Next Article in Journal
Numerical Simulation of Dynamic Mechanical Properties of Concrete under Uniaxial Compression
Next Article in Special Issue
Correction: Ungson, Y. et al. Filling of Irregular Channels with Round Cross-Section: Modeling Aspects to Study the Properties of Porous Materials. Materials 2018, 11, 1901
Previous Article in Journal
Bio-Based Polymers with Antimicrobial Properties towards Sustainable Development
Previous Article in Special Issue
Stochastic Constitutive Model of Isotropic Thin Fiber Networks Based on Stochastic Volume Elements
Article Menu
Issue 4 (February-2) cover image

Export Article

Open AccessArticle

Adaptivity in Bayesian Inverse Finite Element Problems: Learning and Simultaneous Control of Discretisation and Sampling Errors

1
School of Engineering, Cardiff University, The Parade, Cardiff CF243AA, UK
2
MINES ParisTech, PSL University, Centre des Matériaux, BP87 91003 Evry, France
3
Department of Computer Science, University of Copenhagen, Universitetsparken 1, 2100 Copenhagen, Denmark
*
Author to whom correspondence should be addressed.
Materials 2019, 12(4), 642; https://doi.org/10.3390/ma12040642
Received: 23 December 2018 / Revised: 2 February 2019 / Accepted: 5 February 2019 / Published: 20 February 2019
(This article belongs to the Special Issue Randomness and Uncertainty)
  |  
PDF [7023 KB, uploaded 25 February 2019]
  |  

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. View Full-Text
Keywords: finite element inverse problems; Bayesian statistics; data-driven modelling; error estimation; MCMC; machine learning finite element inverse problems; Bayesian statistics; data-driven modelling; error estimation; MCMC; machine learning
Figures

Figure 1

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
SciFeed

Share & Cite This Article

MDPI and ACS Style

Kerfriden, P.; Kundu, A.; Claus, S. Adaptivity in Bayesian Inverse Finite Element Problems: Learning and Simultaneous Control of Discretisation and Sampling Errors. Materials 2019, 12, 642.

Show more citation formats Show less citations formats

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Related Articles

Article Metrics

Article Access Statistics

1

Comments

[Return to top]
Materials EISSN 1996-1944 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top