Regularized symmetric positive definite matrix factorizations for linear systems arising from RBF interpolation and differentiation

Scattered data interpolation using Radial Basis Functions involves solving an ill-conditioned symmetric positive definite (SPD) linear system (with appropriate selection of basis function) when the direct method is used to evaluate the problem. The standard algorithm for solving a SPD system is a Cholesky factorization. Severely ill-conditioned theoretically SPD matrices may not be numerically SPD (NSPD) in which case a Cholesky factorization fails. An alternative symmetric matrix factorization, the square root free Cholesky factorization, has the same flop count as a Cholesky factorization and is successful even when a matrix ceases to be NSPD. A regularization method can be used to prevent the failure of the Cholesky factorization and to improve the accuracy of both SPD matrix factorizations when the matrices are severely ill-conditioned. The specification of the regularization parameter is discussed as well as convergence/stopping criteria for the algorithm. The formation of differentiation matrices with the regularized SPD factorizations is demonstrated to improve eigenvalue stability properties of RBF methods for hyperbolic PDEs.