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Analysis of quasi-Monte Carlo methods for elliptic eigenvalue problems with stochastic coefficients by Gilbert, Alexander D.; Graham, Ivan G.; Kuo, Frances Y.; Scheichl, Robert; Sloan, Ian H. is a scholarly article available to read on EtoBox.

What is Analysis of quasi-Monte Carlo methods for elliptic eigenvalue problems with stochastic coefficients about?

We consider the forward problem of uncertainty quantification for the generalised Dirichlet eigenvalue problem for a coercive second order partial differential operator with random coefficients, motivated by problems in structural mechanics, photonic crystals and neutron diffusion. The PDE coefficients are assumed to be uniformly bounded random fields, represented as infinite series parametrised by uniformly distributed i.i.d. random variables. The expectation of the fundamental eigenvalue of this problem is computed by (a) truncating the infinite series which define the coefficients; (b) approximating the resulting truncated problem using lowest order conforming finite elements and a sparse matrix eigenvalue solver; and (c) approximating the resulting finite (but high dimensional) integral by a randomly shifted quasi-Monte Carlo lattice rule, with specially chosen generating vector. We prove error estimates for the combined error, which depend on the truncation dimension $s$, the finite element mesh diameter $h$, and the number of quasi-Monte Carlo samples $N$. Under suitable regularity assumptions, our bounds are of the particular form $\mathcal{O}(h^2+N^{-1+\delta})$, where $\de

Author

Gilbert, Alexander D.; Graham, Ivan G.; Kuo, Frances Y.; Scheichl, Robert; Sloan, Ian H.

Published

2018

Language

EN