论文标题
关于谐波概括的barycentric坐标的局部性及其在泊松方程的解决方案
On Locality of Harmonic Generalized Barycentric Coordinates and Their Application to Solution of the Poisson Equation
论文作者
论文摘要
我们首先根据Polygon $ω$的边界的顶点扩展了广义Barycentric坐标（GBC），以基于利息$ω$内部的顶点的新型GBCS构建。为了清楚起见，标准GBC称为边界GBC，而新的GBC则称为内部GBC。然后，我们对这两种谐波GBC进行了分析，以表明每个GBC函数的值为$ 1 $在顶点（边界或内部顶点$ω$）的$ 1 $）衰减至零，远离其支撑顶点的速度快速快速，除了一个琐事示例。基于指数衰减特性，我们说明了如何在本地近似谐波GBC函数。也就是说，由于这两种GBC的位置，一个人可以近似 这些GBC通过其本地版本的功能，该版本得到了$ω$的子域的支持。这些GBC函数的本地版本将有助于减少图形设计中形状变形的计算时间。接下来，借助手头这两种GBC功能，我们可以使用它们来近似泊松方程的Dirichlet问题的解决方案。这可能提供了一种更有效的方法来通过使用具有图形处理单元（GPU）的计算机，该计算机比使用具有一个或几个CPU内核的计算机的计算机具有数千个或更多过程的计算机。
We first extend the construction of generalized barycentric coordinates (GBC) based on the vertices on the boundary of a polygon $Ω$ to a new kind of GBCs based on vertices inside the $Ω$ of interest. For clarity, the standard GBCs are called boundary GBCs while the new GBCs are called interior GBCs. Then we present an analysis on these two kinds of harmonic GBCs to show that each GBC function whose value is $1$ at a vertex (boundary or interior vertex of $Ω$) decays to zero away from its supporting vertex exponentially fast except for a trivial example. Based on the exponential decay property, we explain how to approximate the harmonic GBC functions locally. That is, due to the locality of these two kinds of GBCs, one can approximate each of these GBC functions by its local versions which is supported over a sub-domain of $Ω$. The local version of these GBC function will help reduce the computational time for shape deformation in graphical design. Next, with these two kinds of GBC functions at hand, we can use them to approximate the solution of the Dirichlet problem of the Poisson equation. This may provide a more efficient way to solve the Poisson equation by using a computer which has graphical processing unit(GPU) with thousands or more processes than the standard methods using a computer with one or few CPU kernels.
