Lattice-based integration algorithms: Kronecker sequences and rank-1 lattices
Lattice-based integration algorithms: Kronecker sequences and rank-1 lattices
复制标题
基于格的积分算法：克罗内克序列和 1 阶格
DOI：
10.1007/s10231-017-0670-3
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发表时间：
2017
影响因子：
1
通讯作者：
Kosuke Suzuki
中科院分区：
文献类型：
--
作者：
Josef Dick;Friedrich Pillichshammer;Kosuke Suzuki
在单位立方体上具有齐次边界条件的控制混合光滑函数空间中，证明了数值积分格基算法收敛阶的上界。更确切地说，我们研究了具有控制混合光滑性的Besov空间$$\mathring{\mathbf {B}}^s_{p，\theta }$$的最坏情况积分误差，它也包含了作为特殊情况的具有控制混合光滑性的Sobolev空间$$\mathring{\mathbf {H}}^s_{p}$$的概念.所考虑的算法是准蒙特卡罗规则，其基础节点来自于，其中是大小为的真实的可逆生成矩阵。对于这样的规则，最坏情况下的错误可以在格的Zaremba指数方面有界。我们将这一结果应用于克罗内克格和秩1格点集，这两个导致最佳误差界的任意光滑性的因素。Kronecker格和经典格点集的优点是生成这些点集的算法的运行时间非常短。
We prove upper bounds on the order of convergence of lattice-based algorithms for numerical integration in function spaces of dominating mixed smoothness on the unit cube with homogeneous boundary condition. More precisely, we study worst-case integration errors for Besov spaces of dominating mixed smoothness $$\mathring{\mathbf {B}}^s_{p,\theta }$$, which also comprise the concept of Sobolev spaces of dominating mixed smoothness $$\mathring{\mathbf {H}}^s_{p}$$ as special cases. The considered algorithms are quasi-Monte Carlo rules with underlying nodes from, whereis a real invertible generator matrix of sized. For such rules, the worst-case error can be bounded in terms of the Zaremba index of the lattice. We apply this result to Kronecker lattices and to rank-1 lattice point sets, which both lead to optimal error bounds up to-factors for arbitrary smoothnesss. The advantage of Kronecker lattices and classical lattice point sets is that the run-time of algorithms generating these point sets is very short.