An Efficient ADI-Solver for Scattered Data Problems with Global Smoothing

Abstract

For the approximate representation of large data sets over a parameter domain inR2, many geological and other applications require the construction of surfaces which have minimal energy, i.e., minimal curvature. One way to achieve this is by solving a fourth-order elliptic partial differential equation. Its discretization by a difference scheme makes it particularly easy to incorporate (appropriate approximations of) known data points. In this paper, we investigate the performance of different solution methods for the resulting symmetric linear system of equations since this is the most CPU-demanding step in the scattered data approximation procedure. Specifically, we test first the performance of a preconditioned conjugate gradient method with an SSOR and an RILU preconditioner. However, since the partial differential operator does not contain mixed derivatives, using an alternating-direction-implicit scheme (ADI method) which has been employed successfully in the past for second-order problems, together with a Cholesky factorization of the corresponding one-dimensional operators has also been tried for the problem at hand. The computational studies that we have performed here show that for our problem an instationary ADI method is superior to the above-mentioned preconditioned conjugate gradient solvers both with respect to work load and accuracy of the solution. For the fourth-order model problem considered in this paper, the instationary ADI method with Wachspress parameters results in a number of iterations that is essentially independent of the number of variables.