Statistical Numerics

这个项目的重点是将统计思想应用于高维数值问题，如逼近、噪声或非光滑优化。这项工作是在早期一体化成功的基础上进行的。标准蒙特卡罗抽样具有缓慢减小的误差。确定性的准蒙特卡罗抽样可以得到更准确的答案，但没有实际的误差估计。重新注入一些随机性可以让人估计误差，并在答案质量上带来令人惊讶的进一步大幅改善。第一个问题是在近似问题上使用积分方法。人们在基数(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。对这些系数的估计，加上统计不确定性，可以用来给出带有误差估计的近似值。还可以解决定性问题，例如：函数的有效维度、函数的平稳性、重要输入的数量等。第二个问题是在某些变量具有随机性的情况下，对函数的期望值进行优化。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的二元函数。例如，是否持有或行使美式打字选择权。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，用于气候模型，以及金融风险管理。在任何给定的任务中，提取隐藏在该软件中的相关知识都可能是一个巨大的挑战。还有必要将不确定性估计附加到调查结果中。即使只有几十个输入因素，也有必要采用本项目所研究的那种统计方法。该项目还考虑了依赖于一百万或更多输入因子的函数。计算机能力的进步将引起人们对这些功能的更多关注，将需要新的方法，如本项目中所研究的方法。

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.