A probabilistic data assimilation framework to reconstruct finite element error fields from sparse error estimates : application to sub-modelling

https://onlinelibrary.wiley.com/doi/10.1002/nme.7090

In the present work, we propose a computational pipeline to recover full finite element error fields from a few  estimates of errors in scalar quantities of interest (QoI). The approach is weakly intrusive, as it is motivated by large-scale industrial applications wherby modifying the finite element models is undesirable. The goal-oriented error estimation methodology that is chosen is the traditional Zhu-Zienkiewicz (ZZ) approach, which is coupled with the adjoint methodology to deliver goal-oriented results. The novelty of the work is that we consider a set of computed error estimates in QoI as partial observations of an underlying error field, which is to be recovered. We then deploy a Bayesian probabilistic estimation framework, introducing a sparse Gaussian prior for the error field by means of linear stochastic partial differential equations (SPDE), with two adjustable parameters that may be tuned via maximum likelihood (which is made tractable by the SPDE approach). As estimating the posterior state of the error field is a numerical bottleneck, despite the employment of the SPDE-based prior, we propose a projection-based reduced order modelling strategy to reduce the cost of using the SPDE model. The projection basis is constructed adaptively, using a goal-oriented divisive clustering approach that is subsequently used to construct a family of radial basis functions satisfying the partition-of-unity property over the computational domain. We show that the Bayesian reconstruction approach, accelerated by the proposed model reduction technology, yields good probabilistic estimates of full error fields, with a computational complexity that is acceptable compared to the evaluation of the ZZ goal-oriented error estimates that must be provided as input to the algorithm. The strategy is applied to submodelling, whereby the global model is solved using a relatively coarse finite element discretisation, and the effect of the numerical error onto submodelling results is to be controlled. To achieve this, we probabilistically recover full error fields over boundaries of submodelling regions, which we propagate to the submodels using a standard Monte-Carlo approach. Future improvements of the method include the optimal selection of goal-oriented error measures to be acquired prior to the error field reconstruction.