### 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 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 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 (from-to) | 863-915 |

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

### Cite this

*Numerische Mathematik*,

*142*(4), 863-915. https://doi.org/10.1007/s00211-019-01046-6

**Analysis of quasi-Monte Carlo methods for elliptic eigenvalue problems with stochastic coefficients.** / Gilbert, Alexander D.; Graham, Ivan G.; Kuo, Frances Y.; Scheichl, Robert; Sloan, Ian H.

Research output: Contribution to journal › Article

*Numerische Mathematik*, vol. 142, no. 4, pp. 863-915. https://doi.org/10.1007/s00211-019-01046-6

}

TY - JOUR

T1 - Analysis of quasi-Monte Carlo methods for elliptic eigenvalue problems with stochastic coefficients

AU - Gilbert, Alexander D.

AU - Graham, Ivan G.

AU - Kuo, Frances Y.

AU - Scheichl, Robert

AU - Sloan, Ian H.

PY - 2019/8/1

Y1 - 2019/8/1

N2 - 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 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.

AB - 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 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.

UR - http://www.scopus.com/inward/record.url?scp=85065676979&partnerID=8YFLogxK

U2 - 10.1007/s00211-019-01046-6

DO - 10.1007/s00211-019-01046-6

M3 - Article

VL - 142

SP - 863

EP - 915

JO - Numerische Mathematik

JF - Numerische Mathematik

SN - 0029-599X

IS - 4

ER -