Incomplete series expansion for function approximation

Abstract

We present an incomplete series expansion (ISE) as a basis for function approximation. The ISE is expressed in terms of an approximate Hessian matrix which may contain second, third and even higher order ‘main ’ or diagonal terms, but which excludes ‘interaction ’ or off-diagonal terms. From the ISE, a family of approximate interpolating functions may be derived. The interpolating functions may be based on an arbitrary number of previously sampled points, and any of the function and gradient values at previously sampled points may be enforced when deriving the approximate interpolating functions. When function values only are enforced, the approximations are spherical, and the storage requirements are minimal. Irrespective of the conditions enforced, the approximate Hessian matrix is a sparse diagonal matrix. Hence the proposed interpolating functions are very well suited for use in sequential approximate optimization (SAO), based on computationally expensive simulations. In turn, computationally expensive simulations are often required in, for example, optimal structural design problems. We derive a selection of approximations from the family of ISE approximating functions herein; these include approximations based on the substitution of (reciprocal) intervening variables. A comparison with popular approximating functions previously proposed, illustrates the accuracy and flexibility of the new family of interpolatin