Kernel Methods for Numerical Computation

PI的研究为多元函数逼近问题的核方法提供了更深入的理解。有五个主要的研究重点。首先，推导了基于有噪声和无噪声数据的核方法的维数无关的误差边界。其次是调查哪些设计（数据站点的安排）达到了这些误差界限。第三种方法是使用Green的函数来更好地理解与所使用的内核相关的固有本地空间。第四是利用核特征函数展开构造核逼近的数值稳定计算算法。最后的重点是开发核近似的快速评估算法，再次使用特征函数展开。该理论的发展为学术界和工业界的实践者在材料工程、复杂流体流动模拟和核反应堆模拟等应用领域的数值模拟算法的发展提供了洞察力和支持。研究人员与Matlab、NAG和JMP统计软件等软件开发商合作，将他们的算法包含在这些软件包的未来版本中。这项研究正在数学、统计和工程社区之间传播，以建立他们之间的桥梁。特别是，研究人员正在向国内和国际观众介绍核方法的教程。研究结果在几个研究生课程中讲授，这些课程通常吸引应用数学、工程和商业专业的学生。一个关键的优先事项是尽早以REU经验的形式让学生参与计算数学研究，从而为学术界或工业界培养年轻的计算数学家。调查人员强调，包括来自代表性不足的少数民族和芝加哥地区不向学生提供计算研究机会的大学的学生。计算是解决各种科学、工程和社会问题不可或缺的工具。然而，准确和及时的答案需要很好地理解和正确应用的计算算法。这项研究的重点是推断与多个输入和输出相关的函数的基本问题，例如，数十个工程设计参数的值决定核反应堆内部温度的方式。核方法在某些情况下是灵活和准确的，但是它们对于大量输入的适用性，如刚才给出的例子，还不清楚。PI的研究解决了这个问题。成功意味着理解复杂过程所需的时间密集型计算机模拟的数量可以减少，并被通过内核方法构造的代理所取代。本研究展示了如何规划计算机模拟以获得最大的精度。此外，还开发了更快地构建此代理的方法。由于这项研究的基本性质，研究结果预计将影响许多涉及能源、制造业和纳米技术领域的工程师和科学家使用的通用数值计算包。这个研究项目不仅包括博士生，还包括硕士和学士学位的学生，正在为下一代的计算科学家做准备，他们是支持我们持续的技术和经济增长所需要的。

The PI's research provides a deeper understanding of kernel methods for multivariate function approximation problems. There are five main research thrusts. The first is to derive dimension-independent error bounds for kernel methods based on noisy and noiseless data. The second is to investigate which designs (arrangements of data sites) achieve these error bounds. The third is to use Green's functions to develop a better understanding of the inherent native spaces associated with the kernels used. The fourth is to use the kernel eigenfunction expansions to construct numerically stable evaluation algorithms for kernel approximation. The final thrust is to develop fast evaluation algorithms for kernel approximation, again using the eigenfunction expansions. The theoretical development provides practitioners in academia and industry insight and support for the development of numerical simulation algorithms in such application areas as materials engineering, complex fluid flow simulations, and nuclear reactor simulation. The investigators partner with software developers such as Matlab, NAG and JMP statistical software to have their algorithms included in future releases of these software packages. This research is being disseminated among the mathematics, statistical, and engineering communities to build bridges between them. In particular, the investigators are presenting tutorial courses on kernel methods to national and international audiences. The research findings are taught in several graduate courses that routinely draw students from applied mathematics, engineering and business. One key priority is to engage students in computational mathematics research as early as possible in the form of an REU experience and thereby develop a pipeline of young computational mathematicians for academia or industry. The investigators stress the inclusion of students from underrepresented minorities and from universities in the Chicago area that do not provide computational research opportunities to their students.Computation is an indispensable tool for solving a variety of scientific, engineering, and societal problems. However, accurate and timely answers require computational algorithms that are well understood and properly applied. This research focuses on the fundamental problem of inferring the function that relates multiple inputs to an output, e.g., the way in which values of tens of engineering design parameters determine the temperature inside a nuclear reactor. Kernel methods are flexible and accurate in certain settings, but their applicability for large numbers of inputs, as in the example just given, is not understood. The PI's research addresses this issue. Success means that the number of time-intensive computer simulations needed to understand complex processes can be reduced, and be replaced by a surrogate constructed via kernel methods. This research shows how to plan the computer simulations for maximum accuracy. Moreover, the methods for constructing this surrogate more quickly are developed. Because of the fundamental nature of this research, the findings are expected to influence general purpose numerical computation packages used by many engineers and scientists involved in the fields of energy, manufacturing, and nanotechnology. By including not only PhD students, but also MS and BS students, this research project is preparing the next generation of computational scientists, who are needed to support our continued technological and economic growth.