In this paper we design support-tree preconditioners for n × n matrices with m nonzeros that aresymmetric and diagonally-dominant with a nonnegative diagonal (SDD matrix). This reduces to de-signing such a preconditioner for a Laplacian matrix, A, which can be interpreted as an undirectednonnegatively-weighted graph, G with n vertices and m edges. Preconditioners accelerate the ...

OF DISSERTATION RELATIVE PERTURBATION THEORY FOR DIAGONALLY DOMINANT MATRICES Diagonally dominant matrices arise in many applications. In this work, we exploit the structure of diagonally dominant matrices to provide sharp entrywise relative perturbation bounds. We first generalize the results of Dopico and Koev to provide relative perturbation bounds for the LDU factorization with a well condi...

We give the first O(mpolylog(n)) time algorithms for approximating maximum flows in undirected graphs and constructing polylog(n)-quality cut-approximating hierarchical tree decompositions. Our algorithm invokes existing algorithms for these two problems recursively while gradually incorporating size reductions. These size reductions are in turn obtained via ultra-sparsifiers, which are key too...

An M ∨-matrix has the form A = sI − B, where s ≥ ρ(B) ≥ 0 and B is eventually nonnegative; i.e., B k is entrywise nonnegative for all sufficiently large integers k. A theory of M ∨-matrices is developed here that parallels the theory of M-matrices, in particular as it regards exponential nonnegativity, spectral properties, semipositivity, monotonicity, inverse nonnegativity and diagonal dominance.

We present an algorithm that on input of an n-vertex m-edge weighted graph G and a value k, produces an incremental sparsifier Ĝ with n− 1+m/k edges, such that the condition number of G with Ĝ is bounded above by Õ(k log n), with probability 1− p. The algorithm runs in time Õ((m logn+ n log n) log(1/p)). As a result, we obtain an algorithm that on input of an n × n symmetric diagonally dominant...

In this paper, we propose a general smoothing framework for graph kernels by taking structural similarity into account, and apply it to derive smoothed variants of popular graph kernels. Our framework is inspired by state-of-the-art smoothing techniques used in natural language processing (NLP). However, unlike NLP applications that primarily deal with strings, we show how one can apply smoothi...

In this paper, a sequential design method is proposed for analyses and designs of multivariable feedback control systems. The overall compensator is implemented systematiccally with a cascaded layer-wrapped structure and diagonal compensators. The method can take into consideration of stability, integrity, diagonal dominance and performance in a sequential manner. Roots of stability-equations a...

An M∨ matrix has the form A = sI − B, where s ≥ ρ(B) ≥ 0 and B is eventually nonnegative; i.e., Bk is entrywise nonnegative for all sufficiently large integers k. A theory of M∨ matrices is developed here that parallels the theory of M-matrices, in particular as it regards exponential nonnegativity, spectral properties, semipositivity, monotonicity, inverse nonnegativity and diagonal dominance.

We extend graph embedding techniques for bounding the spectral condition number of preconditioned systems involving symmetric, irreducibly diagonally dominant M-matrices to systems where the preconditioner is not diagonally dominant. In particular, this allows us to bound the spectral condition number when the preconditioner is based on an incomplete factorization. We provide a review of previo...