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.
scattered data interpolation radial basis functions eigenvalue stability numerical pdes regularization techniques