Statistical Integration and Approximation

本研究探讨了统计抽样思想在高维数值问题中的应用，特别是数值积分和近似。统计思维的出现是因为在高维问题中，人们最多只能得到一个输入空间的稀疏样本。令人惊讶的好处可以积累：例如，我们知道随机化一个低差异序列可以提高它的准确性。对于数值积分的情况，本项目着眼于自适应重要性采样技术，其中函数值本身用于使采样适应于输入空间的最相关区域，同时保持基于数据的积分精度估计。在数值近似的情况下，本项目将应用混合数值积分方法的最新进展，寻找基于数据确定误差估计的函数的基于样本的近似的方法。数值积分是一项基本的计算任务，在物理、化学、金融、统计学和数值近似中都有应用。数值近似也是基础，而强大而灵活的科学和工程现象计算机模拟器的出现只会增加对基于近似的计算机实验技术的需求。目前，这些计算机模型广泛应用于计算机芯片、汽车零部件、飞机的设计以及全球气候模型。这些函数可能需要数小时甚至数天的计算机时间，因此找到快速近似值（并估计其准确性）非常重要。更好地探索这些功能，称为功能挖掘，可以更快地引入新产品并提高其可靠性。它还可用于深入探讨环境变化和核废料储存模型的预测。

Owen 9704495 This research investigates the use of statistical sampling ideas in high dimensional numerical problems, particularly numerical integration and approximation. Statistical thinking enters the picture because in high dimensional problems one can get at most a sparse sample of the input space. Surprising benefits can accrue: for example it is known that randomizing a low discrepancy sequence can increase it's accuracy. For the case of numerical integration, this project looks at adaptive importance sampling techniques in which the function values themselves are used to adapt the sampling towards the most relevant region of the input space, while maintaining a data based estimate of integral accuracy. In the case of numerical approximation, this project will apply recent progress in hybrid numerical integration methods to finding ways of building sample based approximations of functions with data determined error estimates. Numerical integration is a fundamental computational task with applications in physics, chemistry, finance, statistics, and numerical approximation. Numerical approximation is also fundamental, and the advent of powerful and flexible computer simulators of scientific and engineering phenomena can only increase the demand for computer experiment techniques based on approximation. At present these computer models are widely used in the design of computer chips, automobile parts, airplanes, and in global climate modeling. These functions may take hours or even days of computer time and so finding fast approximations (and estimating their accuracy) is important. Better exploration of these functions, call it function mining, can lead to faster introduction and greater reliability for new products. It can also be used to thoroughly explore the predictions of models for environmental change and storage of nuclear wastes.