Articlegreen2023

Theoretical Guarantees for the Statistical Finite Element Method

Yanni Papandreou · Jon Cockayne · Mark Girolami · A. Duncan

9 citations0/0 files storedSIAM/ASA Journal on Uncertainty Quantification

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Probabilistic and Robust Engineering Design Advanced Numerical Methods in Computational Mathematics Model Reduction and Neural Networks

Abstract

Abstract. The statistical finite element method (StatFEM) is an emerging probabilistic method that allows observations of a physical system to be synthesized with the numerical solution of a PDE intended to describe it in a coherent statistical framework, to compensate for model error. This work presents a new theoretical analysis of the StatFEM demonstrating that it has similar convergence properties to the finite element method on which it is based. Our results constitute a bound on the 2-Wasserstein distance between the ideal prior and posterior and the StatFEM approximation thereof, and show that this distance converges at the same mesh-dependent rate as finite element solutions converge to the true solution. Several numerical examples are presented to demonstrate our theory, including an example which tests the robustness of StatFEM when extended to nonlinear quantities of interest.

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References

2011An Explicit Link between Gaussian Fields and Gaussian Markov Random Fields: The Stochastic Partial Differential Equation Approach 2010Numerical Methods for Stochastic Computations: A Spectral Method Approach 2008The stochastic finite element method: Past, present and future 2014An Introduction to Computational Stochastic PDEs 1997Uncertainties in probabilistic numerical analysis of structures and solids-Stochastic finite elements 2020The statistical finite element method (statFEM) for coherent synthesis of observation data and model predictions 2016Statistical analysis of differential equations: introducing probability measures on numerical solutions 2018Structural reliability and stochastic finite element methods

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