相关论文: Auxiliary Space Multigrid Method Based on Additive…

In this paper, we consider a classical form of optimal algebraic multigrid (AMG) interpolation that directly minimizes the two-grid convergence rate and compare it with the so-called ideal form that minimizes a certain weak approximation…

Multilevel techniques are efficient approaches for solving the large linear systems that arise from discretized partial differential equations and other problems. While geometric multigrid requires detailed knowledge about the underlying…

The numerical simulation of structural mechanics applications via finite elements usually requires the solution of large-size and ill-conditioned linear systems, especially when accurate results are sought for derived variables interpolated…

This paper introduces a novel approach to algebraic multigrid methods for large systems of linear equations coming from finite element discretizations of certain elliptic second order partial differential equations. Based on a discrete…

Efficient numerical solvers for sparse linear systems are crucial in science and engineering. One of the fastest methods for solving large-scale sparse linear systems is algebraic multigrid (AMG). The main challenge in the construction of…

We present augmented Lagrangian Schur complement preconditioners and robust multigrid methods for incompressible Stokes problems with extreme viscosity variations. Such Stokes systems arise, for instance, upon linearization of nonlinear…

Problems from graph drawing, spectral clustering, network flow and graph partitioning can all be expressed in terms of graph Laplacian matrices. There are a variety of practical approaches to solving these problems in serial. However, as…

We consider the solution of elliptic problems on the tensor product of two physical domains as e.g. present in the approximation of the solution covariance of elliptic partial differential equations with random input. Previous sparse…

Algebraic multigrid (AMG) solvers and preconditioners are some of the fastest numerical methods to solve linear systems, particularly in a parallel environment, scaling to hundreds of thousands of cores. Most AMG methods and theory assume a…

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…

We consider effective preconditioners for solving Laplacians of general weighted graphs. Theoretically, spectral sparsifiers (SSs) provide preconditioners of optimal computational complexity. However, they are not easy to use for real-world…

Large linear systems with sparse, non-symmetric matrices arise in the modeling of Markov chains or in the discretization of convection-diffusion problems. Due to their potential to solve sparse linear systems with an effort that is linear…

Graph learning models have been widely deployed in collaborative filtering (CF) based recommendation systems. Due to the issue of data sparsity, the graph structure of the original input lacks potential positive preference edges, which…

In some cases, computational benefit can be gained by exploring the hyper parameter space using a deterministic set of grid points instead of a Markov chain. We view this as a numerical integration problem and make three unique…

Scalable spatial GPs for massive datasets can be built via sparse Directed Acyclic Graphs (DAGs) where a small number of directed edges is sufficient to flexibly characterize spatial dependence. The DAG can be used to devise fast algorithms…

Algebraic multigrid (AMG) is conventionally applied in a black-box fashion, agnostic to the underlying geometry. In this work, we propose that using geometric information -- when available -- to assist with setting up the AMG hierarchy is…

We consider the problem of learning a sparse undirected graph underlying a given set of multivariate data. We focus on graph Laplacian-related constraints on the sparse precision matrix that encodes conditional dependence between the random…

We show how a scalable preconditioner for the primal discontinuous Petrov-Galerkin (DPG) method can be developed using existing algebraic multigrid (AMG) preconditioning techniques. The stability of the DPG method gives a norm equivalence…

This work proposes a scheme for significantly reducing the computational complexity of discretized problems involving the non-smooth forward propagation of uncertainty by combining the adaptive hierarchical sparse grid stochastic…

We introduce a novel Unsmoothed Aggregation (UA) Algebraic Multigrid (AMG) method combined with Preconditioned Conjugate Gradient (PCG) to overcome the limitations of Extended Position-Based Dynamics (XPBD) in high-resolution and…