Matrix algebras in optimal preconditioning

NILPOTENT GRAPHS OF MATRIX ALGEBRAS

Let $R$ be a ring with unity. The undirected nilpotent graph of $R$, denoted by $Gamma_N(R)$, is a graph with vertex set ~$Z_N(R)^* = {0neq x in R | xy in N(R) for some y in R^*}$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xy in N(R)$, or equivalently, $yx in N(R)$, where $N(R)$ denoted the nilpotent elements of $R$. Recently, it has been proved that if $R$ is a left A...

Capacitance Matrix Preconditioning

Preconditioning with Matrix Factorization

Pietsch Factorization and Grothendieck Factorization are the two landmark theorems in modern functional analysis. They were first introduced to the numerical linear algebra community by the work of Joel A. Tropp in the column subset selection problem, which seeks to extract from a matrix a column submatrix that has lower spectral norm. Despite their broad application in functional analysis, the...

Preconditioning of Optimal Transport

A preconditioning procedure is developed for the L2 and more general optimal transport problems. The procedure is based on a family of affine map pairs which transforms the original measures into two new measures that are closer to each other while preserving the optimality of solutions. It is proved that the preconditioning procedure minimizes the remaining transportation cost among all admiss...

Matrix - Free Preconditioning Using Partial Matrix Estimation ∗

We consider matrix-free solver environments where information about the underlying matrix is available only through matrix vector computations which do not have access to a fully assembled matrix. We introduce the notion of partial matrix estimation for constructing good algebraic preconditioners used in Krylov iterative methods in such matrix-free environments, and formulate three new graph co...