Statistical Integration and Approximation

欧文9704495这项研究调查了统计抽样思想在高维数值问题中的使用，特别是数值积分和近似。统计思维之所以进入图景，是因为在高维问题中，人们最多只能得到输入空间的稀疏样本。可能会带来令人惊讶的好处：例如，众所周知，将低差异序列随机化可以提高其准确性。对于数值积分的情况，本项目着眼于自适应重要性抽样技术，其中使用函数值本身来调整抽样以适应输入空间中最相关的区域，同时保持基于数据的积分精度估计。在数值逼近的情况下，这个项目将应用混合数值积分方法的最新进展来寻找建立基于样本的函数逼近的方法，其中数据决定了误差估计。数值积分是一项基本的计算任务，在物理、化学、金融、统计学和数值逼近等领域都有应用。数值近似也是基本的，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，这些计算机模型被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拟。这些函数可能需要几个小时甚至几天的计算机时间，因此找到快速的近似(并估计其精度)是很重要的。更好地探索这些功能，称为功能挖掘，可以为新产品带来更快的推出速度和更高的可靠性。它还可以用来彻底探索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.