Optimization of the prediction of second refined wavelet coefficients in electron structure calculations

Abstract

Abstract In wavelet-based solution of eigenvalue-type differential equations, like the Schrödinger equation, refinement in the resolution of the solution is a costly task, as the number of the potential coefficients in the wavelet expansion of the solution increases exponentially with the resolution. Predicting the magnitude of the next resolution level coefficients from an already existing solution in an economic way helps to either refine the solution,or to select the coefficients, which are to be included into the next resolution level calculations, or to estimate the magnitude of the error of the solution. However, after accepting a solution with a predicted refinement as a basis, the error can still be estimated by a second prediction, i.e., from a prediction to the second finer resolution level coefficients. These secondary predicted coefficients are proven to be oscillating around the values of the wavelet expansion coefficients of the exact solution. The optimal averaging of these coefficients is presented in the following paper using a sliding average with three optimized coefficients for simple, one-dimensional electron structures.