随机多层级与变维算法的下界误差
Lower Error Bounds for Randomized Multilevel and Changing Dimension Algorithms
AI总结：
针对无限维积分问题，在加权希尔伯特空间中，研究随机算法在两种代价模型下的下界误差，证明变维算法收敛率可任意接近最优。
摘要：
我们为随机算法提供了下界误差，这些算法近似计算依赖于无限制甚至无限多个变量的函数的积分。更精确地说，我们考虑具有底层锚定分解和任意权重的加权希尔伯特空间上的无限维积分问题。我们关注随机算法和随机最坏情况误差。我们研究了两种函数评估代价模型，它们取决于所选样本点的活跃变量数量。多层级算法在第一种代价模型中表现非常好，而变维算法以及基于类似思想的逐维求积方法可以利用更宽松的第二种代价模型。我们证明了在这些代价模型中随机算法的第一个非平凡下界误差，并在乘积权重的情况下展示了其质量。特别地，我们表明[L. Plaskota, G. W. Wasilkowski, J. Complexity 27 (2011), 505--518]中提供的随机变维算法实现了任意接近最优收敛率的收敛速度。
英文摘要：
We provide lower error bounds for randomized algorithms that approximate integrals of functions depending on an unrestricted or even infinite number of variables. More precisely, we consider the infinite-dimensional integration problem on weighted Hilbert spaces with an underlying anchored decomposition and arbitrary weights. We focus on randomized algorithms and the randomized worst case error. We study two cost models for function evaluation which depend on the number of active variables of the chosen sample points. Multilevel algorithms behave very well with respect to the first cost model, while changing dimension algorithms and also dimension-wise quadrature methods, which are based on a similar idea, can take advantage of the more generous second cost model. We prove the first non-trivial lower error bounds for randomized algorithms in these cost models and demonstrate their quality in the case of product weights. In particular, we show that the randomized changing dimension algorithms provided in [L. Plaskota, G. W. Wasilkowski, J. Complexity 27 (2011), 505--518] achieve convergence rates arbitrarily close to the optimal convergence rate.
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