Statistical Finite Elements via Langevin Dynamics

The recent statistical finite element method (statFEM) provides a coherent framework to synthesize models with observed data. Through embedding uncertainty inside of the governing equations, solutions are updated give posterior distribution which quantifies all sources associated model. However incorporate uncertainty, one must integrate over model parameters, known forward problem quantification. In this paper, we make use Langevin dynamics solve statFEM problem, studying utility unadjusted algorithm (ULA), Metropolis-free Markov chain Monte Carlo sampler, build sample-based characterization otherwise intractable measure. Due structure these methods able without explicit full PDE solves, requiring only sparse matrix-vector products. ULA is also gradient-based, and hence scalable approach up high degrees-of-freedom. Leveraging theory behind Langevin-based samplers, provide theoretical guarantees on sampler performance, demonstrating convergence, for both prior posterior, in Kullback–Leibler divergence Wasserstein-2, further results effect preconditioning. Numerical experiments provided, demonstrate efficacy Python package included.