Abstract
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 quasiMonte 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 quasiMonte Carlo samples N. Under suitable regularity assumptions, our bounds are of the particular form O(h ^{2}+ N ^{} ^{1} ^{+} ^{δ}) , where δ> 0 is arbitrary and the hidden constant is independent of the truncation dimension, which needs to grow as h→ 0 and N→ ∞. As for the analogous PDE source problem, the conditions under which our error bounds hold depend on a parameter p∈ (0 , 1) representing the summability of the terms in the series expansions of the coefficients. Although the eigenvalue problem is nonlinear, which means it is generally considered harder than the source problem, in almost all cases (p≠ 1) we obtain error bounds that converge at the same rate as the corresponding rate for the source problem. The proof involves a detailed study of the regularity of the fundamental eigenvalue as a function of the random parameters. As a key intermediate result in the analysis, we prove that the spectral gap (between the fundamental and the second eigenvalues) is uniformly positive over all realisations of the random problem.
Original language  English 

Pages (fromto)  863915 
Number of pages  53 
Journal  Numerische Mathematik 
Volume  142 
Issue number  4 
Early online date  10 May 2019 
DOIs  
Publication status  Published  1 Aug 2019 
ASJC Scopus subject areas
 Computational Mathematics
 Applied Mathematics
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Profiles

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