Likelihood-free Bayesian computation for structural model calibration: a feasibility study

Finite element (FE) model updating is often used to associate FE models with corresponding existing structures for the condition assessment. FE model updating is an inverse problem and prone to be ill-posed and ill-conditioning when there are many errors and uncertainties in both an FE model and its corresponding measurements. In this case, it is important to quantify these uncertainties properly. Bayesian FE model updating is one of the well-known methods to quantify parameter uncertainty by updating our prior belief on the parameters with the available measurements. In Bayesian inference, likelihood plays a central role in summarizing the overall residuals between model predictions and corresponding measurements. Therefore, likelihood should be carefully chosen to reflect the characteristics of the residuals. It is generally known that very little or no information is available regarding the statistical characteristics of the residuals. In most cases, the likelihood is assumed to be the independent identically distributed Gaussian distribution with the zero mean and constant variance. However, this assumption may cause biased and over/underestimated estimates of parameters, so that the uncertainty quantification and prediction are questionable. To alleviate the potential misuse of the inadequate likelihood, this study introduced approximate Bayesian computation (i.e., likelihood-free Bayesian inference), which relaxes the need for an explicit likelihood by analyzing the behavior similarities between model predictions and measurements. We performed FE model updating based on likelihood-free Markov chain Monte Carlo (MCMC) without using the likelihood. Based on the result of the numerical study, we observed that the likelihood-free Bayesian computation can quantify the updating parameters correctly and its predictive capability for the measurements, not used in calibrated, is also secured.

[1]  D. Rubin,et al.  Inference from Iterative Simulation Using Multiple Sequences , 1992 .

[2]  Costas Papadimitriou,et al.  On prediction error correlation in Bayesian model updating , 2013 .

[3]  D. Rubin.  Bayesianly Justifiable and Relevant Frequency Calculations for the Applied Statistician , 1984 .

[4]  Hyung-Jo Jung,et al.  A new multi-objective approach to finite element model updating , 2014 .

[5]  Guido De Roeck,et al.  Damage detection and parameter identification by finite element model updating , 2005 .

[6]  J. Vrugt,et al.  Toward diagnostic model calibration and evaluation: Approximate Bayesian computation , 2013 .

[7]  Costas Papadimitriou,et al.  Variability of updated finite element models and their predictions consistent with vibration measurements , 2012 .

[8]  I. Smith,et al.  Structural identification with systematic errors and unknown uncertainty dependencies , 2013 .

[9]  J. Beck,et al.  Bayesian Model Updating Using Hybrid Monte Carlo Simulation with Application to Structural Dynamic Models with Many Uncertain Parameters , 2009 .

[10]  P. Donnelly,et al.  Inferring coalescence times from DNA sequence data. , 1997, Genetics.

[11]  R. Wilkinson.  Approximate Bayesian computation (ABC) gives exact results under the assumption of model error , 2008, Statistical applications in genetics and molecular biology.

[12]  P. Guillaume,et al.  Finite element model updating taking into account the uncertainty on the modal parameters estimates , 2006 .

[13]  J. Beck,et al.  Updating Models and Their Uncertainties. I: Bayesian Statistical Framework , 1998 .

[14]  A. Emin Aktan,et al.  Structural identification of constructed systems : approaches, methods, and technologies for effective practice of St-Id , 2013 .

[15]  P. Diggle,et al.  Monte Carlo Methods of Inference for Implicit Statistical Models , 1984 .

[16]  He-sheng Tang,et al.  Differential evolution strategy for structural system identification , 2008 .

[17]  K. F. Gauss,et al.  Theoria combinationis observationum erroribus minimis obnoxiae , 1823 .

[18]  M. Beaumont.  Approximate Bayesian Computation in Evolution and Ecology , 2010 .

[19]  D. Higdon,et al.  Accelerating Markov Chain Monte Carlo Simulation by Differential Evolution with Self-Adaptive Randomized Subspace Sampling , 2009 .

[20]  Andreas Huth,et al.  Statistical inference for stochastic simulation models--theory and application. , 2011, Ecology letters.

[21]  John E. Mottershead,et al.  The sensitivity method in finite element model updating: A tutorial (vol 25, pg 2275, 2010) , 2011 .

[22]  Ka-Veng Yuen,et al.  Bayesian Methods for Updating Dynamic Models , 2011 .

[23]  Brandon M. Turner,et al.  Journal of Mathematical Psychology a Tutorial on Approximate Bayesian Computation , 2022 .

[24]  Chan Ghee Koh,et al.  Modified genetic algorithm strategy for structural identification , 2006 .

[25]  J. Beck,et al.  Bayesian Updating of Structural Models and Reliability using Markov Chain Monte Carlo Simulation , 2002 .

[26]  J. Ching,et al.  Transitional Markov Chain Monte Carlo Method for Bayesian Model Updating, Model Class Selection, and Model Averaging , 2007 .

[27]  K. D. Murphy,et al.  Bayesian identification of a cracked plate using a population-based Markov Chain Monte Carlo method , 2011 .

[28]  Brandon M. Turner,et al.  Approximate Bayesian computation with differential evolution , 2012 .