Dimension-adaptive tensor-product quadrature
Dimension-adaptive tensor-product quadrature
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DOI：
10.1007/s00607-003-0015-5
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发表时间：
2003-09-01
期刊：
影响因子：
3.7
通讯作者：
Griebel, M
中科院分区：
文献类型：
--
作者：
Gerstner, T;Griebel, M
本文研究单位超立方体上多元函数的数值积分。在这里，我们特别解决了高维的情况下，在一般的维数灾难遇到。由于测度的集中现象，这类函数通常可以很好地近似为低维项的和。然而，问题是要找到一个很好的展开式给出很少的知识的被积本身。维数自适应求积方法，这是本文开发和提出的目的是自动找到这样的展开式。它是基于稀疏网格方法，已被证明给低，中维问题的良好效果。维数自适应求积方法试图找到重要的维度，并在这方面的指导下，适当的误差估计自适应地完善。这导致了一种基于广义稀疏网格索引集的方法。我们提出了有效的数据结构的存储和遍历的索引集，并讨论了一个有效的实现算法的性能，从计算物理和金融的几个数值例子，其中降维得到的布朗桥离散化的基础随机过程说明了该方法的性能。
We consider the numerical integration of multivariate functions defined over the unit hypercube. Here, we especially address the high-dimensional case, where in general the curse of dimension is encountered. Due to the concentration of measure phenomenon, such functions can often be well approximated by sums of lower-dimensional terms. The problem, however, is to find a good expansion given little knowledge of the integrand itself.The dimension-adaptive quadrature method which is developed and presented in this paper aims to find such an expansion automatically. It is based on the sparse grid method which has been shown to give good results for low- and moderate-dimensional problems. The dimension-adaptive quadrature method tries to find important dimensions and adaptively refines in this respect guided by suitable error estimators. This leads to an approach which is based on generalized sparse grid index sets. We propose efficient data structures for the storage and traversal of the index sets and discuss an efficient implementation of the algorithm.The performance of the method is illustrated by several numerical examples from computational physics and finance where dimension reduction is obtained from the Brownian bridge discretization of the underlying stochastic process.