论文标题
Hodge分解和嵌入式简单复合物的一般拉普拉斯求解器
Hodge Decomposition and General Laplacian Solvers for Embedded Simplicial Complexes
论文作者
论文摘要
We describe a nearly-linear time algorithm to solve the linear system $L_1x = b$ parameterized by the first Betti number of the complex, where $L_1$ is the 1-Laplacian of a simplicial complex $K$ that is a subcomplex of a collapsible complex $X$ linearly embedded in $\mathbb{R}^{3}$.我们的算法概括了Black等人的工作。我们的算法适用于$ k $的$ k $，具有任意第一同源性的运行时间，相对于第一个betti数字，与复杂的大小和多项式相对于多项式的运行时间几乎是线性的。我们求解器的关键是一种用于计算几乎线性时间内1链$ K $的Hodge分解的新算法。此外，我们的算法意味着几乎二次求解器和几乎二次的Hodge分解，用于任何简单的复合物$ k $嵌入在$ \ Mathbb {r}^{3} $中的$ k $，因为$ k $始终可以扩展到杂质的复杂性的含量的复杂性。
We describe a nearly-linear time algorithm to solve the linear system $L_1x = b$ parameterized by the first Betti number of the complex, where $L_1$ is the 1-Laplacian of a simplicial complex $K$ that is a subcomplex of a collapsible complex $X$ linearly embedded in $\mathbb{R}^{3}$. Our algorithm generalizes the work of Black et al.~[SODA2022] that solved the same problem but required that $K$ have trivial first homology. Our algorithm works for complexes $K$ with arbitrary first homology with running time that is nearly-linear with respect to the size of the complex and polynomial with respect to the first Betti number. The key to our solver is a new algorithm for computing the Hodge decomposition of 1-chains of $K$ in nearly-linear time. Additionally, our algorithm implies a nearly quadratic solver and nearly quadratic Hodge decomposition for the 1-Laplacian of any simplicial complex $K$ embedded in $\mathbb{R}^{3}$, as $K$ can always be expanded to a collapsible embedded complex of quadratic complexity.
