发表机构
Institute for Applied Mathematics, University of Bonn; Heart Center Bonn, Department of Internal Medicine II, University Hospital Bonn; Department of Mathematics, University of Trento（波恩大学应用数学研究所; 波恩大学医院第二内科心脏中心; 特伦托大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明分段线性有限元离散化在二阶总广义变分泛函中的收敛性，并推广至时空设置，收敛速率为h^{1/4}，数值实验验证了理论，并在心电图成像逆问题中改进现有技术。
AI 中文摘要
高阶正则化方法在图像处理等逆问题中已显示出优于一阶方法（如全变分）的效果。本文针对二阶总广义变分泛函，证明了分段线性有限元离散化的收敛性。此外，在全面引入时空函数空间后，该收敛性结果被推广到时空设置中。在这两种情况下，离散极小元以 $h^{1/4}$ 的速率收敛到连续极小元，这与全变分有限元离散化在相同一般设置下的最佳已知速率相匹配。数值实验证实了收敛速率，并表明实际收敛速度更快。与相关空间图像重建算法的比较显示，重建结果相当。时空情况以心电图成像中的逆问题为例进行讨论，所提出的泛函改进了现有技术水平。
英文摘要
Higher-order regularization has shown improved results over first-order methods such as total variation. In this paper, convergence of a piecewise linear finite element discretization is shown for the second-order total generalized variation functional. Additionally, this convergence result is extended to a spatiotemporal setting after the spatiotemporal function space is comprehensively introduced. In both cases, discrete minimizers converge to the continuous minimizer with rate $h^{1/4}$, matching the best known rate for finite element discretizations of total variation in the same general setting. Numerical experiments confirm the convergence rates and indicate faster convergence in practice. Comparisons with related spatial image reconstruction algorithms show comparable reconstruction results. The spatiotemporal case is discussed using the example of the inverse problem in electrocardiographic imaging, where the proposed functional improves the state of the art.
