Constructing QMC finite element methods for elliptic PDEs with random coefficients by a reduced CBC construction

2019/02/28 by Ebert, Adrian, Kritzer, Peter, Nuyens, Dirk
#65D30 #65D32 #65N30 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1902.11068

Abstract

In the analysis of using quasi-Monte Carlo (QMC) methods to approximate expectations of a linear functional of the solution of an elliptic PDE with random diffusion coefficient the sensitivity w.r.t. the parameters is often stated in terms of product-and-order-dependent (POD) weights. The (offline) fast component-by-component (CBC) construction of an N-point QMC method making use of these POD weights leads to a cost of O(s N log(N) + s² N) with s the parameter truncation dimension. When s is large this cost is prohibitive. As an alternative Herrmann and Schwab introduced an analysis resulting in product weights to reduce the construction cost to O(s N log(N)). We here show how the reduced CBC method can be used for POD weights to reduce the cost to O(∑_j=1^min\s,s^*\ (m-w_j+j) b^m-w_j), where N=b^m with prime b, w₁ ≤ ⋯ ≤ w_s are nonnegative integers and s^* can be chosen much smaller than s depending on the regularity of the random field expansion as such making it possible to use the POD weights directly. We show a total error estimate for using randomly shifted lattice rules constructed through the reduced CBC construction.

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