Infinite-Dimensional Quadrature和量化
Infinite-Dimensional Quadrature and Quantization
摘要：
研究了巴拿赫空间上Lipschitz泛函的确定性和随机（Monte Carlo）算法的数值积分.这个正交问题被证明是密切相关的问题的量化的基本概率测度。除了一般的设置，我们分析特别集成w.r.t.扩散过程的高斯测度和分布。我们推导出每一个算法的计算成本方面的最坏情况下的错误的下界，我们提出匹配的上界，直到最优算法，和相应的几乎最优算法。作为辅助结果，我们确定的扩散过程的量子化数和柯尔莫哥洛夫宽度的渐近行为。
英文摘要：
We study numerical integration of Lipschitz functionals on a Banach space by means of deterministic and randomized (Monte Carlo) algorithms. This quadrature problem is shown to be closely related to the problem of quantization of the underlying probability measure. In addition to the general setting we analyze in particular integration w.r.t. Gaussian measures and distributions of diffusion processes. We derive lower bounds for the worst case error of every algorithm in terms of its computational cost, and we present matching upper bounds, up to logarithms, and corresponding almost optimal algorithms. As auxiliary results we determine the asymptotic behaviour of quantization numbers and Kolmogorov widths for diffusion processes.
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