Generation and application of multivariate polynomial quadrature rules
Generation and application of multivariate polynomial quadrature rules
复制标题
DOI：
10.1016/j.cma.2018.04.009
复制
发表时间：
2017-09
影响因子：
7.2
通讯作者：
J. Jakeman;A. Narayan
中科院分区：
文献类型：
--
作者：
J. Jakeman;A. Narayan
寻找具有指定多项式精度的最小尺寸的多元求积规则一直是多年研究的主题。找到这样的规则可以实现矩的精确积分，矩在复杂模型的科学计算的许多方面发挥着核心作用。本文的贡献是双重的。首先，我们提供了多项式求积问题的新颖数学分析，该分析为具有指定精度的多项式规则中的最小可能节点数提供了下界。我们给出了具体但简单的多元示例，其中可以设计最小求积规则来实现此下界，以及展示何时不可能实现此下界的情况。我们的第二个贡献是制定了一种算法，该算法能够有效地生成在非张量域上具有正权重的多元求积规则。我们的测试表明此过程在多达 20 个维度上是成功的。我们测试了我们的方法在降维和化学动力学问题中的应用，包括与稀疏网格、蒙特卡罗和准蒙特卡罗序列以及斯特劳德规则等流行替代方案的比较。在几乎所有情况下，本文计算的正交规则都优于这些替代方案。
The search for multivariate quadrature rules of minimal size with a specified polynomial accuracy has been the topic of many years of research. Finding such a rule allows accurate integration of moments, which play a central role in many aspects of scientific computing with complex models. The contribution of this paper is twofold. First, we provide novel mathematical analysis of the polynomial quadrature problem that provides a lower bound for the minimal possible number of nodes in a polynomial rule with specified accuracy. We give concrete but simplistic multivariate examples where a minimal quadrature rule can be designed that achieves this lower bound, along with situations that showcase when it is not possible to achieve this lower bound. Our second contribution is the formulation of an algorithm that is able to efficiently generate multivariate quadrature rules with positive weights on non-tensorial domains. Our tests show success of this procedure in up to 20 dimensions. We test our method on applications to dimension reduction and chemical kinetics problems, including comparisons against popular alternatives such as sparse grids, Monte Carlo and quasi Monte Carlo sequences, and Stroud rules. The quadrature rules computed in this paper outperform these alternatives in almost all scenarios.