In this work, we address the interpolation of values at given points using bivariate splines, focusing on the Lagrange interpolation problem. One approach to solving the Lagrange interpolation problem for scattered points is interpolation with continuous linear splines over a triangulation. Another approach, presented in this work, involves finding a spline that minimizes its energy. A spline obtained in this way is called a minimal energy interpolating spline. Two examples of such splines are the Argyris and Powell-Sabin minimal energy splines. The latter is constructed on a special refinement known as the Powell-Sabin refinement. In both macro-element spaces, there exist uniquely determined splines that solve the Hermite interpolation problem. If the derivatives at the nodes of the triangulation are unknown, they can be estimated with radial basis functions using two-stage methods. In the first stage, the derivatives at the nodes are estimated using radial basis functions, and in the second stage, the desired method for solving the Hermite interpolation problem is applied.
|
