Fast radial basis function interpolation via preconditioned Krylov iteration

TitleFast radial basis function interpolation via preconditioned Krylov iteration
Publication TypeJournal Articles
Year of Publication2010
AuthorsGumerov NA, Duraiswami R
JournalSIAM Journal on Scientific Computing
Pagination1876 - 1876
Date Published2010///

We consider a preconditioned Krylov subspace iterative algorithm presented by Faul,Goodsell, and Powell (IMA J. Numer. Anal. 25 (2005), pp. 1–24) for computing the coefficients of
a radial basis function interpolant over N data points. This preconditioned Krylov iteration has
been demonstrated to be extremely robust to the distribution of the points and the iteration rapidly
convergent. However, the iterative method has several steps whose computational and memory
costs scale as O(N2), both in preliminary computations that compute the preconditioner and in the
matrix-vector product involved in each step of the iteration. We effectively accelerate the iterative
method to achieve an overall cost of O(N log N). The matrix vector product is accelerated via the use
of the fast multipole method. The preconditioner requires the computation of a set of closest points
to each point. We develop an O(N log N) algorithm for this step as well. Results are presented for
multiquadric interpolation in R2 and biharmonic interpolation in R3. A novel FMM algorithm for
the evaluation of sums involving multiquadric functions in R2 is presented as well.