Recent advancements in Markov chain Monte Carlo (MCMC) sampling and surrogate modelling have significantly enhanced the feasibility of Bayesian analysis across engineering fields. However, the selection and integration of surrogate models and cutting-edge MCMC algorithms, often depend on ad-hoc decisions. A systematic assessment of their combined influence on accuracy and efficiency is notably lacking. The present work offers a comprehensive comparative study, employing a scalable case study in computational mechanics focused on the inference of spatially varying material parameters, that sheds light on the impact of methodological choices for surrogate modelling and sampling. We show that a priori training of the surrogate model introduces large errors in the posterior estimation even in low to moderate dimensions. We introduce a simple active learning strategy based on the path of the MCMC algorithm that is superior to all a priori trained models, and determine its training data requirements. We demonstrate that the choice of the MCMC algorithm has only a small influence on the amount of training data but no significant influence on the accuracy of the resulting surrogate model. Further, we show that the accuracy of the posterior estimation largely depends on the surrogate model, but not even a tailored surrogate guarantees convergence of the MCMC. Finally, we identify the forward model as the bottleneck in the inference process, not the MCMC algorithm. While related works focus on employing advanced MCMC algorithms, we demonstrate that the training data requirements render the surrogate modelling approach infeasible before the benefits of these gradient-based MCMC algorithms on cheap models can be reaped.

A key challenge in structural health monitoring is the inference of material properties from measurements. The challenge is particularly acute for cases that involve spatial distributions of uncertain material parameters. These spatially distributed parameters form a random field with high stochastic dimensionality. Bayesian analysis offers a robust method to infer these parameter distributions. However, its dependence on approximations such as Markov chain Monte Carlo (MCMC) sampling, which require many model evaluations, limits the applicability of Bayesian analysis, especially when the model, e.g. a finite element model, is expensive to evaluate. To address this challenge, we investigate the applicability of the Zig-Zag sampling process, a Markov process-based sampler with linear deterministic dynamics, for inferring material parameter fields from displacements measurements.

In theory, the Zig-Zag process offers excellent mixing and low autocorrelation in high-dimensional parameter spaces. However, its application has been limited to simple distributions due to the need for a global upper bound on the gradient of the posterior, a quantity typically unavailable in non-linear Bayesian inverse problems. To address this, we employ a surrogate model to approximate the posterior gradient, allowing us to globally estimate this upper bound and simulate the process efficiently. The bias introduced by the surrogate model is then alleviated with Poisson thinning of the approximate process.

This study marks the first application of a Markov process sampler to Bayesian inference in computational mechanics, yielding promising results. Our methodology demonstrates that the Zig-Zag sampler outperforms traditional MCMC methods, particularly in terms of full model evaluations needed to reach the same accuracy in the posterior moments. Nonetheless, our findings underscore the challenges introduced by the bias of the surrogate model. We present strategies to reduce the impact of correcting for this bias on the efficiency of the sampler.

MUDE stands for Modelling, Uncertainty and Data for Engineers, a required module in the MSc programs from the faculty of Civil Engineering and Geosciences at Delft University of Technology in the Netherlands.

The current version of the MUDE Textbook can be found at mude.citg.tudelft.nl/book and the most recent "complete" version is mude.citg.tudelft.nl/book/2024. Additional information about the book and its contents can be found on the Credits Page from 2024; technical information about the book and its source code can be found in the README of the GitHub repository TUDelft-MUDE/book. General information about MUDE can be found at mude.citg.tudelft.nl.

This Zenodo record archives the HTML files and provides a DOI for the MUDE Textbook. In general, the GitHub repository github.com/TUDelft-MUDE/book and book URL mude.citg.tudelft.nl/book should be used as primary links for the book, whereas Zenodo is used as an archive and DOI publisher, providing a "permanent" URL. The book is registrered in TU Delft's Research Portal PURE too.

The recommended citation for the MUDE Textbook is provided on the Credits page of the book (link above); the Zenodo recommendation on the side of this page should not be used (neither should the citation in the source code record).