Good point sets and corresponding weights for bivariate discrete least squares approximation. (English) Zbl 1372.65044

Van Barel, Marc; Humet, Matthias

Dolomites Res. Notes Approx. 8, Spec. Iss., 37-50 (2015).

Summary: An algorithm is presented to compute good point sets and weights for discrete least squares polynomial approximation on a geometry \(\Omega\subset\mathbb R^2\). The criterion that is used is the minimisation of the Lebesgue constant of the corresponding least squares operator. In order to approximate the Lebesgue constant, we evaluate the Lebesgue function in a point set generated by a refinement method that is based on Delaunay triangulation. The algorithm is greedy in the sense that points are added where the Lebesgue function is largest. It also uses a new updating algorithm for the weights such that the corresponding Lebesgue constant is made smaller. Advantages of the method are that it works for a general geometry \(\Omega\) and that the point sets are nested. Numerical experiments show that the Lebesgue constant corresponding to the least squares operator is low and grows slower than polynomially in function of the total degree of the polynomial approximation space. It follows that the computed points are point sets of a weakly admissible mesh (WAM).

MSC:

65D10	Numerical smoothing, curve fitting
41A10	Approximation by polynomials
41A60	Asymptotic approximations, asymptotic expansions (steepest descent, etc.)