Quasi--Monte Carlo Integration for Affine-Parametric, Elliptic PDEs: Local Supports and Product Weights

2018/01/01 by Robert N. Gantner, Lukas Herrmann, Christoph Schwab · 2 citations
Mathematics · Decision Sciences · #Mathematical Approximation and Integration #Probabilistic and Robust Engineering Design #Mathematical functions and polynomials

paper · doi:10.1137/16m1082597

Abstract

We analyze convergence rates of first-order quasi--Monte Carlo (QMC) integration with randomly shifted lattice rules and for higher-order, interlaced polynomial lattice rules for a class of countably parametric integrands that result from linear functionals of solutions of linear, elliptic diffusion equations with affine-parametric, uncertain coefficient function a(x,y) = a(x) + ∑_{j≥ 1} y_j ψ_j(x) in a bounded domain D⊂ ℝ^d. Extending the result in [F. Y. Kuo, C. Schwab, and I. H. Sloan, SIAM J. Numer. Anal., 50 (2012), pp. 3351--3374], where ψ_j was assumed to have global support in the domain D, we assume in the present paper that supp(ψ_j) is localized in D and that we have control on the overlaps of these supports. Under these conditions we prove dimension-independent convergence rates in [1/2,1) of randomly shifted lattice rules with product weights and corresponding higher-order convergence rates by higher-order, interlaced polynomial lattice rules with product weights. The product structure of the QMC weights facilitates work bounds for the fast, component-by-component constructions of [D. Nuyens and R. Cools, Math. Comp., 75 (2006), pp. 903--920] which scale linearly with respect to the parameter dimension s. The dimension-independent convergence rates are only limited by the degree of digit interlacing used in the construction of the higher-order QMC quadrature rule and, for locally supported coefficient functions, by the summability of the locally supported coefficient sequence in the affine-parametric coefficient.

Citations

• Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces
• Quasi-Monte Carlo Finite Element Methods for a Class of Elliptic Partial Differential Equations with Random Coefficients

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