RANDOM INTERPOLATION

Summary of Proposal "INTERPOLATION MODEL (Dirac-Monte Carlo Method)"

Part (1) This Small Business Innovation Research Phase I project is to develop a workable and efficient solution to multivariate interpolation. In the interpolation field, a general problem exists in the form of the so-called curse of dimensionality (COD). The convergence rate of mean-squared error of the interpolant (estimator) found depends on the dimension number D and has lower bound proportional to N-4/(4+D) where N is the input sample number. Thus, N should be a sufficiently large number in order to yield good convergence rate for high D. The second problem in the interpolation field is its inability to handle non-convex domain. The research objective of this proposal is threefold: (1) To scale up the current Random Data Interpolation Center (RDIC), which is on the web at http://www.fanginc.com/main.htm, from the present 4-D capability to 10-D and from the present capacity of input sample number 72 to 10,000; (2) To demonstrate convergence performance of the interpolation calculations by using the proposed Dirac-Monte Carlo (DMC) method via quasi-Monte Carlo sampling; (3) To demonstrate interpolation calculation in non-Cartesian coordinates (3-D) in non-convex domain. This SBIR Phase I research effort will primarily consist of software development of upgrading RDIC. Interpolation testing cases will be constructed, tested and documented. The anticipated results of this proposed project are: (a) To provide proof of the efficiency and the scalability of the proposed method so that, in SBIR Phase II, further extension to 100-dimension with N equal to 1 million can be carried out; (b) To provide an unique web portal of interpolation computation and online training to research and education communities as well as commercial industries; (c) To demonstrate the advantage of Dirac-Monte Carlo method to the interpolation community for stimulating further mathematical research. Part (2) The proposed Phase I project is in compliance with NSF SBIR program goals and it is aimed at securing subsequent progression to Phase II and Phase III. The commercial aspects of the proposed activity can be grouped in two areas, (1) Near term - RDIC shall be able to provide a dedicated web computing facility to interpolation users, and users need to pay subscription fee for accessing RDIC. Online visualization graphics as well as customized programming services shall be provided; (2) Long term - RDIC shall be further upgraded and expanded to provide more features and data services to the user community as a dedicated web computing portal, and the commercialization of "interpolation integrated-circuit (IC) chip" shall be implemented to solve problems with higher dimension D>100 and/or large input sample number N>106 for real-time applications. It is known that interpolation is fundamental and generic to all disciplines that require the work of data analysis. The algorithms formulated in the proposal are strictly based on mathematics and they are independent of the nature of the application. Therefore, the projected software establishment at RDIC can be directly applied to, engineering design, information data mining, climatology and environmental modeling, social and psychological studies, econometrics, biometrics, geostatistics, astronomy, data assimilation, computer graphics, geometric surface reconstruction, artificial intelligence (machine learning), robotics and humanoid robot, CNC machine ruled surface interpolation, visual recognition, neuroscience, hurricane and flood forecast, seismology, oceanography, etc.