Discrete least-squares radial basis functions approximations

Abstract

We consider discrete least-squares methods using radial basis functions. A general ℓ2-Tikhonov regularization with W2m-penalty is considered. We provide error estimates that are comparable to kernel-based interpolation in cases which the function it is approximating is within and is outside of the native space of the kernel. Our proven theories concern the denseness condition of collocation points and selection of regularization parameters. In particular, the unregularized least-squares method is shown to have W2μ(Ω) convergence for μ > d/2 on smooth domain Ω⊂Rd. For any properly regularized least-squares method, the same convergence estimates hold for a large range of μ ≥ 0. These results are extended to the case of noisy data. Numerical demonstrations are provided to verify the theoretical results. In terms of applications, we also apply the proposed method to solve a heat equation whose initial condition has huge oscillation in the domain.