A Krylov subspace algorithm for multiquadric interpolation in many dimensions

We consider the version of multiquadric interpolation wherethe interpolation conditions are the equations s(x_i) = f_i, i= 1,2,..., n, and where the interpolant has the form s(x) =_j=1ⁿ _j (||x – x_j ||² + c²)^1/2 + x R^d, subject to theconstraint _j=1ⁿ _j = 0. The points x_i R^d, the right-hand sidesf_i, i = 1,2,...,n, and the constant c are data. The equationsand the constraint define the parameters _j, j = 1,2,...,n, and. The resultant approximation s f is useful in many applications,but the calculation of the parameters by direct methods requiresO (n³) operations, and n may be large. Therefore iterative proceduresfor this calculation have been studied at Cambridge since 1993,the main task of each iteration being the computation of s(x_i),i = 1,2,...,n, for trial values of the required parameters.These procedures are based on approximations to Lagrange functions,and often they perform very well. For example, ten iterationsusually provide enough accuracy in the case d = 2 and c = 0,for general positions of the data points, but the efficiencydeteriorates if d and c are increased. Convergence can be guaranteedby the inclusion of a Krylov subspace technique that employsthe native semi-norm of multiquadric functions. An algorithmof this kind is specified, its convergence is proved, and carefulattention is given to the choice of the operator that definesthe Krylov subspace, which is analogous to pre-conditioningin the conjugate gradient method. Finally, some numerical resultsare presented and discussed, for values of d and n from theintervals 2,40] and 200,10 000], respectively.