Kernel Multi-Grid on Manifolds

Abstract: Kernel methods for solving partial differential equations on surfaces have the advantage that those methods work intrinsically on the surface and yield high approximation rates if the solution to the partial differential equation is smooth enough. Localized Lagrange bases have proven to alleviate the computational complexity of usual kernel methods to some extent, although the efficient numerical solution of the ill-conditioned linear systems of equations arising from kernel-based Galerkin solutions to PDEs has not been addressed in the literature so far. In this article we apply the framework of the geometric multigrid method with a $\tau\ge 2$-cycle to scattered, quasi-uniform point clouds on the surface. We show that the resulting linear algebra can be accelerated by using the Lagrange function decay, with convergence rates which are obtained by a rigorous analysis. In particular, we can show that the computational cost to solve the linear system scales log-linear in the degrees of freedom.