Related papers: A Proposal of Multigrid Methods for Hermitian Posi…
This study develops identification methods for linear continuous-time symmetric systems, such as electrical network systems, multi-agent network systems, and temperature dynamics in buildings. To this end, we formulate three system…
This paper is to give an overview of AMG methods for solving large scale systems of equations such as those from the discretization of partial differential equations. AMG is often understood as the acronym of "Algebraic Multi-Grid", but it…
The present work develops hybrid multigrid methods for high-order discontinuous Galerkin discretizations of elliptic problems. Fast matrix-free operator evaluation on tensor product elements is used to devise a computationally efficient PDE…
Inspired by multigrid methods for linear systems of equations, multilevel optimization methods have been proposed to solve structured optimization problems. Multilevel methods make more assumptions regarding the structure of the…
This work develops a nonlinear multigrid method for diffusion problems discretized by cell-centered finite volume methods on general unstructured grids. The multigrid hierarchy is constructed algebraically using aggregation of degrees of…
Nonlinear matrix equations arise in many practical contexts related to control theory, dynamical programming and finite element methods for solving some partial differential equations. In most of these applications, it is needed to compute…
The construction of multigrid operators for disordered linear lattice operators, in particular the fermion matrix in lattice gauge theories, by means of algebraic multigrid and block LU decomposition is discussed. In this formalism, the…
We provide an alternative Fourier analysis for multigrid applied to the Poisson problem in 1D, based on explicit derivation of spectra of the iteration matrix. The new Fourier analysis has advantages over the existing one. It is easy to…
We present novel improvements in the context of symbol-based multigrid procedures for solving large block structured linear systems. We study the application of an aggregation-based grid transfer operator that transforms the symbol of a…
Computing at the exascale level is expected to be affected by a significantly higher rate of faults, due to increased component counts as well as power considerations. Therefore, current day numerical algorithms need to be reexamined as to…
We develop a simple algorithmic framework to solve large-scale symmetric positive definite linear systems. At its core, the framework relies on two components: (1) a norm-convergent iterative method (i.e. smoother) and (2) a preconditioner.…
Algebraic multigrid (AMG) is one of the most efficient iterative methods for solving large sparse system of equations. However, how to build/check restriction and prolongation operators in practical of AMG methods for nonsymmetric {\em…
Standard gradient-based iteration algorithms for optimization, such as gradient descent and its various proximal-based extensions to nonsmooth problems, are known to converge slowly for ill-conditioned problems, sometimes requiring many…
The goal of this primer is to provide a relatively short exposition of the basics of multigrid methods, simplified by focusing on fundamental concepts in a variational setting. This is done by way of a quadratic energy minimization…
We propose some multigrid methods for solving the algebraic systems resulting from finite element approximations of space fractional partial differential equations (SFPDEs). It is shown that our multigrid methods are optimal, which means…
In this paper, several two-grid finite element algorithms for solving parabolic integro-differential equations (PIDEs) with nonlinear memory are presented. Analysis of these algorithms is given assuming a fully implicit time discretization.…
An interior point method for the structural topology optimization is proposed. The linear systems arising in the method are solved by the conjugate gradient method preconditioned by geometric multigrid. The resulting method is then compared…
We present a first step towards a multigrid method for solving the min-cost flow problem. Specifically, we present a strategy that takes advantage of existing black-box fast iterative linear solvers, i.e. algebraic multigrid methods. We…
In the literature, there exist several studies on symbol-based multigrid methods for the solution of linear systems having structured coefficient matrices. In particular, the convergence analysis for such methods has been obtained in an…
We devise a method that reduces the problem of classifying systems of forms and linear mappings to the problem of classifying systems of linear mappings. Canonical matrices of (i) bilinear or sesquilinear forms, (ii) pairs of symmetric,…