Dimension-adaptive sparse grid quadrature for integrals with boundary singularities
Dimension-adaptive sparse grid quadrature for integrals with boundary singularities
复制标题
具有边界奇点的积分的维度自适应稀疏网格求积
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
Jens Oettershagen
中科院分区：
文献类型：
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作者：
M. Griebel;Jens Oettershagen
经典高斯求积规则可实现无限平滑且所有导数均一致有界的单变量函数的指数收敛。本文的目的是构建基于非多项式基函数的广义高斯求积规则，即使对于具有（可积）边界奇点且其确切类型未知的被积函数也能产生指数收敛。此外，我们使用这些新公式的稀疏张量积通过维度自适应方法来计算具有边界奇点的 d 维被积函数。作为应用，除了标准模型问题之外，我们还考虑使用 Genz 算法来逼近多元正态概率。
Classical Gaussian quadrature rules achieve exponential convergence for univariate functions that are infinitely smooth and where all derivatives are uniformly bounded. The aim of this paper is to construct generalized Gaussian quadrature rules based on non-polynomial basis functions, which yield exponential convergence even for integrands with (integrable) boundary singularities whose exact type is not a-priori known. Moreover, we use sparse tensor-products of these new formulae to compute d-dimensional integrands with boundary singularities by means of a dimension-adaptive approach. As application, we consider, besides standard model problems, the approximation of multivariate normal probabilities using the Genz-algorithm.