非对称线性方程组的高斯置信传播求解器
Gaussian belief propagation solvers for nonsymmetric systems of linear equations
摘要：
在本文中，我们讨论了确定性推理在数值线性代数的经典问题，即求解线性系中的效用。我们将展示如何高斯信念传播求解器，已知的工作对称矩阵可以修改处理非对称矩阵。此外，我们引入了一个新的算法矩阵求逆，对应于来自集群变分方法（或菊池近似）的广义信念传播。我们将这些算法的LU和块LU分解，并提供一定的保证的基础上，从理论上的信念传播的定理。所有提出的算法与经典的求解器（例如，Gauss-Seidel，BiCGSTAB）及其在线性椭圆型方程中的应用。我们还展示了如何高斯信念传播可以被用来作为多重网格平滑，从而在一个更强大的求解器比一个基于高斯-赛德尔迭代法。
英文摘要：
In this paper, we argue for the utility of deterministic inference in the classical problem of numerical linear algebra, that of solving a linear system. We show how the Gaussian belief propagation solver, known to work for symmetric matrices can be modified to handle nonsymmetric matrices. Furthermore, we introduce a new algorithm for matrix inversion that corresponds to the generalized belief propagation derived from the cluster variation method (or Kikuchi approximation). We relate these algorithms to LU and block LU decompositions and provide certain guarantees based on theorems from the theory of belief propagation. All proposed algorithms are compared with classical solvers (e.g., Gauss-Seidel, BiCGSTAB) with application to linear elliptic equations. We also show how the Gaussian belief propagation can be used as multigrid smoother, resulting in a substantially more robust solver than the one based on the Gauss-Seidel iterative method.
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