Low-rank statistical finite elements for scalable model-data synthesis

• Published in: Journal of computational physics 2022-08, Vol.463, p.111261, Article 111261
• Main Authors: Duffin, Connor, Cripps, Edward, Stemler, Thomas, Girolami, Mark
• Format: Article
• Language: English
• Subjects: Bayesian analysis; Bayesian filtering; Bayesian inverse problems; Complex systems; Computational physics; Covariance matrix; Embedding; Finite element method; Finite element methods; Mathematical analysis; Mathematical models; Reaction-diffusion; Statistical analysis
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2022.111261

Statistical learning additions to physically derived mathematical models are gaining traction in the literature. A recent approach has been to augment the underlying physics of the governing equations with data driven Bayesian statistical methodology. Coined statFEM, the method acknowledges a priori model misspecification, by embedding stochastic forcing within the governing equations. Upon receipt of additional data, the posterior distribution of the discretised finite element solution is updated using classical Bayesian filtering techniques. The resultant posterior jointly quantifies uncertainty associated with the ubiquitous problem of model misspecification and the data intended to represent the true process of interest. Despite this appeal, computational scalability is a challenge to statFEM's application to high-dimensional problems typically experienced in physical and industrial contexts. This article overcomes this hurdle by embedding a low-rank approximation of the underlying dense covariance matrix, obtained from the leading order modes of the full-rank alternative. Demonstrated on a series of reaction-diffusion problems of increasing dimension, using experimental and simulated data, the method reconstructs the sparsely observed data-generating processes with minimal loss of information, in both the posterior mean and variance, paving the way for further integration of physical and probabilistic approaches to complex systems. •Computationally scalable statistical FEM, with uncertainty quantification.•Online, sequential Bayesian updating of FEM solutions.•Low-rank approximation to Kalman filtering.•Illustrated on reaction-diffusion processes.•Makes use of parallelisation and GPU computing.