Kernel multigrid on manifolds

IF 1.8 2区 数学 Q1 MATHEMATICS
Thomas Hangelbroek , Christian Rieger
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引用次数: 0

Abstract

Kernel methods for solving partial differential equations work coordinate-free on the surface and yield high approximation rates for smooth solutions. Localized Lagrange bases have proven to alleviate the computational complexity of usual kernel methods for data fitting problems, but the efficient numerical solution of the ill-conditioned linear systems of equations arising from kernel-based Galerkin solutions to PDEs is a challenging problem which has not been addressed in the literature so far. In this article we apply the framework of the geometric multigrid method with a τ2-cycle to scattered, quasi-uniform point clouds on the surface. We show that the resulting solver can be accelerated by using the Lagrange function decay and obtain satisfying convergence rates by a rigorous analysis. In particular, we show that the computational cost of the linear solver scales log-linear in the degrees of freedom.
流形上的核多网格
求解偏微分方程的核方法在曲面上是无坐标工作的,对平滑解具有很高的逼近率。事实证明,局部拉格朗日基减轻了通常核方法在数据拟合问题上的计算复杂性,但如何高效地数值求解基于核的 Galerkin 求解偏微分方程所产生的无条件线性方程组,是一个极具挑战性的问题,迄今为止尚未有文献解决这个问题。在本文中,我们将带有 τ≥2 周期的几何多网格方法框架应用于表面上的散乱准均匀点云。我们通过严格的分析表明,利用拉格朗日函数衰减可以加速由此产生的求解器,并获得令人满意的收敛率。我们特别指出,线性求解器的计算成本与自由度成对数线性关系。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Journal of Complexity
Journal of Complexity 工程技术-计算机:理论方法
CiteScore
3.10
自引率
17.60%
发文量
57
审稿时长
>12 weeks
期刊介绍: The multidisciplinary Journal of Complexity publishes original research papers that contain substantial mathematical results on complexity as broadly conceived. Outstanding review papers will also be published. In the area of computational complexity, the focus is on complexity over the reals, with the emphasis on lower bounds and optimal algorithms. The Journal of Complexity also publishes articles that provide major new algorithms or make important progress on upper bounds. Other models of computation, such as the Turing machine model, are also of interest. Computational complexity results in a wide variety of areas are solicited. Areas Include: • Approximation theory • Biomedical computing • Compressed computing and sensing • Computational finance • Computational number theory • Computational stochastics • Control theory • Cryptography • Design of experiments • Differential equations • Discrete problems • Distributed and parallel computation • High and infinite-dimensional problems • Information-based complexity • Inverse and ill-posed problems • Machine learning • Markov chain Monte Carlo • Monte Carlo and quasi-Monte Carlo • Multivariate integration and approximation • Noisy data • Nonlinear and algebraic equations • Numerical analysis • Operator equations • Optimization • Quantum computing • Scientific computation • Tractability of multivariate problems • Vision and image understanding.
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