Some Results on the Complexity of Numerical Integration

This is a survey (21 pages, 124 references) written for the MCQMC 2014 conference in Leuven, April 2014. We start with the seminal paper of Bakhvalov (1959) and end with new results on the curse of dimension and on the complexity of oscillatory integrals. Some small errors of earlier versions are corrected

Clustering-based collocation for uncertainty propagation with multivariate correlated inputs

In this article, we propose the use of partitioning and clustering methods as an alternative to Gaussian quadrature for stochastic collocation (SC). The key idea is to use cluster centers as the nodes for collocation. In this way, we can extend the use of collocation methods to uncertainty propagation with multivariate, correlated input. The approach is particularly useful in situations where the probability distribution of the input is unknown, and only a sample from the input distribution is available. We examine several clustering methods and assess their suitability for stochastic collocation numerically using the Genz test functions as benchmark. The proposed methods work well, most notably for the challenging case of nonlinearly correlated inputs in higher dimensions. Tests with input dimension up to 16 are included. Furthermore, the clustering-based collocation methods are compared to regular SC with tensor grids of Gaussian quadrature nodes. For 2-dimensional uncorrelated inputs, regular SC performs better, as should be expected, however the clustering-based methods also give only small relative errors. For correlated 2-dimensional inputs, clustering-based collocation outperforms a simple adapted version of regular SC, where the weights are adjusted to account for input correlatio

Differential equations with general highly oscillatory forcing terms

The concern of this paper is in expanding and computing initial-value problems of the form y' = f(y) + hw(t) where the function hw oscillates rapidly for w >> 1. Asymptotic expansions for such equations are well understood in the case of modulated Fourier oscillators hw(t) = Σm am(t)eim!t and they can be used as an organising principle for very accurate and aordable numerical solvers. However, there is no similar theory for more general oscillators and there are sound reasons to believe that approximations of this kind are unsuitable in that setting. We follow in this paper an alternative route, demonstrating that, for a much more general family of oscillators, e.g. linear combinations of functions of the form ei!gk(t), it is possible to expand y(t) in a different manner. Each rth term in the expansion is for some & > 0 and it can be represented as an r-dimensional highly oscillatory integral. Since computation of multivariate highly oscillatory integrals is fairly well understood, this provides a powerful method for an effective discretisation of a numerical solution for a large family of highly oscillatory ordinary differential equations