Multilevel interpolation of scattered data using ${\mathcal{H}}$-matrices

Abstract

Scattered data interpolation can be used to approximate a multivariate function by a linear combination of positive definite radial basis functions (RBFs). In practice, the approximation error stagnates (due to numerical instability) even if the function is smooth and the number of data centers is increased. A smaller approximation error can be obtained using multilevel interpolation on a sequence of nested subsets of the initial set of centers. For the construction of these nested subsets, we compare two thinning algorithms from the literature, a greedy algorithm based on nearest neighbor computations and a Poisson point process. The main novelty of our approach lies in the use of ${\mathcal{H}}$ -matrices both for the solution of linear systems and for the evaluation of residual errors at each level. For the solution of linear systems, we use GMRes combined with a domain decomposition preconditioner. Using ${\mathcal{H}}$ -matrices allows us to solve larger problems more efficiently compared with multilevel interpolation based on dense matrices. Numerical experiments with up to 50,000 scattered centers in two and three spatial dimensions demonstrate that the computational time required for the construction of the multilevel interpolant using ${\mathcal{H}}$ -matrices is of almost linear complexity with respect to the number of centers.