book

Bayesian Inverse Problems

Name: Bayesian Inverse Problems
ISBN: 9781351869652

by Juan Chiachio-Ruano, Manuel Chiachio-Ruano, Shankar Sankararaman

November 2021

Intermediate to advanced

248 pages

8h 12m

English

CRC Press

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Cover
Title Page
Copyright Page
Dedication
Preface
Table of Contents
List of Figures
List of Tables
Contributors
Part I Fundamentals

1. Introduction to Bayesian Inverse Problems
1.1 Introduction1.2 Sources of uncertainty1.3 Formal definition of probability1.4 Interpretations of probability1.4.1 Physical probability1.4.2 Subjective probability1.5 Probability fundamentals1.5.1 Bayes’ Theorem1.5.2 Total probability theorem1.6 The Bayesian approach to inverse problems1.6.1 The forward problem1.6.2 The inverse problem1.7 Bayesian inference of model parameters1.7.1 Markov Chain Monte Carlo methods1.7.1.1 Metropolis-Hasting algorithm1.8 Bayesian model class selection1.8.1 Computation of the evidence of a model class1.8.2 Information-theory approach to model-class selection1.9 Concluding remarks
2. Solving Inverse Problems by Approximate Bayesian Computation
2.1 Introduction to the ABC method2.2 Basis of ABC using Subset Simulation2.2.1 Introduction to Subset Simulation2.2.2 Subset Simulation for ABC2.3 The ABC-SubSim algorithm2.4 Summary
3. Fundamentals of Sequential System Monitoring and Prognostics Methods
3.1 Fundamentals3.1.1 Prognostics and SHM3.1.2 Damage response modelling3.1.3 Interpreting uncertainty for prognostics3.1.4 Prognostic performance metrics3.2 Bayesian tracking methods3.2.1 Linear Bayesian Processor: The Kalman Filter3.2.2 Unscented Transformation and Sigma Points: The Unscented Kalman Filter3.2.3 Sequential Monte Carlo methods: Particle Filters3.2.3.1 Sequential importance sampling3.2.3.2 Resampling3.3 Calculation of EOL and RUL3.3.1 The failure prognosis problem3.3.2 Future state prediction3.4 Summary
4. Parameter Identification Based on Conditional Expectation
4.1 Introduction4.1.1 Preliminaries—basics of probability and information4.1.1.1 Random variables4.1.2 Bayes’ theorem4.1.3 Conditional expectation4.2 The Mean Square Error Estimator4.2.1 Numerical approximation of the MMSE4.2.2 Numerical examples4.3 Parameter identification using the MMSE4.3.1 The MMSE filter4.3.2 The Kalman filter4.3.3 Numerical examples4.4 Conclusion
Part II Engineering Applications
5. Sparse Bayesian Learning and its Application in Bayesian System Identification
5.1 Introduction5.2 Sparse Bayesian learning5.2.1 General formulation of sparse Bayesian learning with the ARD prior5.2.2 Bayesian Ockham's razor implementation in sparse Bayesian learning5.3 Applying sparse Bayesian learning to system identification5.3.1 Hierarchical Bayesian model class for system identification5.3.2 Fast sparse Bayesian learning algorithm5.3.2.1 Formulation5.3.2.2 Proposed fast SBL algorithm for stiffness inversion5.3.2.3 Damage assessment5.4 Case studies5.5 Concluding remarksAppendicesAppendix A: Derivation of MAP estimation equations for α and β
6. Ultrasonic Guided-waves Based Bayesian Damage Localisation and Optimal Sensor Configuration
6.1 Introduction6.2 Damage localisation6.2.1 Time-frequency model selection6.2.1.1 Stochastic embedding of TF models6.2.1.2 Model parameters estimation6.2.1.3 Model class assessment6.2.2 Bayesian damage localisation6.2.2.1 Probabilistic description of ToF model6.2.2.2 Model parameter estimation6.3 Optimal sensor configuration6.3.1 Value of information for optimal design6.3.2 Expected value of information6.3.2.1 Algorithmic implementation6.4 Summary
7. Fast Bayesian Approach for Stochastic Model Updating using Modal Information from Multiple Setups
7.1 Introduction7.2 Probabilistic consideration of frequency-domain responses7.2.1 PDF of multivariate FFT coefficients7.2.2 PDF of PSD matrix7.2.3 PDF of the trace of the PSD matrix7.3 A two-stage fast Bayesian operational modal analysis7.3.1 Prediction error model connecting modal responses and measurements7.3.2 Spectrum variables identification using FBSTA7.3.3 Mode shape identification using FBSDA7.3.4 Statistical modal information for model updating7.4 Bayesian model updating with modal data from multiple setups7.4.1 Structural model class7.4.2 Formulation of Bayesian model updating7.4.2.1 The introduction of instrumental variables system mode shapes7.4.2.2 Probability model connecting 'system mode shapes' and measured local mode shape7.4.2.3 Probability model for the eigenvalue equation errors7.4.2.4 Negative log-likelihood function for model updating7.4.3 Solution strategy7.5 Numerical example7.5.1 Robustness test of the probabilistic model of trace of PSD matrix7.5.2 Bayesian operational modal analysis7.5.3 Bayesian model updating7.6 Experimental study7.6.1 Bayesian operational modal analysis7.6.2 Bayesian model updating7.7 Concluding remarks
8. A Worked-out Example of Surrogate-based Bayesian Parameter and Field Identification Methods
8.1 Introduction8.2 Numerical modelling of seabed displacement8.2.1 The deterministic computation of seabed displacements8.2.2 Modified probabilistic formulation8.3 Surrogate modelling8.3.1 Computation of the surrogate by orthogonal projection8.3.2 Computation of statistics8.3.3 Validating surrogate models8.4 Efficient representation of random fields8.4.1 Karhunen-Loeve Expansion (KLE)8.4.2 Proper Orthogonal Decomposition (POD)8.5 Identification of the compressibility field8.5.1 Bayes’ Theorem8.5.2 Sampling-based procedures—the MCMC method8.5.3 The Kalman filter and its modified versions8.5.3.1 The Kalman filter8.5.3.2 The ensemble Kalman filter8.5.3.3 The PCE-based Kalman filter8.5.4 Non-linear filters8.6 Summary, conclusion, and outlook
Appendices
Appendix A: FEM computation of seabed displacements
Appendix B: Hermite polynomialsB.1 Generation of Hermite PolynomialsB.2 Calculation of the normsB.3 Quadrature points and weightsAppendix C: Galerkin solution of the Karhunen Loève eigenfunction problemAppendix D: Computation of the PCE Coefficients by Orthogonal projection
Bibliography
Index

Overview

This book is devoted to a special class of engineering problems called Bayesian inverse problems. These problems comprise not only the probabilistic Bayesian formulation of engineering problems, but also the associated stochastic simulation methods needed to solve them.

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