Some preconditioners for harmonic spherical spline problems

When harmonic spherical splines are used to interpolate and predict discret ely given data we are confronted with the problem of solving symmetric posi tive definite systems involving as many equations as the number of data. Du e to harmonicity, these systems are dense and thus iterative methods should be preferred to direct ones for large data sets. Nevertheless iterative so lvers may converge very slowly or fail to converge. This paper develops a c lass of preconditioners based on sparse (banded) symmetric positive definit e approximations to the Gram matrices of harmonic kernels, the sparse appro ximations being defined as Gram matrices of locally supported approximation s to the kernels. After recalling the basic framework of harmonic spherical splines, truncated Legendre coefficients with closed-form expressions are presented. Numerical results on the use of the sparse approximates as preco nditioners for the conjugate gradient method are shown and demonstrate the efficiency of these preconditioners.