Leevan Ling

Preconditioning for radial basis functions with domain decomposition methods

In our previous work, an effective preconditioning scheme that is based upon constructing least-squares approximation cardinal basis functions (ACBFs) from linear combinations of the RBF-PDE matrix elements has shown very attractive numerical results. The preconditioner costs O(N^2) flops to set up and O(N) storage. The preconditioning technique is sufficiently general that it can be applied to different types of different operators. This was applied to the 2D multiquadric method, with c~1/@/N on the Poisson test problem, the preconditioned GMRES converges in tens of iterations. In this paper, we combine the RBF methods and the ACBF preconditioning technique with the domain decomposition method (DDM). We studied different implementations of the ACBF-DDM scheme and provide numerical results for N > 10,000 nodes. We shall demonstrate that the efficiency of the ACBF-DDM scheme improves dramatically as successively finer partitions of the domain are considered.

A least-squares preconditioner for radial basis functions collocation methods

Abstract Although meshless radial basis function (RBF) methods applied to partial differential equations (PDEs) are not only simple to implement and enjoy exponential convergence rates as compared to standard mesh-based schemes, the system of equations required to find the expansion coefficients are typically badly conditioned and expensive using the global Gaussian elimination (G-GE) method requiring

Identification of source locations in two-dimensional heat equations

\mathcal{O}(N^{3})

Numerical simulations of 2D fractional subdiffusion problems

flops. We present a simple preconditioning scheme that is based upon constructing least-squares approximate cardinal basis functions (ACBFs) from linear combinations of the RBF-PDE matrix elements. The ACBFs transforms a badly conditioned linear system into one that is very well conditioned, allowing us to solve for the expansion coefficients iteratively so we can reconstruct the unknown solution everywhere on the domain. Our preconditioner requires

Stable and Convergent Unsymmetric Meshless Collocation Methods

\mathcal{O}(mN^{2})

On convergent numerical algorithms for unsymmetric collocation

flops to set up, and

Inverse source identification for Poisson equation

\mathcal{O}(mN)

Confidence intervals for a difference between proportions based on paired data

storage locations where m is a user define parameter of order of 10. For the 2D MQ-RBF with the shape parameter

Finding numerical derivatives for unstructured and noisy data by multiscale kernels

c\sim1/\sqrt{N}

An adaptive greedy technique for inverse boundary determination problem

, the number of iterations required for convergence is of order of 10 for large values of N, making this a very attractive approach computationally. As the shape parameter increases, our preconditioner will eventually be affected by the ill conditioning and round-off errors, and thus becomes less effective. We tested our preconditioners on increasingly larger c and N. A more stable construction scheme is available with a higher set up cost.