Related papers: Sparse solution of the Lyapunov equation for large…
In this paper, we present, evaluate and analyse the performance of parallel synchronous Jacobi algorithms by different partitioned procedures including band-row splitting, band-row sparsity pattern splitting and substructuring splitting,…
Although some preconditioners are available for solving dense linear systems, there are still many matrices for which preconditioners are lacking, in particular in cases where the size of the matrix $N$ becomes very large. There remains…
We present a distributed asynchronous algorithm for approximating a single component of the solution to a system of linear equations $Ax = b$, where $A$ is a positive definite real matrix, and $b \in \mathbb{R}^n$. This is equivalent to…
We investigate the SPAI and PSAI preconditioning procedures and shed light on two important features of them: (i) For the large linear system $Ax=b$ with $A$ irregular sparse, i.e., with $A$ having $s$ relatively dense columns, SPAI may be…
In this article we investigate model order reduction of large-scale systems using time-limited balanced truncation, which restricts the well known balanced truncation framework to prescribed finite time intervals. The main emphasis is on…
We consider the problem of finding a sparse solution for an underdetermined linear system of equations when the known parameters on both sides of the system are subject to perturbation. This problem is particularly relevant to…
In this paper, applying the Newton method, we transform the complex continuous-time algebraic Riccati matrix equation into a Lyapunov equation. Then, we introduce an efficient general alternating-direction implicit (GADI) method to solve…
We present the submatrix method, a highly parallelizable method for the approximate calculation of inverse p-th roots of large sparse symmetric matrices which are required in different scientific applications. We follow the idea of…
The computation of the entire Lyapunov spectrum for extended dynamical systems is a very time consuming task. If the system is in a chaotic spatio-temporal regime it is possible to approximately reconstruct the Lyapunov spectrum from the…
Sparse recovery is widely applied in many fields, since many signals or vectors can be sparsely represented under some frames or dictionaries. Most of fast algorithms at present are based on solving $l^0$ or $l^1$ minimization problems and…
Sequences of parametrized Lyapunov equations can be encountered in many application settings. Moreover, solutions of such equations are often intermediate steps of an overall procedure whose main goal is the computation of…
A class of splitting alternating algorithms is proposed for finding the sparse solution of linear systems with concatenated orthogonal matrices. Depending on the number of matrices concatenated, the proposed algorithms are classified into…
One of the most computationally expensive steps of the low-rank ADI method for large-scale Lyapunov equations is the solution of a shifted linear system at each iteration. We propose the use of the extended Krylov subspace method for this…
Some important applicative problems require the evaluation of functions $\Psi$ of large and sparse and/or \emph{localized} matrices $A$. Popular and interesting techniques for computing $\Psi(A)$ and $\Psi(A)\mathbf{v}$, where $\mathbf{v}$…
The computation of a matrix function $f(A)$ is an important task in scientific computing appearing in machine learning, network analysis and the solution of partial differential equations. In this work, we use only matrix-vector products…
This paper is concerned with the problem of approximating the determinant of A for a large sparse symmetric positive definite matrix A. It is shown that an efficient solution of this problem is obtained by using a sparse approximate inverse…
The approximate solution of large-scale algebraic Riccati equations is considered. We are interested in approximate solutions which yield a Riccati residual matrix of a particular small rank. It is assumed that such approximate solutions…
Inversion of sparse matrices with standard direct solve schemes is robust, but computationally expensive. Iterative solvers, on the other hand, demonstrate better scalability; but, need to be used with an appropriate preconditioner (e.g.,…
We consider the numerical solution of large-scale symmetric differential matrix Riccati equations. Under certain hypotheses on the data, reduced order methods have recently arisen as a promising class of solution strategies, by forming…
We study the problem of approximating a matrix $\mathbf{A}$ with a matrix that has a fixed sparsity pattern (e.g., diagonal, banded, etc.), when $\mathbf{A}$ is accessed only by matrix-vector products. We describe a simple randomized…