Multi-level Quasi-Monte Carlo Finite Element Methods for a Class of Elliptic PDEs with Random Coefficients

Abstract

This paper is a sequel to our previous work (Kuo et al. in SIAM J Numer Anal, 2012) where quasi-Monte Carlo (QMC) methods (specifically, randomly shifted lattice rules) are applied to finite element (FE) discretizations of elliptic partial differential equations (PDEs) with a random coefficient represented by a countably infinite number of terms. We estimate the expected value of some linear functional of the solution, as an infinite-dimensional integral in the parameter space. Here, the (singlelevel) error analysis of our previous work is generalized to a multi-level scheme, with the number of QMC points depending on the discretization level and with a level-dependent dimension truncation strategy. In some scenarios, it is shown that the overall error (i.e., the root-mean-square error averaged over all shifts) is of order O(h 2 ), where h is the finest FE mesh width, or O(N −1+δ ) for arbitrary δ > 0, where N denotes the maximal number of QMC sampling points in the parameter space. For these scenarios, the total work for all PDE solves in the multi-level QMC FE method is essentially of the order of one single PDE solve at the finest FE discretization level, for spatial dimension d = 2 with linear elements. The analysis exploits regularity of the parametric solution with respect to both the physical variables (the variables in Communicated by Philippe Ciarlet. 123Found Comput Math the physical domain) and the parametric variables (the parameters corresponding to randomness). As in our previous work, families of QMC rules with "POD weights" ("product and order dependent weights") which quantify the relative importance of subsets of the variables are found to be natural for proving convergence rates of QMC errors that are independent of the number of parametric variables.

Approximation of high-dimensional parametric PDEs

“…These results establish for the first time convergence rates immune to the curse of dimensionality, in the sense that they hold with infinitely many variables, see also [44] for a survey dealing in particular with these issues. In a similar infinite dimensional framework, and not covered in our paper, let us mention the following related works: (i) similar holomorphy and approximation results are established in [47,48,58] for specific type of PDEs and control problems, (ii) approximation of integrals by quadratures is discussed in [60,61], (iii) inverse problems are discussed in [78,79,82], following the Bayesian perspective from [87], and (iv) diffusion problems with lognormal coefficients are treated in [50,43,45].…”

“…In §8, we discuss an elementary greedy strategy for the offline selection of the instances v i = u(a i ), that consists in picking the n-th instance which deviates the most from the space V n−1 generated from the n − 1 previously selected ones. The approximation error 61) produced by such spaces may be significantly larger than the ideal benchmark of the n-width of the solution manifold for a given value of n. However, a striking result is that both are comparable in terms of rate of decay: for any s > 0, there is a constant C s such that…”

“…Since these spaces are out of reach, both from a theoretical and computational point of view, we build sub-optimal approximations y → u n (y) based on best n-term truncations of polynomial expansions. This approach leads us to quantitative convergence results such as in Corollaries 3.10 and 3.11, and in turn to estimates for the decay of the n-widths d n (M) V of solution manifolds as discussed in §4, for example by using the estimate 61) in the case when Assumption A holds. A legitimate question is to evaluate the possible lack of optimality of the convergence rates, obtained by our polynomial approximation approach, in comparison to the rates which could be achieved by using the optimal n-dimensional spaces V n .…”

Multilevel Accelerated Quadrature for PDEs with Log-Normally Distributed Diffusion Coefficient

“…This idea has already been proposed for different quadrature strategies in case of uniformly elliptic diffusion coefficients in [23]. The well-known Multilevel Monte Carlo Method (MLMC), as introduced in [3,15,16,26,27], and also the Randomized Multilevel Quasi-Monte Carlo Method, as introduced in [30], only provide probabilistic error estimates in the mean-square sense. To avoid this drawback, two fully deterministic methods have been proposed in [23], namely the Multilevel Quasi-Monte Carlo Method (MLQMC) and the Multilevel Polynomial Chaos Method (MLPC).…”

A multigrid multilevel Monte Carlo method for transport in the Darcy–Stokes system

“…More recently, the Monte Carlo method has been generalized to multiple grid levels, exhibiting an exceptional improvement over the standard MC [20,21]. The improved efficiency of these multilevel Monte Carlo (MLMC) methods comes from building the estimate for the QoI, [18,22].…”