Fast and Accurate High Dimensional Function Approximation

本研究将开发计算方法，以近似的函数值，可能会被噪声污染的基础上的许多变量的函数。 重点将放在基于核的方法上，如径向基函数方法、平滑样条、回归样条和移动最小二乘法。 非各向同性内核，空间填充设计，和新的收敛性证明将用于使这些方法在更高的维度实用。 快速变换和迭代算法将被开发以减少核方法的计算负担。 计算方法和统计方法都将用来解决这个问题。 理论的发展将为从业者提供深入了解设计（样本点）的质量如何影响近似的准确性。 开发的算法将作为软件包出版，供广泛使用。在这个信息时代，有大量的数据来自仪器测量和计算机模拟。 战略性的企业规划和盈利性的产品设计都依赖于精确的数学模型来描述这些数据。 随着观测和变量数量的增加，现有的计算最佳模型的方法无法捕捉复杂的关系，也无法在合理的时间内计算出答案。 这项研究将开发新一代的计算方法来建模数据，克服这些缺点。这些新方法可使制造业更迅速地制造产品原型，从而提高其竞争力。此外，这些新方法亦可使服务业更迅速地评估不断转变的市场，并作出回应。

This research will develop computational methods to approximate functions of many variables based on function values which may be contaminated by noise. The focus will be on kernel-based methods, such as radial basis function methods, smoothing splines, regression splines and moving least squares methods. Non-isotropic kernels, space filling designs, and new convergence proofs will be used to make these methods practical in higher dimensions. Fast transform and iterative algorithms will be developed to reduce the computational burden of kernel methods. Both computational and statistical approaches will be brought to bear on this problem. The theoretical development will provide practitioners insight into how the quality of the designs (sample points) affects the accuracy of approximation. The algorithms developed will be published as software packages for widespread use. In this information age there is an abundance of data generated from instrumental measurements and computer simulations. Strategic corporate planning and profitable product design both rely on accurate mathematical models to describe this data. As the numbers of observations and variables increase, existing methods for computing the best models fail to capture the complex relationships and fail to compute an answer in a reasonable amount of time. This research will develop a new generation of computational methods for modeling data that overcome these drawbacks. These new methods will make our manufacturing industry more competitive by facilitating more rapid prototyping of products, and they will enable our service industry to assess and respond more quickly to changing markets.