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.

翻译：用于求解曲面上偏微分方程的核方法具有以下优势：这些方法在曲面上内在地工作，并且当偏微分方程的解足够光滑时，能提供较高的逼近精度。局部化拉格朗日基函数已被证明能在一定程度上缓解常规核方法的计算复杂度，然而，迄今为止，文献中尚未解决由基于核的偏微分方程伽辽金解法所产生的病态线性方程组的高效数值求解问题。在本文中，我们将几何多重网格方法的框架与$\tau\ge 2$循环相结合，应用于曲面上的散乱、拟均匀点云。我们证明，利用拉格朗日函数的衰减特性可以加速所得的线性代数运算，并通过严格的分析获得了收敛速率。特别地，我们能够证明，求解该线性系统的计算成本在自由度数量上呈对数线性增长。