Date of Award
Doctor of Philosophy (PhD)
Mathematics and Statistics
Computational and Applied Mathematics
Philip W. Smith
James L. Schwing
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's 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's 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's work on minimal norm unconstrained interpolation.
Irvine, Larry D..
"Minimal Norm Constrained Interpolation"
(1985). Doctor of Philosophy (PhD), dissertation, Mathematics and Statistics, Old Dominion University, DOI: 10.25777/k9fd-rm76