Abstract
The local size of computational grids used in partial differential equation (PDE)-based probabilistic inverse problems can have a tremendous impact on the numerical results. As a consequence, numerical model identification procedures used in structural or material engineering may yield erroneous, mesh-dependent result. In this work, we attempt to connect the field of adaptive methods for deterministic and forward probabilistic finite-element (FE) simulations and the field of FE-based Bayesian inference. In particular, our target setting is that of exact inference, whereby complex posterior distributions are to be sampled using advanced Markov Chain Monte Carlo (MCMC) algorithms. Our proposal is for the mesh refinement to be performed in a goal-oriented manner. We assume that we are interested in a finite subset of quantities of interest (QoI) such as a combination of latent uncertain parameters and/or quantities to be drawn from the posterior predictive distribution. Next, we evaluate the quality of an approximate inversion with respect to these quantities. This is done by running two chains in parallel: (i) the approximate chain and (ii) an enhanced chain whereby the approximate likelihood function is corrected using an efficient deterministic error estimate of the error introduced by the spatial discretisation of the PDE of interest. One particularly interesting feature of the proposed approach is that no user-defined tolerance is required for the quality of the QoIs, as opposed to the deterministic error estimation setting. This is because our trust in the model, and therefore a good measure for our requirement in terms of accuracy, is fully encoded in the prior. We merely need to ensure that the finite element approximation does not impact the posterior distributions of QoIs by a prohibitively large amount. We will also propose a technique to control the error introduced by the MCMC sampler, and demonstrate the validity of the combined mesh and algorithmic quality control strategy.
Highlights
The Bayesian statistical framework has been used extensively in the problem of system identification [1] or model updating based on experimental test data [2,3]
We have presented a methodology to control the various sources of errors arising in finite-element based Bayesian inverse problems
We have focussed on a simple numerical approximation chain consisting of (i) a finite element discretisation of the continuum mechanics problem and (ii) a Markov Chain Monte Carlo (MCMC)
Summary
The Bayesian statistical framework has been used extensively in the problem of system identification [1] or model updating based on experimental test data [2,3]. Coupling engineering uncertainty quantification (UQ) with an adaptive scheme for goal-oriented finite element model refinement remains a sparsely studied domain and presents significant challenges It is so both from the problem formulation perspective, owing to the choice of appropriate candidate estimates based on which adaptive model enrichment can be performed, as well as incorporating it into the general formulation of Bayesian inversion. The main focus of the paper is to develop a robust methodology for the simultaneous control of errors from multiple sources—the goal-oriented finite element error and the uncertainty-driven statistical error—in a Bayesian identification framework for identification system parameters conditional on data (experimental or otherwise).
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