论文标题
拉普拉斯样矩阵的结构和近似特性
Structure and approximation properties of Laplacian-like matrices
论文作者
论文摘要
当今许多问题都需要涉及任意大型系统$ a \ mathbf {x} = \ mathbf {b} $的解决方案的技术。一种流行的数值方法是基于特定张量分解的所谓贪婪排名量更新算法。数值实验支持以下事实：当线性系统的基质类似拉普拉斯时，该算法会特别快。这些遵循拉普拉斯运算符张量结构的矩阵是由遵循特定模式的矩阵的kronecker产物的总和形成的。此外，这组矩阵不仅是线性子空间，而且是矩阵谎言代数的一个谎言子代数。在本文中，我们表征并赋予此特定类矩阵的主要特性。此外，以上结果允许我们提出一种算法，可以将正交投影明确计算到给定的正方形矩阵$ a \ in \ mathbb {r}^r}^{n \ times n}的此子空间上。$
Many of today's problems require techniques that involve the solution of arbitrarily large systems $A\mathbf{x}=\mathbf{b}$. A popular numerical approach is the so-called Greedy Rank-One Update Algorithm, based on a particular tensor decomposition. The numerical experiments support the fact that this algorithm converges especially fast when the matrix of the linear system is Laplacian-Like. These matrices that follow the tensor structure of the Laplacian operator are formed by sums of Kronecker product of matrices following a particular pattern. Moreover, this set of matrices is not only a linear subspace it is a a Lie sub-algebra of a matrix Lie Algebra. In this paper, we characterize and give the main properties of this particular class of matrices. Moreover, the above results allow us to propose an algorithm to explicitly compute the orthogonal projection onto this subspace of a given square matrix $A \in \mathbb{R}^{N\times N}.$
