Estimation of Analytic Surfaces with Applications to Nanoparticle Characterization via Surface Waves

This three-year project will advance functional data analysis by developing a new compound estimation paradigm for simultaneously estimating a mean response and several of its derivatives as functions of a one- or two-dimensional covariate (to be generalized in the future to a covariate of arbitrary dimension). This project will also advance nanoparticle characterization by creating and operationalizing an inversion technique to visualize nanoparticles nonintrusively in real time. A hybrid between local modeling and global modeling, compound estimation will enable data analysts to recover a mean response and several of its derivatives from noisy data while circumventing the difficulties associated with local averaging (e.g., kernel), local modeling (e.g., local regression), and global modeling (e.g., spline) approaches. These difficulties include: the empirical disparity between asymptotic theory and finite-sample performance in local averaging; the pointwise character of local modeling, along with the incompatibilities between mean response estimates and derivative estimates in that setting; and, the unrealistic assumptions tacitly imposed in global modeling, such as there being a finite number of nonzero derivatives with discontinuities in the last nonzero derivative. Compound estimation will be particularly useful for problems in which physical phenomena are described by differential equations, problems in which modeling velocities and accelerations is of scientific importance, and pattern recognition problems in which features of higher-order derivatives can be exploited for classification. One such pattern recognition problem has remained an outstanding challenge in nanoparticle characterization, and its solution is of independent interest. Given surface wave scattering data obtained at various scattering angles and/or wavelengths, we want to infer the configuration of nano-size metallic particles, agglomerates, and structures on or near the surface. Being able to characterize nano-size particles and agglomerates accurately from surface wave scattering data overcomes a major hurdle to building realtime diagnostic tools, which are indispensable for future nanomanufacturing efforts. This pattern recognition problem will be solved through the creation of an inversion technique that objectively assigns probabilities to the different possible configurations ofnanoparticles. The inversion technique will employ mean response estimates along with derivative estimates, as the latter may be crucial to classification when mean responses associated with different configurations look similar but have disparities that are amplified upon repeated differentiation. The ability of compound estimation to recover derivatives accurately is thus central to the inversion. In addition, an algorithm will be designed to detern1ine what limited amount of experimental data should be gathered for effective visualization. This will ensure the feasibility of performing the inversion in real time. Broader Impacts. Nanomanufacturing is a very important extension of nanotechnology, and reliable control of nanoparticle growth and surface assembly is a key building block for future applications in nanomanu facturing. There is a need for advanced instrumentation to allow real-time on-line diagnostics of chemical and physical processes relevant to nano-scale engineering, from self-assembly to nanofabrication. These instruments will find applications in electronics, bulk material fabrication, membrane synthesis, nano-mechanical systems, and DNA screening. This project is about developing the "eyes" and "brain" behind such diagnostic tools. In that sense, the success of the compound estimation approach will directly impact the development of instruments crucial to future nanotechnology applications. Empirical and theoretical findings, along with guidelines for implementing the new methodologies, will be published in appropriate venues and presented to local, regional, and national audiences. User-friendly software will be made freely available online so that compound estimation can be employed by data analysts as easily as kernel smoothing, local regression, and spline smoothing are now. This project will have a strong impact on graduate and undergraduate education at the University of Kentucky, both in Statistics and Engineering. This project will directly support two doctoral students in Statistics and one in Mechanical Engineering. These students will participate in the presentation of findings to local, regional, and national audiences. Their involvement will also constitute an exciting example of the multidisciplinary research opportunities that can be anticipated by incoming and prospective students. In addition, three undergraduate students will serve as paid research assistants (one in each year). Currently the College of Engineering has two education programs on Nano-Scale Engineering. In particular, the Nano-Scale Engineering Certificate Program (NECP) was established in early 2004 with the help of an NSF -NU E grant. These programs foster extensive awareness of nano-scale engineering efforts among students (including freshmen) and allow undergraduates to work closely with faculty and graduate students. Through this project we intend to further such interactions.