Variational bivariate interpolating splines with positivity constraints
Positivity preserving interpolation methods are discussed in the context of Sobolev spaces \(H^m(\Omega) \) of order \( m \) for a finite set of interpolation points \(A\) given in a bounded subset \(\Omega \) of the plane \({\mathbb{R}} ^2 . \) For a data function \( f \in H^m(\Omega) \) the authors show the existence of a so-called \textit{positive} interpolation variational spline \(\sigma \in H^m(\Omega) \) which is the unique solution of the minimization problem \(|\sigma |\leq |v|\) where \( v\in H^m(\Omega) \) satisfies the positivity condition \( v(x) \geq 0 \) for all \( x\in \Omega \) and the interpolation condition \( v(a) = f(a) \) for all \( a \in A \); here the semi-norm \(|\cdot|\) is given through the \(L^2 \)-integral of the \(m\)-th derivative. Using finite elements methods a discretized version of the above problem is formulated and solved by an explicit algorithm. Convergence results show that the discretizations \(\sigma_h \) converges in the Sobolev norm to the positive interpolation variational spline \( \sigma . \) The authors illustrate their results by numerical and graphical examples.
scientific article; zbMATH DE number 1879009