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

Abstract

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 Matérn covariance function show that sample recycling is more efficient when the input random field is nonsmooth.

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

“…In applications, usually, the rate of increase in cost, γ j , j =1,…, d , is fixed, and the benefit of the sample reuse will be more pronounced when the rate of decrease in variance, β j , j =1,…, d , is small, reflecting a slow decay of the number of samples as ℓ increases. This is in agreement with previous results obtained for the ML(Q)MC setting 19 …”

“…The randomisation of the index L is the key difference with the classic multi-index estimator. The finite index set I(L) in (20) introduces an additional bias term in the expression for the root mean square error of the estimator, that needs to be controlled. Although adaptive approaches have been proposed to tackle this problem, see, e.g., [19], obtaining a stable and accurate bound for the bias term is nontrivial.…”

“…The argument inside the multiple sum will be nonzero up to only a finite subset of N d 0 . If the probability mass π L is decreasing with L, component-wise, this is mimicking the decreasing multivariate sequence of sample sizes {N } in the classic MIMC estimator from equation (20).…”

Multilevel quasi-Monte Carlo for random elliptic eigenvalue problems II: Efficient algorithms and numerical results

“…In almost all of our numerical tests using the eigenvector of a nearby QMC point as the starting vector reduced the number of RQ iterations to 2. A similar speedup by a factor of 2 was also observed in [41], which recycled samples from the multigrid hierarchy within a MLQMC algorithm for the elliptic source problem.…”

A stochastic collocation method based on sparse grids for a stochastic Stokes-Darcy model

“…A number of efficient numerical methods have been developed to solve stochastic PDEs, such as polynomial chaos [69,116,117], stochastic Galerkin method [8,35,86,98], stochastic collocation method [7,59], sparse grid methods [11,87,90,91], multilevel Monte Carlo method [14,29,42,52,73,97,104], and many others [9,12,27,82,103,107,109,112,114,115,118,121,122]. These methods have also been applied to solve the stochastic optimization and control problems [4,13,34,58,105].…”