The computation of correspondences between shapes is a principal task in shape analysis. To this end, methods based on partial differential equations (PDEs) have been established, encompassing e.g. the classic heat kernel signature as well as numerical solution schemes for geometric PDEs. In this work we focus on the latter approach. We consider here several time stepping schemes. The goal of this investigation is to assess, if one may identify a useful property of methods for time integration for the shape analysis context. Thereby we investigate the dependence on time step size, since the class of implicit schemes that are useful candidates in this context should ideally yield an invariant behaviour with respect to this parameter. To this end we study integration of heat and wave equation on a manifold. In order to facilitate this study, we propose an efficient, unified model order reduction framework for these models. We show that specific $l_0$ stable schemes are favourable for numerical shape analysis. We give an experimental evaluation of the methods at hand of classical TOSCA data sets.

翻译：形状之间的对应关系计算是形状分析中的核心任务。为此，基于偏微分方程的各类方法已被建立，包括经典热核签名以及几何偏微分方程的数值求解方案。本文聚焦于后者，考虑多种时间步进格式。本研究旨在评估能否为形状分析中的时间积分方法识别出有用特性。我们考察了时间步长依赖性，因为此类情境中具有潜在应用价值的隐式格式应理想地展现出对该参数的不变性。为此，我们研究了流形上热方程与波动方程的积分过程。为便于研究，我们提出了一个针对这些模型的高效统一降阶框架。研究表明，特定的$l_0$稳定格式更适用于数值形状分析。最后，我们在经典TOSCA数据集上对各类方法进行了实验评估。