Uncertainty quantification for random domains using periodic random variables

• Published in: Numerische Mathematik Vol. 156; no. 1; pp. 273 - 317
• Main Authors: Hakula, Harri, Harbrecht, Helmut, Kaarnioja, Vesa, Kuo, Frances Y., Sloan, Ian H.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2024; Springer Nature B.V
• Subjects: Domains; Error analysis; Fields (mathematics); Independent variables; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Numerical Analysis; Numerical and Computational Physics; Parameterization; Periodic functions; Random variables; Simulation; Stochastic processes; Theoretical; Uncertainty; 65D30; 35R60; 65D32
• ISSN: 0029-599X; 0945-3245
• DOI: 10.1007/s00211-023-01392-6

We consider uncertainty quantification for the Poisson problem subject to domain uncertainty. For the stochastic parameterization of the random domain, we use the model recently introduced by Kaarnioja et al. (SIAM J. Numer. Anal., 2020) in which a countably infinite number of independent random variables enter the random field as periodic functions. We develop lattice quasi-Monte Carlo (QMC) cubature rules for computing the expected value of the solution to the Poisson problem subject to domain uncertainty. These QMC rules can be shown to exhibit higher order cubature convergence rates permitted by the periodic setting independently of the stochastic dimension of the problem. In addition, we present a complete error analysis for the problem by taking into account the approximation errors incurred by truncating the input random field to a finite number of terms and discretizing the spatial domain using finite elements. The paper concludes with numerical experiments demonstrating the theoretical error estimates.