Projects per year
Abstract
In this paper we analyze the numerical approximation of diffusion problems over polyhedral domains in (Formula presented.) ((Formula presented.)), with diffusion coefficient (Formula presented.) given as a lognormal random field, i.e., (Formula presented.) where (Formula presented.) is the spatial variable and (Formula presented.) is a Gaussian random field. The analysis presents particular challenges since the corresponding bilinear form is not uniformly bounded away from (Formula presented.) or (Formula presented.) over all possible realizations of (Formula presented.). 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 (Formula presented.) (leading to a possibly highdimensional parametrized deterministic diffusion problem); and (3) latticebased quasiMonte 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 on a recent result on randomly shifted lattice rules for highdimensional integrals over the unbounded domain of Euclidean space, which shows that (under suitable conditions) the quadrature error decays with (Formula presented.) with respect to the number of quadrature points (Formula presented.), where (Formula presented.) is arbitrarily small and where the implied constant in the asymptotic error bound is independent of the dimension of the domain of integration.
Original language  English 

Pages (fromto)  329368 
Number of pages  40 
Journal  Numerische Mathematik 
Volume  131 
Issue number  2 
Early online date  2 Dec 2014 
DOIs  
Publication status  Published  Oct 2015 
Fingerprint Dive into the research topics of 'QuasiMonte Carlo finite element methods for elliptic PDEs with lognormal random coefficients'. Together they form a unique fingerprint.
Projects
 2 Finished

Multilevel Monte Carlo Methods for Elliptic Problems
Engineering and Physical Sciences Research Council
1/07/11 → 30/06/14
Project: Research council

Adaptive Multiscale Methods for Approximation and Preconditioning
Engineering and Physical Sciences Research Council
1/05/10 → 31/07/11
Project: Research council
Profiles

Ivan Graham
 Department of Mathematical Sciences  Professor
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
Person: Research & Teaching