发表机构
Zhejiang University（浙江大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文首次为Bänsch型四面体细化算法（AMP和原始Bänsch）证明了无条件累积闭包估计，通过因果森林和加权打包论证，解决了该家族长期存在的复杂度问题。
AI 中文摘要
我们证明了，据我们所知，这是首个针对Arnold--Mukherjee--Pouly（AMP）细化算法和Bänsch原始面标记四面体算法在任意一致初始四面体网格上的无条件累积闭包估计。设$T_0,\ldots,T_L$为由任一算法生成的适应性网格序列，其中$M_\ell\subseteq T_\ell$表示第$\ell$步的标记集。则$$\\#T_L-\\#T_0\leq C_{\mathrm{clos}}(T_0)\sum_{\ell=0}^{L-1}\\#M_\ell.$$该证明是物理三维网格固有的，既不需要初始相容性条件，也不需要高维嵌入。它将一致性细化组织成一个因果森林，并结合均匀水平传播估计与加权打包论证，得到显式闭包常数。对于原始Bänsch算法，初始两条边歧义的每个历史依赖解都由有限多个AMP历史之一表示。因此，该估计对任意确定性或非确定性选择都一致成立。这解决了Bänsch--AMP家族长期存在的复杂度问题。
英文摘要
We prove, to our knowledge, the first unconditional cumulative closure estimates for the Arnold--Mukherjee--Pouly (AMP) refinement algorithm and the original face-marked tetrahedral algorithm of Bänsch on arbitrary conforming initial tetrahedral meshes. Let $T_0,\ldots,T_L$ be an adaptive mesh sequence generated by either algorithm, with $M_\ell\subseteq T_\ell$ denoting the marking set at step $\ell$. Then $$\#T_L-\#T_0\leq C_{\mathrm{clos}}(T_0)\sum_{\ell=0}^{L-1}\#M_\ell.$$ The proof is intrinsic to the physical three-dimensional mesh and requires neither an initial compatibility condition nor a higher-dimensional embedding. It organizes conformity refinements into a causal forest and combines a uniform horizontal-propagation estimate with a weighted packing argument to obtain an explicit closure constant. For the original Bänsch algorithm, every history-dependent resolution of the initial two-edge ambiguity is represented by one of finitely many AMP histories. The estimate therefore holds uniformly for arbitrary deterministic or nondeterministic choices. This resolves a long-standing complexity question for the Bänsch--AMP family.
Comments38 pages
