Multilevel quasi-Monte Carlo for random elliptic eigenvalue problems II: Efficient algorithms and numerical results

Abstract

Stochastic PDE eigenvalue problems often arise in the field of uncertainty quantification, whereby one seeks to quantify the uncertainty in an eigenvalue, or its eigenfunction. In this paper we present an efficient multilevel quasi-Monte Carlo (MLQMC) algorithm for computing the expectation of the smallest eigenvalue of an elliptic eigenvalue problem with stochastic coefficients. Each sample evaluation requires the solution of a PDE eigenvalue problem, and so tackling this problem in practice is notoriously computationally difficult. We speed up the approximation of this expectation in four ways: 1) we use a multilevel variance reduction scheme to spread the work over a hierarchy of FE meshes and truncation dimensions; 2) we use QMC methods to efficiently compute the expectations on each level; 3) we exploit the smoothness in parameter space and reuse the eigenvector from a nearby QMC point to reduce the number of iterations of the eigensolver; and 4) we utilise a two-grid discretisation scheme to obtain the eigenvalue on the fine mesh with a single linear solve. The full error analysis of a basic MLQMC algorithm is given in the companion paper [Gilbert and Scheichl, 2021], and so in this paper we focus on how to further improve the efficiency and provide theoretical justification of the enhancement strategies 3) and 4). Numerical results are presented that show the efficiency of our algorithm, and also show that the four strategies we employ are complementary.

Efficient stochastic modal decomposition methods for structural stochastic static and dynamic analyses

“…However, it is usually not a simple matter since efficient and accurate numerical algorithms are required to solve stochastic eigenvalue problems. Although a lot of methods can be used to solve stochastic eigenvalue equations, for example, the MCS, 20,21 the perturbation method, [22][23][24] the PC-based method, 25 the subspace iteration approach, [26][27][28][29] the stochastic collocation method, 30 the polynomial/spline dimensional decomposition methods, 31,32 the homotopy approach, 33 the low-rank approximation method 34,35 and so forth, only a few effort has been made to apply stochastic eigenvalue algorithms to modal decomposition-based stochastic dynamic analyses.…”

An efficient reduced‐order method for stochastic eigenvalue analysis

“…A large number of deterministic eigenvalue problems are solved in order to compute high-accuracy stochastic eigenvalues, which are computationally expensive, especially for large-scale problems. Several improvements, for example, quasi-MCS and multilevel MCS, 4 are used to save computational costs of MCS. The second kind of method is perturbation methods.…”

Analyticity of Parametric Elliptic Eigenvalue Problems and Applications to Quasi-Monte Carlo Methods

“…Gantner et al [12], Gilbert et al [15,16,17,18], Graham et al [20,19,21], Herrmann and Schwab [26,25,27], Kazashi [28], Kuo and Nuyens [29,30], Kuo et al [32,33,31], Lemieux [34], Leobacher and Pillichshammer [35], Nguyen and Nuyens [36,37], Nichols and Kuo [38] to mention just a few.…”

“…The eigenvalue problems of parametric or stochastic elliptic differential operators have been of interest for the past fifty years, see [41,40,13,42,1,43,22,11,23,15,16,17,18] and references therein. These problems appear in many areas of engineering and physics, for example, in nuclear reactor physics; photonics; quantum physics; acoustic; or in electromagnetic.…”

“…The analytic dependence of the eigenvalue λ 1 (y) on the parameters y of the parametric EVP (1.4) has been considered in [1]. The analysis of the QMC method for λ 1 (y) was studied in [15,17,18]. However, in these mentioned papers, the authors have not taken into account the amount of overlap between the supports of the functions (a j ) j∈N .…”