On Locality of Harmonic Generalized Barycentric Coordinates and Their Application to Solution of the Poisson Equation

TL;DR

This paper introduces interior harmonic generalized barycentric coordinates (GBCs), analyzes their exponential decay, and demonstrates their local approximation for efficient Poisson equation solutions, enhancing computational performance in graphical applications.

Contribution

It extends boundary GBCs to interior GBCs, analyzes their decay properties, and applies them for efficient local approximation in solving the Poisson equation.

Findings

Interior GBCs decay exponentially away from vertices

Local approximation reduces computational time

Application to GPU-based Poisson solver

Abstract

We first extend the construction of generalized barycentric coordinates (GBC) based on the vertices on the boundary of a polygon $\Omega$ to a new kind of GBCs based on vertices inside the $\Omega$ 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 $\Omega$) 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 $\Omega$. 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.