
Multilevel quasiMonte Carlo for random elliptic eigenvalue problems I: Regularity and error analysis
Random eigenvalue problems are useful models for quantifying the uncerta...
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Stochastic optimization for numerical evaluation of imprecise probabilities
In applications of imprecise probability, analysts must compute lower (o...
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Embedded multilevel Monte Carlo for uncertainty quantification in random domains
The multilevel Monte Carlo (MLMC) method has proven to be an effective v...
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Uncertainty Quantification for Geometry Deformations of Superconducting Cavities using Eigenvalue Tracking
The electromagnetic field distribution as well as the resonating frequen...
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Efficient implementations of the Multivariate Decomposition Method for approximating infinitevariate integrals
In this paper we focus on efficient implementations of the Multivariate ...
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Multilevel Sequential^2 Monte Carlo for Bayesian Inverse Problems
The identification of parameters in mathematical models using noisy obse...
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Multilevel quasi Monte Carlo methods for elliptic PDEs with random field coefficients via fast white noise sampling
When solving partial differential equations with random fields as coeffi...
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Multilevel quasiMonte Carlo for random elliptic eigenvalue problems II: Efficient algorithms and numerical results
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 quasiMonte 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 twogrid 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.
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