We study multiscale scattered data interpolation schemes for globally supported radial basis functions with focus on the Mat\'ern class. The multiscale approximation is constructed through a sequence of residual corrections, where radial basis functions with different lengthscale parameters are combined to capture varying levels of detail. We prove that the condition numbers of the the diagonal blocks of the corresponding multiscale system remain bounded independently of the particular level, allowing us to use an iterative solver with a bounded number of iterations for the numerical solution. Employing an appropriate diagonal scaling, the multiscale system becomes well conditioned. We exploit this fact to derive a general error estimate bounding the consistency error issuing from a numerical approximation of the multiscale system. To apply the multiscale approach to large data sets, we suggest to represent each level of the multiscale system in samplet coordinates. Samplets are localized, discrete signed measures exhibiting vanishing moments and allow for the sparse approximation of generalized Vandermonde matrices issuing from a vast class of radial basis functions. Given a quasi-uniform set of $N$ data sites, and local approximation spaces with exponentially decreasing dimension, the samplet compressed multiscale system can be assembled with cost $\mathcal{O}(N \log^2 N)$. The overall cost of the proposed approach is $\mathcal{O}(N \log^2 N)$. The theoretical findings are accompanied by extensive numerical studies in two and three spatial dimensions.

翻译：本文研究了具有全局支撑的径向基函数的多尺度散乱数据插值方案，重点关注Matérn类函数。多尺度近似通过一系列残差修正构建，其中结合了具有不同长度尺度参数的径向基函数以捕捉不同层次的细节。我们证明了相应多尺度系统对角块的条件数独立于特定层级保持有界，这允许我们在数值求解时使用具有有限迭代次数的迭代求解器。通过采用适当的对角缩放，多尺度系统变得良态。我们利用这一事实推导出一般误差估计，该估计界定了由多尺度系统数值近似产生的一致性误差。为将多尺度方法应用于大型数据集，我们建议在多尺度系统的每一层使用样本坐标表示。样本是局部化的离散带符号测度，具有消失矩特性，并允许对源自广泛径向基函数类的广义Vandermonde矩阵进行稀疏近似。给定一个具有$N$个数据点的拟均匀集合，以及维度指数递减的局部近似空间，样本压缩多尺度系统的组装成本为$\mathcal{O}(N \log^2 N)$。所提方法的总体成本为$\mathcal{O}(N \log^2 N)$。理论结果辅以二维和三维空间中的大量数值研究予以验证。