • B.3 Quadrature points and weights
  • Appendix C: Galerkin solution of the Karhunen Loève eigenfunction problem
  • Appendix D: Computation of the PCE Coefficients by Orthogonal projection
• Bibliography
• Index