Bayesian Model Updating Tutorials:

Bayesian Model Updating is a technique which casts the model updating problem in the form of a Bayesian Inference. There have been 3 popular advanced Monte Carlo sampling techniques which are adopted by researchers to address Bayesian Model Updating problems and make the necessary estimations of the epistemic parameter(s). These 3 techniques are:

• Markov Chain Monte Carlo (MCMC)
• Transitional Markov Chain Monte Carlo (TMCMC)
• Sequential Monte Carlo (SMC)

In this repository, 3 tutorials are presented to enable users to understand how the advanced Monte Carlo techniques are implemented in addressing various Bayesian Model Updating problems. The following tutorials are (in order of increasing difficulty):

• 1-Dimensional Linear Static System
• 1-Dimensional Simple Harmonic Oscillator
• 2-Dimensional Eigenvalue Problem

Tutorials:

1) 1-Dimensional Linear Static System:

This tutorial presents a simple static Spring-Mass system. In this set-up, the spring is assumed to obey Hooke's Law model whereby the restoring force of the spring, F, is linearly proportional to the length of its displacement from rest length, d. The elasticity constant of the spring is k. This study, seeks to realize two objectives:

1. To compare the estimation the epistemic parameter k;

2. To compare the model updating results obtained through the use of MCMC, TMCMC, and SMC.

2) 1-Dimensional Simple Harmonic Oscillator:

This tutorial presents a simple harmonic oscillator system. In this set-up, the natural oscillating frequency of the ocillator, F, obeys the Simple Harmonic Frequency model whereby F is defined as the square-root of the ratio between the elasticity constant of the spring, k, and the mass of the body attached to the oscillator, m. This study, seeks to realize two objectives:

1. To compare the estimation the epistemic parameter k;

2. To compare the model updating results obtained through the use of MCMC, TMCMC, and SMC.

3) 2-Dimensional Eigenvalue Problem:

This tutorial presents a 2-by-2 square matrix in which there exists two distinct real eignvalue solutions. The matrix elements here are defined by two epistemic parameters: Theta 1 and Theta 2. This tutorial seeks to achieve three objectives:

1. To observe the performance of each of the advanced Monte Carlo samplers in obtaining samples from a 2-dimensional, bi-modal posterior distribution;

2. To estimate the solutions to the epistemic parameters: Theta 1 and Theta 2;

3. To compare the model updating results obtained through the use of MCMC, TMCMC, and SMC.

4) Alternative TMCMC Transition criteria:

The work explores a possible alternative transitional criteria for the TMCMC sampler involving the use of the Effective sample size metric. The alternative transitional criteria is such that in transiting from one transitional distribution to another, the Effective sample size has to be half the total sample size. The alternative TMCMC sampler is referred to as the TMCMC-II sampler and it will be implemented on a SDoF structure subjected to an unknown Coulomb friction. The code to be executed is named: "example_SDOF_System_Coulomb_Friction_numerical.m"

The work was presented at the 33rd European Safety and Reliability Conference (ESREL 2023) held in Southampton, United Kingdom.

Reference(s):

• A. Lye, A. Cicirello, and E. Patelli (2021). Sampling methods for solving Bayesian model updating problems: A tutorial. Mechanical Systems and Signal Processing, 159, 107760. doi: 10.1016/j.ymssp.2021.107760
• A. Lye, and L. Marino (2023). An investigation into an alternative transition criterion of the Transitional Markov Chain Monte Carlo method for Bayesian model updating. In Proceedings of the 33rd European Safety and Reliability Conference, 1. doi: 10.3850/978-981-18-8071-1_P331-cd

Author:

• Name: Adolphus Lye
• Contact: adolphus.lye@liverpool.ac.uk
• Affiliation: Insitute for Risk and Uncertainty, University of Liverpool