On a construction of a hierarchy of best linear splineapproximations using repeated bisection

摘要

We present a method for the construction of hierarchies ofnsingle-valued functions in one, two, and three variables. The input tonour method is a coarse decomposition of the compact domain of a functionnin the form of an interval (univariate case), triangles (bivariatencase), or tetrahedra (trivariate case). We compute best linear splinenapproximations, understood in an integral least squares sense, fornfunctions defined over such triangulations and refine triangulationsnusing repeated bisection. This requires the identification of theninterval (triangle, tetrahedron) with largest error and splitting itninto two intervals (triangles, tetrahedra). Each bisection step requiresnthe recomputation of all spline coefficients due to the global nature ofnthe best approximation problem. Nevertheless, this can be donenefficiently by bisecting multiple intervals (triangles, tetrahedra) innone step and by reducing the bandwidths of the matrices resulting fromnthe normal equations