Projection Methods and the Curse of Dimensionality

Abstract

We study the ability of three different projection methods to solve high-dimensional state space problems: Galerkin, collocation, and least squares projection. The curse of dimensionality can be reduced substantially for both Least Squares and Galerkin projection methods through the use of monomial formulas. Least Squares are shown to require a good initial value in order to give an accurate solution. Alternatively, we suggest a new ad hoc collocation method for complete polynomials that is fast and easy to implement.