Sequential Function Approximation for the Solution of Differential Equations
Andrew J. Meade, Jr., Michael Kokkolaras and Boris A. Zeldin
Submitted to Communications in Numerical Methods in Engineering, 1997.

Keywords:: sequential function approximation, interpolation functions, optimization, parallel direct search.

Abstract: A computational method for the solution of differential equations is proposed. With this method an accurate approximation is built by incremental additions of optimal local basis functions. The parallel direct search software package (PDS) that supports parallel objective function evaluations is used to efficiently solve the associated optimization problem. The advantage of the method is that although it resembles adaptive methods in computational mechanics, neither an a-priori grid, nor modifications of the implicitly built one (by means of nodes relocation algorithms, for example), are necessary. Moreover, the traditional set up of systems of equations and the associated expensive operations are avoided. Computational cost is reduced while efficiency is enhanced by low-dimensional in-parallel executed optimization. In addition, the method can be applied using a broad class of interpolation functions. Results and convergence rates obtained for one- and two-dimensional boundary value problems are satisfactory compared to these obtained by using conventional finite element methods.

This work was supported under NASA grant number CRA2-35504 and ONR grant N00014-95-1-0741.

15 pages.