用热核Pagerank求解带边界条件的局部线性方程组
Solving Local Linear Systems with Boundary Conditions Using Heat Kernel Pagerank
摘要：
利用与系统相关的连通诱导子图的格林函数，给出了求解带边界条件的局部线性方程组的一个有效算法。我们介绍的方法，使用Dirichlet热核pagerank向量近似局部解的线性系统的图拉普拉斯满足给定的边界条件，在一个特定的顶点子集。我们的局部线性求解器算法通过执行$O(ε^{-5}s^3\log(s^3ε^{-1})\log n)$随机行走步骤来计算具有乘法和加法误差$ε$的近似局部解，其中$n$是完整图中的顶点数，$s$是诱导子图上的局部系统的大小。
英文摘要：
We present an efficient algorithm for solving local linear systems with a boundary condition using the Green's function of a connected induced subgraph related to the system. We introduce the method of using the Dirichlet heat kernel pagerank vector to approximate local solutions to linear systems in the graph Laplacian satisfying given boundary conditions over a particular subset of vertices. With an efficient algorithm for approximating Dirichlet heat kernel pagerank, our local linear solver algorithm computes an approximate local solution with multiplicative and additive error $ε$ by performing $O(ε^{-5}s^3\log(s^3ε^{-1})\log n)$ random walk steps, where $n$ is the number of vertices in the full graph and $s$ is the size of the local system on the induced subgraph.
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