Node Generation for RBF-FD Methods by QR Factorization

Polyharmonic spline (PHS) radial basis functions (RBFs) have been used in conjunction with polynomials to create RBF finite-difference (RBF-FD) methods. In 2D, these methods are usually implemented with Cartesian nodes, hexagonal nodes, or most commonly, quasi-uniformly distributed nodes generated through fast algorithms. We explore novel strategies for computing the placement of sampling points for RBF-FD methods in both 1D and 2D while investigating the benefits of using these points. The optimality of sampling points is determined by a novel piecewise-defined Lebesgue constant. Points are then sampled by modifying a simple, robust, column-pivoting QR algorithm previously implemented to find sets of near-optimal sampling points for polynomial approximation. Using the newly computed sampling points for these methods preserves accuracy while reducing computational costs by mitigating stencil size restrictions for RBF-FD methods. The novel algorithm can also be used to select boundary points to be used in conjunction with fast algorithms that provide quasi-uniformly distributed nodes.

Automated summary

Polyharmonic spline (PHS) radial basis functions (RBFs) are often used in RBF finite-difference (RBF-FD) methods alongside polynomials. Novel strategies for computing the placement of sampling points in both 1D and 2D have been explored, determining the optimality of sampling points through a piecewise-defined Lebesgue constant. By modifying a column-pivoting QR algorithm, sampling points can be selected to reduce computational costs and maintain accuracy in RBF-FD methods.