Preconditioned conjugate gradient method for finding minimal energy surfaces on Powell–Sabin triangulations

We present an iterative proposal, based on the preconditioned conjugate gradient method, to solve the linear system associated to the problem of approximating a data set by a minimal energy surface constructed through a Powell–Sabin finite element over a Δ1Δ1-type triangulation defined on a polygonal domain. These approximation problems give rise to symmetric, banded, and positive definite matrices with a very special block structure that depends on the basis functions of the associated vector space, and also on the numeration of the nodes in the triangulation. In practice, the associated sparse matrices are large and ill-conditioned. The special structure of these matrices allows us to adapt and explore several known preconditioned strategies to improve the performance of the conjugate gradient method.We adapt and explore five different preconditioning strategies, including some well-known direct and also some recent inverse strategies. Special attention is paid to the delicate and difficult task of choosing the related parameters in each case. The quality of each preconditioner is evaluated by observing the clustering of the preconditioned matrix eigenvalues, and the obtained reduction in number of iterations. We report on seven different surfaces, and our results indicate that the best preconditioning strategies for this application are the ones based on incomplete factorizations. Nevertheless, from a computational-cost point of view, all the explored strategies are competitive.