Pointwise optimal multivariate spline method for recovery of twice differentiable functions on a simplex

TL;DR

This paper introduces a pointwise optimal multivariate spline method for recovering twice differentiable functions on a simplex, utilizing function values and gradients at vertices, and characterizes its error function as a piecewise quadratic spline.

Contribution

It develops a new optimal spline recovery method on a simplex using function and gradient data, and characterizes its error function as a multivariate analogue of Euler splines.

Findings

The method is optimal for recovery within the class of twice differentiable functions.

The error function is a piecewise quadratic $C^1$-function over a polyhedral partition.

The method produces a continuous spline of degree two with some degree-one pieces.

Abstract

We obtain the spline recovery method on a $d$-dimensional simplex $T$ that uses as information values and gradients of a function $f$ at the vertices of $T$ and is optimal for recovery of $f({\bf w})$ at every point ${\bf w}$ of an admissible domain $P$ containing $T$ on the class $W^2(P)$ of twice differentiable functions on $P$ with uniformly bounded second order derivatives in any direction. If, in particular, every face of $T$ (of any dimension) contains its circumcenter, we can take $P=T$. We also find the error function of the pointwise optimal method which turns out to be a function in $W^2(P)$ with zero information. The error function is a piecewise quadratic $C^1$-function over a certain polyhedral partition and can be considered as a multivariate analogue of the classical Euler spline $\phi_2$. The pointwise optimal method is a continuous spline of degree two (with some pieces of degree one) over the same partition.