We consider the application of multilevel Monte Carlo methods to elliptic PDEs with random coefficients. We focus on models of the random coefficient that lack uniform ellipticity and boundedness with respect to the random parameter, and that only have limited spatial regularity. We extend the finite element error analysis for this type of equation, carried out in [6], to more difficult problems, posed on non-smooth domains and with discontinuities in the coefficient. For this wider class of model problem, we prove convergence of the multilevel Monte Carlo algorithm for estimating any bounded, linear functional and any continuously Fréchet differentiable non-linear functional of the solution. We further improve the performance of the multilevel estimator by introducing level dependent truncations of the Karhunen-Loève expansion of the random coefficient. Numerical results complete the paper.
In this paper we analyze the numerical approximation of diffusion problems over polyhedral domains in R d (d = 1, 2, 3), with diffusion coefficient a(x, ω) given as a lognormal random field, i.e., a(x, ω) = exp(Z (x, ω)) where x is the spatial variable and Z (x, ·) is a Gaussian random field. The analysis presents particular challenges since the corresponding bilinear form is not uniformly bounded away from 0 or ∞ over all possible realizations of a. Focusing on the problem of computing the expected value of linear functionals of the solution of the diffusion problem, we give a rigorous error analysis for methods constructed from (1) standard continuous and piecewise linear finite element approximation in physical space; (2) truncated Karhunen-Loève expansion for computing realizations of a (leading to a possibly high-dimensional parametrized deterministic diffusion problem); and (3) lattice-based quasi-Monte Carlo (QMC) quadrature rules for computing integrals over parameter space which define the expected values. The paper contains novel error analysis which accounts for the effect of all three types of approximation. The QMC analysis is based
We consider a finite element approximation of elliptic partial differential equations with random coefficients. Such equations arise, for example, in uncertainty quantification in subsurface flow modelling. Models for random coefficients frequently used in these applications, such as log-normal random fields with exponential covariance, have only very limited spatial regularity, and lead to variational problems that lack uniform coercivity and boundedness with respect to the random parameter. In our analysis we overcome these challenges by a careful treatment of the model problem almost surely in the random parameter, which then enables us to prove uniform bounds on the finite element error in standard Bochner spaces. These new bounds can then be used to perform a rigorous analysis of the multilevel Monte Carlo method for these elliptic problems that lack full regularity and uniform coercivity and boundedness. To conclude, we give some numerical results that confirm the new bounds.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.