Liberating the Dimension for Function Approximation and Integration
Liberating the Dimension for Function Approximation and Integration
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解放函数逼近和积分的维度
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
G. Wasilkowski
中科院分区：
文献类型：
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作者：
G. Wasilkowski
我们讨论了处理这些问题的问题，尤其是路径积分的问题，包括数学金融，量子物理和化学和随机微分方程。 - 由于两个问题之间的差异降低了d的差异，因此变量仅具有D变量。基于信息的复杂性研究已分析了任意大但固定的问题，为了获得最佳的结果，D的特定值应成为有效算法的一部分。在本文中，这种选择称为“自由”的选择。在问题中，最佳算法来自一个变化算法的家族，该算法通过特殊功能的组合近似∞变化函数，每个函数都取决于不同的变量。 Mathcal {O}（LN（1/Epsilon）/LN（Ln（1/Epsilon）））变量。功能在d中指数。
We discuss recent results on the complexity and tractability of problems dealing with ∞-variate functions. Such problems, especially path integrals, arise in many areas including mathematical finance, quantum physics and chemistry, and stochastic differential equations. It is possible to replace the ∞-variate problem by one that has only d variables since the difference between the two problems diminishes with d approaching infinity. Therefore, one could use algorithms obtained in the Information-Based Complexity study, where problems with arbitrarily large but fixed d have been analyzed. However, to get the optimal results, the choice of a specific value of d should be a part of an efficient algorithm. This is why the approach discussed in the present paper is called liberating the dimension. Such a choice should depend on the cost of sampling d-variate functions and on the error demand (epsilon ). Actually, as recently observed for a specific class of problems, optimal algorithms are from a family of changing dimension algorithms which approximate ∞-variate functions by a combination of special functions, each depending on a different set of variables. Moreover, each such set contains no more than (d(epsilon ) = mathcal{O}(ln (1/epsilon )/ln (ln (1/epsilon )))) variables. This is why the new algorithms have the total cost polynomial in (1/epsilon ) even if the cost of sampling a d-variate function is exponential in d.