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

Series: 
Applied and Computational Mathematics Seminar
Monday, September 21, 2009 - 1:00pm
1 hour (actually 50 minutes)
Location: 
Skiles 255
,  
Department of Mathematics and Statistics, Sam Houston State University
Organizer: 
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.