Statistical Numerics

摘要本课题的重点是将统计思想应用于高维数值问题，如近似和有噪声或非光滑优化。这项工作是在早期集成成功的基础上进行的。标准蒙特卡罗采样集成了一个缓慢减小的误差。确定性的准蒙特卡罗采样可以获得更准确的答案，但没有实际的误差估计。重新注入一些随机性使人们能够估计误差，并使答案的质量得到惊人的进一步提高。第一个问题是在逼近问题上使用积分方法。在一组基（多项式、傅立叶函数或小波）中展开函数，发现系数是高维积分。这些系数的估计，加上统计不确定性，可以用来给出带有误差估计的近似值。也可以解决定性问题，如：函数的有效维度，函数的平滑度，重要输入的数量，等等。第二个问题是在面对其他一些变量的随机性时，优化函数对某些变量的期望值。一个例子是如何设计一个非线性模型的实验。第三个问题是预测从数据中学习到的二元函数。一个例子是是否持有或执行美式期权。依赖于大量输入的计算机代码正变得无处不在。它们被用于半导体、飞机和汽车的设计、气候模型和金融风险管理。在任何给定的任务中，提取埋藏在该软件中的相关知识都是一个巨大的挑战。对研究结果进行不确定性估计也是必要的。即使只有几十个输入因素，也有必要采用本项目所研究的统计方法。该项目还考虑依赖于一百万或更多输入因素的函数。计算机能力的进步将使人们更多地关注这些功能，因此将需要新的方法，例如本项目所研究的方法。

Owen0072445AbstractThe focus of this project is the application of statistical ideas to high dimensional numerical problems, such as approximation and noisy or nonsmooth optimization. This work follows on earlier successes in integration. Standard Monte Carlo sampling integrates with a slowly decreasing error. Deterministic quasi-Monte Carlo sampling can achieve a much more accurate answer, but without a practical error estimate. Re-injecting some randomness allows one to estimate the error, and gave rise to a surprising further large improvement in the quality of the answer. The first problem is to use integration methods on approximation problems. One expands the function in a basis (polynomials, Fourier functions, or wavelets), and finds that the coefficients are high dimensional integrals. Estimates of these coefficients, with statistical uncertainty attached, can be used to give approximations with error estimates. It is also possible to address qualitative issues such as: effective dimension of the function, smoothness of the function, number of important inputs, and so on. The second problem is to optimize the expected value of a function over some variables in the face of randomness in some others. An example is how to design an experiment for a nonlinear model. The third problem is to predict binary functions learned from data. An example is whether to hold or exercise an American type option.Computer codes that depend on a great many inputs are becoming ubiquitous. They are used in the design of semiconductors, airplanes and automobiles, in climate models, and in financial risk management. On any given task, it can be a great challenge to extract the relevant knowledge buried within this software. It is also necessary to attach uncertainty estimates to the findings. For even a few dozen input factors, it becomes necessary to employ statistical methods, of the type being researched in this project. This project also considers functions that depend on one million or more input factors. Advances in computer power will bring more attention to such functions, and new methods, such as those investigated in this project, will be required.