Weighted geometric discrepancies and numerical integration on reproducing kernel Hilbert spaces
Weighted geometric discrepancies and numerical integration on reproducing kernel Hilbert spaces
复制标题
DOI：
10.1016/j.jco.2011.02.003
复制
发表时间：
2012-02-01
影响因子：
1.7
通讯作者：
Gnewuch, Michael
中科院分区：
文献类型：
--
作者：
Gnewuch, Michael
我们扩展了 [E.1] 中引入的 L-2-B 差异的概念。 Novak, H. Wozniakowski，L-2 差异和多元积分，见：W.W.L. Chen，W.T. Cowers，H. Halberstam，W.M.施密特和 R.C.沃恩（编辑），解析数论。 Essays in Honor of Klaus Roth，剑桥大学出版社，剑桥，2009 年，第 359-388 页]我们称之为加权几何 L-2 差异。这一扩展使我们能够考虑权重，以调节不同变量组的重要性，并考虑与勒贝格测度不同的体积测度，以及与欧几里德空间可测量子集不同的测试集类别。我们将加权几何 L-2 差异与在加权再生核希尔伯特空间上定义的数值积分联系起来，并以这种方式解决了 Novak 和 Wozniakowski。此外，我们证明了使用允许样本点的体积公式的数值积分误差的上限。可接受的样本点集实际上可能是零测度积分域的子集。我们说明，特别是在无限维数值积分中，区分整个积分域和算法实际可以使用的样本点集至关重要。 (C) 2011 Elsevier Inc. 保留所有权利。
We extend the notion of L-2-B-discrepancy introduced in [E. Novak, H. Wozniakowski, L-2 discrepancy and multivariate integration, in: W.W.L. Chen, W.T. Cowers, H. Halberstam, W.M. Schmidt, and R.C. Vaughan (Eds.), Analytic Number Theory. Essays in Honour of Klaus Roth, Cambridge University Press, Cambridge, 2009, pp. 359-388] to what we shall call weighted geometric L-2-discrepancy. This extension enables us to consider weights in order to moderate the importance of different groups of variables, as well as to consider volume measures different from the Lebesgue measure and classes of test sets different from measurable subsets of Euclidean spaces.We relate the weighted geometric L-2-discrepancy to numerical integration defined over weighted reproducing kernel Hilbert spaces and settle in this way an open problem posed by Novak and Wozniakowski.Furthermore, we prove an upper bound for the numerical integration error for cubature formulas that use admissible sample points. The set of admissible sample points may actually be a subset of the integration domain of measure zero. We illustrate that particularly in infinite-dimensional numerical integration it is crucial to distinguish between the whole integration domain and the set of those sample points that actually can be used by the algorithms. (C) 2011 Elsevier Inc. All rights reserved.