Abstract
Building on previous research which generalized multilevel Monte Carlo methods using either sparse grids or Quasi-Monte Carlo methods, this paper considers the combination of all these ideas applied to elliptic PDEs with finite-dimensional uncertainty in the coefficients. It shows the potential for the computational cost to achieve an \(O(\varepsilon )\) r.m.s. accuracy to be \(O(\varepsilon ^{-r})\) with \(r\!<\!2\), independently of the spatial dimension of the PDE.
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Acknowledgements
The authors acknowledge the support of the Australian Research Council under the projects FT130100655 and DP150101770.
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Giles, M.B., Kuo, F.Y., Sloan, I.H. (2018). Combining Sparse Grids, Multilevel MC and QMC for Elliptic PDEs with Random Coefficients. In: Owen, A., Glynn, P. (eds) Monte Carlo and Quasi-Monte Carlo Methods. MCQMC 2016. Springer Proceedings in Mathematics & Statistics, vol 241. Springer, Cham. https://doi.org/10.1007/978-3-319-91436-7_14
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