Orthogonal Global/Local Approximation in N-dimensions: Applications to Input-Output Approximation

Several approximation ideas are presented. The main idea discussed is a weighting function technique that generates a global family of overlapping preliminary approximations whose centroids of validity lie on at the vertices of an N-dimensional grid, with vertices separated by a uniform step h. These preliminary approximations are constructed so they represent the behavior in hypercubes with a volume (2h)N centered on a typical vertex in the grid. These volumes, where the preliminary approximations are valid, overlap by 50% and are averaged in the overlapped hN volume hypercubes, interior to each contiguous set of 2N vertices, to determine final approximations. We establish an averaging method that ensures these final approximations are globally piecewise continuous with adjacent approximations determined in an analogous averaging process, to some prescribed order of partial differentiation. The continuity conditions are enforced by using a unique set of weighting functions in the averaging process. The weight functions are designed to guarantee the global continuity conditions while retaining near complete freedom on the selection of the generating local approximations. However, if the preliminary local approximations are chosen as linear combinations of a set of basis functions constructed such that they are orthogonal with respect to the weight functions, then many advantages are realized, as demonstrated in the paper. Construction of the new set of orthogonal polynomials, and several properties of these functions are novel results presented in this paper. This paper enables a first: piecewise continuous least square approximation in N-dimensions, using orthogonal functions. Several applications are given which provide a basis for optimism on the practical value of the ideas presented. ∗Distinguished Professor, holder of George J. Eppright Chair, Department of Aerospace Engineering, Texas A&M University. †Graduate Research Assistant, Ph. D. Candidate, Dept. of Aerospace Engg., Texas A&M University. ‡Graduate Research Assistant, Ph. D. Candidate, Dept. of Aerospace Engg., Texas A&M University. §Graduate Research Assistant, M.S. Candidate, Dept. of Aerospace Engg., Texas A&M University