Enhanced multi‐index Monte Carlo by means of multiple semicoarsened multigrid for anisotropic diffusion problems

• Published in: Numerical linear algebra with applications 2021-05, Vol.28 (3), p.n/a
• Main Authors: Robbe, Pieterjan, Nuyens, Dirk, Vandewalle, Stefan
• Format: Article
• Language: English
• Subjects: anisotropic diffusion problems; Cost analysis; Covariance; Fields (mathematics); Geophysics; Material properties; multiple semicoarsened multigrid; multi‐index Monte Carlo; Porous media; Robustness (mathematics)
• ISSN: 1070-5325; 1099-1506
• DOI: 10.1002/nla.2281

Summary In many models used in engineering and science, material properties are uncertain or spatially varying. For example, in geophysics and porous media flow, in particular, the uncertain permeability of the material is modeled as a random field. These random fields can be highly anisotropic. Efficient solvers, such as the multiple semicoarsened multigrid (MSG) method are required to compute solutions for various realizations of the uncertain material. The MSG method is an extension of the classic multigrid method, which 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 to multi‐index Monte Carlo (MIMC). We present an unbiased MIMC method that reuses the MSG coarse solutions. 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 unifies the previous work on adaptive MIMC and unbiased estimation. We analyze the cost of this new estimator theoretically 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 unbiased MIMC without sample reuse. We combine multiple semicoarsened multigrid (MSG) with unbiased multi‐index Monte Carlo (MIMC) and show how coarse solutions from the MSG solver can be reused in the estimator. We analyze the cost of this new estimator theoretically 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 unbiased MIMC without sample reuse.