Optimal Incremental Approximation for the Solution of Differential Equations
Andrew J. Meade, Jr., Michael Kokkolaras, and Boris A. Zeldin
Submitted to International Journal for Numerical Methods in Engineering, 1998.

Keywords:: Incremental approximation, basis functions, variational principles, adaptivity, optimization, parallel direct search.

Abstract: A method for optimal incremental function approximation is proposed for the solution of differential equations. The basis functions and associated coefficients of a series expansion, representing the solution, are optimally selected at each step of the algorithm according to appropriate error minimization criteria; the solution is built sequentially. In this manner, the computational technique is adaptive in nature, although a grid is neither built nor adapted in the traditional sense using a-posteriori error estimates. Variational principles are utilized for the definition of the objective function to be extremized in the associated optimization problems, ensuring that the problems are well-posed. Complicated data structures, expensive remeshing algorithms, and systems solvers are avoided. Computational efficiency is increased by using low-order basis functions and the parallel direct search optimization technique. Numerical results and convergence rates are reported for linear nonself-adjoint and nonlinear problems associated with general boundary conditions. Generalization aspects of the method are discussed.

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

33 pages.