Multilevel quasi-Monte Carlo for random elliptic eigenvalue problems I: Regularity and error analysis

Abstract

Random eigenvalue problems are useful models for quantifying the uncertainty in several applications from the physical sciences and engineering, e.g., structural vibration analysis, the criticality of a nuclear reactor or photonic crystal structures. In this paper we present a simple multilevel quasi-Monte Carlo (MLQMC) method for approximating the expectation of the minimal eigenvalue of an elliptic eigenvalue problem with coefficients that are given as a series expansion of countably-many stochastic parameters. The MLQMC algorithm is based on a hierarchy of discretisations of the spatial domain and truncations of the dimension of the stochastic parameter domain. To approximate the expectations, randomly shifted lattice rules are employed. This paper is primarily dedicated to giving a rigorous analysis of the error of this algorithm. A key step in the error analysis requires bounds on the mixed derivatives of the eigenfunction with respect to both the stochastic and spatial variables simultaneously. An accompanying paper , focusses on practical extensions of the MLQMC algorithm to improve efficiency, and presents numerical results.

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

“…Note that similar results also hold for general N , but with N replaced by the Euler Totient function, which counts the number of integers less than and coprime to N , see, e.g., [13,Theorem 5.10]. For more details on the general theory of lattice rules see [13], and for a theoretical analysis of randomly shifted lattice rules for MLQMC applied to (1.1) see [21].…”

“…In the companion paper [21], it is shown that for h sufficiently small 1 the spectral gap of the FE eigenvalue problem (2.11) satisfies the uniform lower bound…”

“…The focus of this paper is on developing the above practical strategies to give an efficient MLQMC method. A rigorous analysis of MLQMC methods for the stochastic EVP (1.1) is the focus of the separate paper [21]. However, we do give a theoretical justification of the enhancement strategies 3) and 4).…”

“…Using a QMC rule to approximate the expectation on each level in (1.2) instead of Monte Carlo gives a Multilevel quasi-Monte Carlo (MLQMC) method. MLQMC methods were first developed in [23] for option pricing, and since then have also had great success for UQ in stochastic PDE problems [21,34,33]. The gains are complementary, so that for several problems MLQMC methods can be shown to result in faster convergence than either MLMC methods or single level QMC approximations.…”

“…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.…”