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.
A function may be thought of as a rule that takes many inputs and provides an output, e.g., the temperatture of a nuclear reactor core (output) depends on several inputs, such as the dimensions of the core, the spacing of the fuel rods, and the size of the fuel rods. One may be able to observe (by physical measurement or computer simulation) the values of a function for a modest number of different sets of input values. Based on this knowledge, one would like to quickly and accurately approximate the value of a function for an arbitrary set of input values. This is the goal of our project. Intellectual Merit: Function approximation from data has been studied for a long time, but unresolved issues remain. One method that provides an optimal answer in some sense is based on a kernel, i.e., a special function of two different sets of inputs. Unfortunately, computations with nice kernels for approximation involve catstrophic errors due to the finite precision of computer calculations. Our new algorithms avoid those errors by reformulating the problem in a better way. Although there are theoretical upper bounds on the errors of kernel methods, they are impractical for estimating the actual error observed. We have found ways of reliably estimating error for related problems that may in the future be used for kernel methods. Kernels have parameters that need to be tuned, and the error bounds depend on the number of inputs as well as the number of data. We have shown that under certain conditions on the parameters, kernel methods can still be practical for functions that depend on a very large number of inputs. Broader Impacts: Function approximation is used in a variety of applications where expensive computer simulations are run, and the practioners need to construct an approximation to the output of the simulation as a function of the simulation inputs. Our results have clarified the limits of kernel methods and shown ways to overcome some of those limits. This will assist non-mathematicians in their use of mathematical and computational methods for approximating functions. We have involved two high school students, more than ten undergraduate students, and over a dozen graduate students in our research, along with some visiting scholars and early-career faculty. They have been given the opportunity to learn how to identify an interesting and tractable research problem, to try a variety of approaches, to obtain help from books, articles, and other group members, to organize their accomplishments, to translate their mathematical results into quality mathematical software, and to present their work to friends and to strangers. Thus, they have learned about the scientific process and are better prepared for careers in academia, business, government, or industry.
