基于ANOVA型函数空间分解的无穷维积分最优随机多水平算法
Optimal randomized multilevel algorithms for infinite-dimensional integration on function spaces with ANOVA-type decomposition
AI总结：
针对加权再生核希尔伯特空间中的无穷维积分问题，提出新的随机多水平算法并证明其最优性，显著改进了已有误差界。
摘要：
本文考虑加权再生核希尔伯特空间上的无穷维积分问题，其范数由ANOVA型函数空间分解诱导。权重模拟不同变量组的重要性。我们提出新的随机多水平算法来解决该积分问题，并证明其随机误差的上界。此外，我们在该设定下首次给出一般随机算法（特别地，可以是自适应的或非线性的）的非平凡下界。这些下界表明我们的多水平算法是最优的。我们的分析细化和扩展了[F. J. Hickernell, T. Müller-Gronbach, B. Niu, K. Ritter, J. Complexity 26 (2010), 229-254]中的分析，并且我们的误差界显著改进了那里给出的误差界。作为说明性示例，我们讨论了无锚Sobolev空间，并采用了基于加扰多项式格规则的随机拟蒙特卡罗多水平算法。
英文摘要：
In this paper, we consider the infinite-dimensional integration problem on weighted reproducing kernel Hilbert spaces with norms induced by an underlying function space decomposition of ANOVA-type. The weights model the relative importance of different groups of variables. We present new randomized multilevel algorithms to tackle this integration problem and prove upper bounds for their randomized error. Furthermore, we provide in this setting the first non-trivial lower error bounds for general randomized algorithms, which, in particular, may be adaptive or non-linear. These lower bounds show that our multilevel algorithms are optimal. Our analysis refines and extends the analysis provided in [F. J. Hickernell, T. Müller-Gronbach, B. Niu, K. Ritter, J. Complexity 26 (2010), 229-254], and our error bounds improve substantially on the error bounds presented there. As an illustrative example, we discuss the unanchored Sobolev space and employ randomized quasi-Monte Carlo multilevel algorithms based on scrambled polynomial lattice rules.
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