Minimal Norm Constrained Interpolation

In computational fluid dynamics and in CAD/CAM a physical boundary, usually known only discreetly (say, from a set of measurements), must often be approximated. An acceptable approximation must, of course, preserve the salient features of the data (convexity, concavity, etc.) In this dissertation we compute a smooth interpolant which is locally convex where the data are locally convex and is locally concave where the data are locally concave. Such an interpolant is found by posing and solving a minimization problem. The solution is a piecewise cubic polynomial. We actually solve this problem indirectly by using the Peano kernel theorem to recast this problem into an equivalent minimization problem having the second derivative of the interpolant as the solution. We are then led to solve a nonlinear system of equations. We show that with Newton\u27s method we have an exceptionally attractive and efficient method for solving this nonlinear system of equations. We display examples of such interpolants as well as convergence results obtained by using Newton\u27s method. We list a FORTRAN program to compute these shape-preserving interpolants. Next we consider the problem of computing the interpolant of minimal norm from a convex cone in a normed dual space. This is an extension of de Boor\u27s work on minimal norm unconstrained interpolation