A Measure-Theoretic Interpretation of Sample Based Numerical Integration with Applications to Inverse and Prediction Problems under Uncertainty
A Measure-Theoretic Interpretation of Sample Based Numerical Integration with Applications to Inverse and Prediction Problems under Uncertainty
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基于样本的数值积分的测度理论解释及其在不确定性下逆向和预测问题中的应用
DOI：
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发表时间：
2017
影响因子：
3.1
通讯作者：
Scott N. Walsh
中科院分区：
文献类型：
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作者：
T. Butler;L. Graham;S. Mattis;Scott N. Walsh
函数在可测集合上的积分是计算科学中的一个基本问题。当可测集合属于高维空间或函数计算复杂时，基于有限样本集合中函数值的加权和来估计积分可能只是实用的。蒙特卡罗、准蒙特卡罗和其他(伪)随机方案是确定一组样本的常见选择。这些计划之所以吸引人，是因为它们在概念上很容易，而且有能力规避所谓的维度诅咒，并取得了不同程度的成功。然而，收敛通常是缓慢的，并且用概率来描述。我们考虑任何基于样本的积分数值逼近算法的一般测度论解释。证明了先验误差界，为定义解决误差优化问题的自适应采样算法提供了洞察力。我们使用这些界来改进正演和逆演的积分逼近。
The integration of functions over measurable sets is a fundamental problem in computational science. When the measurable sets belong to high-dimensional spaces or the function is computationally complex, it may only be practical to estimate integrals based on weighted sums of function values from a finite collection of samples. Monte Carlo, quasi--Monte Carlo, and other (pseudo-)random schemes are common choices for determining a set of samples. These schemes are appealing for their conceptual ease and ability to circumvent, with various degrees of success, the so-called curse of dimensionality. However, convergence is often slow and described in terms of probability. We consider a general measure-theoretic interpretation of any sample based algorithm for numerically approximating an integral. A priori error bounds are proven that provide insight into defining adaptive sampling algorithms solving error optimization problems. We use these bounds to improve integral approximations for both forward and inver...