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

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