Recycling Samples in the Multigrid Multilevel (Quasi-)Monte Carlo Method
The Multilevel Monte Carlo method is an efficient variance reduction technique. It uses a sequence of coarse approximations to reduce the computational cost in uncertainty quantification applications. The method is nowadays often considered to be the method of choice for solving PDEs with random coe...
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| Main Authors | , , |
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| Format | Journal Article |
| Language | English |
| Published |
14.06.2018
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| Subjects | |
| Online Access | Get full text |
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| Summary: | The Multilevel Monte Carlo method is an efficient variance reduction
technique. It uses a sequence of coarse approximations to reduce the
computational cost in uncertainty quantification applications. The method is
nowadays often considered to be the method of choice for solving PDEs with
random coefficients when many uncertainties are involved. When using Full
Multigrid to solve the deterministic problem, coarse solutions obtained by the
solver can be recycled as samples in the Multilevel Monte Carlo method, as was
pointed out by Kumar, Oosterlee and Dwight [Int. J. Uncertain. Quantif., 7
(2017), pp. 57--81]. In this article, an alternative approach is considered,
using Quasi-Monte Carlo points, to speed up convergence. Additionally, our
method comes with an improved variance estimate which is also valid in case of
the Monte Carlo based approach. The new method is illustrated on the example of
an elliptic PDE with lognormal diffusion coefficient. Numerical results for a
variety of random fields with different smoothness parameters in the Mat\'ern
covariance function show that sample recycling is more efficient when the input
random field is nonsmooth. |
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| DOI: | 10.48550/arxiv.1806.05619 |