Kernel Methods for Numerical Computation

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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.