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Recycling Samples in the Multigrid Multilevel (Quasi-)Monte Carlo Method

Tijdschriftbijdrage - Tijdschriftartikel

In many models used in engineering and science, material properties are uncertain or spatially varying. For example, in geophysics, and porous media in particular, the uncertain permeability of the material is modelled as a random field. Depending on the material, these random fields can be highly anisotropic. Efficient solvers, such as the Multiple Semi-Coarsened Multigrid (MSG) method, see [11, 12, 13], are required to compute solutions for various realisations of the uncertain material. The MSG method is an extension of the classic Multigrid method that uses additional coarse grids that are coarsened in only a single coordinate direction. In this sense, it closely resembles the extension of Multilevel Monte Carlo (MLMC), see [4], to Multi-Index Monte Carlo (MIMC), see [7]. We present an unbiased MIMC method that reuses the MSG coarse samples, similar to the work in [9] and [16]. Our formulation of the estimator can be interpreted as the problem of learning the unknown distribution of the number of samples across all indices, and, in this sense, unifies the previous work on adaptive MIMC from [15] and unbiased estimation from [14]. We analyse the cost of this new estimator and present numerical experiments with various anisotropic random fields, where the unknown coefficients in the covariance model are considered as hyperparameters. We illustrate its robustness and superiority over standard MIMC without sample reuse.
Tijdschrift: SIAM Journal on Scientific Computing
ISSN: 1064-8275
Issue: 5
Volume: 41
Pagina's: S37 - S60
Jaar van publicatie:2019
BOF-keylabel:ja
IOF-keylabel:ja
BOF-publication weight:2
CSS-citation score:1
Authors from:Higher Education
Toegankelijkheid:Open