Algorithms for optimal discontinuous piecewise linear and constant 𝐿₂ fits to continuous functions with adjustable nodes in one and two dimensions

Abstract

Abstract. In this paper a direct variational approach (with nonstandard variations) is used to generate algorithms to determine optimal discontinuous piecewise linear and piecewise constant L2 fits to a continuous function of one or two variables with adjustable nodes. In the one-variable case the algorithm is fast and robust, the mesh cannot tangle, and the resulting fits are continuous a.e. In the two-variable case, on an adjustable triangular grid, the algorithm is less robust but gives good results for particular functions possessing a single steep feature. The extension to higher dimensions is straightforward.