Adaptive spline interpolation: asymptotics of the error and construction of the partitions

Applied and Computational Mathematics Seminar
Monday, September 21, 2009 - 13:00
1 hour (actually 50 minutes)
Skiles 255
Department of Mathematics and Statistics, Sam Houston State University
In this talk we first present the exact asymptotics of the optimal error in the weighted L_p-norm, 1\leq p \leq \infty, of linear spline interpolation of an arbitrary bivariate function f \in C^2([0,1]^2). We further discuss the applications to numerical integration and adaptive mesh generation for finite element methods, and explore connections with the problem of approximating the convex bodies by polytopes. In addition, we provide the generalization to asymmetric norms. We give a brief review of known results and introduce a series of new ones. The proofs of these results lead to algorithms for the construction of asymptotically optimal sequences of triangulations for linear interpolation. Moreover, we derive similar results for other classes of splines and interpolation schemes, in particular for splines over rectangular partitions. Last but not least, we also discuss several multivariate generalizations.