Multilevel quasi Monte Carlo methods for elliptic PDEs with random field coefficients via fast white noise sampling

@article{Croci2021MultilevelQM,
  title={Multilevel quasi Monte Carlo methods for elliptic PDEs with random field coefficients via fast white noise sampling},
  author={Matteo Croci and Michael B. Giles and Patrick E. Farrell},
  journal={ArXiv},
  year={2021},
  volume={abs/1911.12099}
}
When solving partial differential equations with random fields as coefficients the efficient sampling of random field realisations can be challenging. In this paper we focus on the fast sampling of Gaussian fields using quasi-random points in a finite element and multilevel quasi Monte Carlo (MLQMC) setting. Our method uses the SPDE approach combined with a new fast (ML)QMC algorithm for white noise sampling. We express white noise as a wavelet series expansion that we divide in two parts. The… 

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References

SHOWING 1-10 OF 73 REFERENCES
Efficient White Noise Sampling and Coupling for Multilevel Monte Carlo with Nonnested Meshes
TLDR
A new sampling technique is presented that can be used to efficiently compute white noise samples in a finite element method and multilevel Monte Carlo setting and a good coupling is enforced and the telescoping sum is respected.
Multi-level Quasi-Monte Carlo Finite Element Methods for a Class of Elliptic PDEs with Random Coefficients
TLDR
Families of QMC rules with “POD weights” (“product and order dependent weights’) which quantify the relative importance of subsets of the variables are found to be natural for proving convergence rates of Q MC errors that are independent of the number of parametric variables.
Multilevel quasi-Monte Carlo integration with product weights for elliptic PDEs with lognormal coefficients
  • L. Herrmann, C. Schwab
  • Mathematics, Computer Science
    ESAIM: Mathematical Modelling and Numerical Analysis
  • 2019
TLDR
Comparison of the ε-complexity for a class of Matérn-like GRF inputs indicates, for input GRFs with low sample regularity, superior performance of the present MLQMC-FEM with locally supported representation functions over alternative representations, e.g. of Karhunen–Loève type.
Circulant embedding with QMC: analysis for elliptic PDE with lognormal coefficients
TLDR
A convergence analysis for the quasi-Monte Carlo method in the case when the QMC method is a specially designed randomly shifted lattice rule is provided, which can be independent of the number of stochastic variables under certain assumptions.
Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients
TLDR
It is proved that convergence of the multilevel Monte Carlo algorithm for estimating any bounded, linear functional and any continuously Fréchet differentiable non-linear functional of the solution is convergence.
Multilevel Quasi-Monte Carlo methods for lognormal diffusion problems
TLDR
A rigorous cost and error analysis of a multilevel estimator based on randomly shifted Quasi-Monte Carlo lattice rules for lognormal diffusion problems and shows that in practice theMultilevel QMC method consistently outperforms both the multileVEL MC method and the single-level variants even for non-smooth problems.
Finite Element Error Analysis of Elliptic PDEs with Random Coefficients and Its Application to Multilevel Monte Carlo Methods
TLDR
A finite element approximation of elliptic partial differential equations with random coefficients is considered, which is used to perform a rigorous analysis of the multilevel Monte Carlo method for these elliptic problems that lack full regularity and uniform coercivity and boundedness.
Quasi-Monte Carlo integration
The standard Monte Carlo approach to evaluating multidimensional integrals using (pseudo)-random integration nodes is frequently used when quadrature methods are too difficult or expensive to
Scalable hierarchical PDE sampler for generating spatially correlated random fields using nonmatching meshes
TLDR
The scalability of the sampling method with nonmatching mesh embedding, coupled with a parallel forward model problem solver, for large‐scale 3D MLMC simulations with up to 1.9·109 unknowns is demonstrated.
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