Multi-level biharmonic and bi-Helmholtz interpolation with application to the boundary element method

Abstract

The scattered data interpolation problem is investigated. Instead of the direct use of radial basis functions, the interpolation function is sought as a solution of a higher order partial differential equation supplied with the interpolation equations as special boundary conditions. In this paper the methods based on the biharmonic and the bi-Helmholtz equations are analysed. The interpolation problem is reformulated in variational forms. Existence and uniqueness theorems are proved in Sobolev spaces. The approximation properties of this interpolation are also investigated. A representation theorem is proved which shows the similarity to the method of radial basis functions based on the fundamental solution of the applied partial differential operator. To solve the appearing biharmonic/bi-Helmholtz equation, a multi-level method is presented which is based on a quadtree/octtree cell system generated by the interpolation points. It is shown that the overall computational cost of the presented method is much less than that of the traditional method of radial basis functions. The method makes it possible to avoid the solution of large, fully populated and often ill-conditioned systems of linear equations. Finally, some applications to solving partial differential equations are outlined. The biharmonic/bi-Helmholtz interpolation technique immediately defines a grid-free method, but can be combined with the boundary element method as well. A possible application in the dual reciprocity method is also presented.