Constrained Polynomial Approximation for Inverse Problems in Engineering

Abstract

This paper presents the derivation, implementation and testing of a series of algorithms for the least squares approximation of perturbed data by polynomials subject to arbitrary constraints. These approximations are applied to the solution of inverse problems in engineering applications. The generalized nature of the constraints considered enables the generation of vector basis sets which correspond to admissible functions for the solution of inverse initial-, internal- and boundary-value problems. The selection of the degree of the approximation polynomial corresponds to spectral regularization using incomplete sets of basis functions. When applied to the approximation of data, all algorithms yield the vector of polynomial coefficients \(\varvec{\alpha }\), together with the associated covariance matrix \(\mathsf {\Lambda }_{\varvec{\alpha }}\). A matrix algebraic approach is taken to all the derivations. A numerical application example is presented for each of the constraint types presented. Furthermore, a new approach to performing constrained polynomial approximation with constraints on the coefficients is presented.