Mathematical Sciences: Miltivariate Interpolation and Approximation

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Combining techniques from disparate areas (Algebra, Geometry,and Combinatorics) the investigator and his coworkers recently determined the generic dimension of trivariate spline spaces (of sufficiently high degree) defined on tetrahedral decompositions. In this project the techniques will be further developed and applied to multivariate spline research, particularly as regards the following problems: exact dimension formulas, improved upper and lower bounds on dimensions, dependence of the dimension on geometric degeneracies, existence and construction of local bases,identification of useful sub- and superspaces of the full spline space, solvability of interpolation problems, effects of degeneracies on numerical stability and solvability of interpolation problems, algorithms, surface design, approximation of functions defined on surfaces, and investigation of domain partitions other than simplicial ones. Splines are smooth piecewise polynomial functions. They are used ubiquitously to solve problems involving functions of one independent variable. Regarding functions of several independent variables, splines (which in this context are called "multivariate splines") are much less understood and their structure is vastly more complicated. Yet they have great potential for applications such as the solution of partial differential equations (e.g., modeling fluid flow, heat distribution, combustion, or radioactive decay), interpolation and approximation of data, and the design of shapes (e.g., of a vehicle). They are also fundamental objects that deserve to be studied in their own right.