This paper presents a multigrid algorithm for the computation of the rank-R canonical decomposition of a tensor for low rank R. Standard alternating least squares (ALS) is used as the relaxation method. Transfer operators and coarse-level tensors are constructed in an adaptive setup phase based on multiplicative correction and on Bootstrap algebraic multigrid. An accurate solution is then computed by an additive solve phase based on the Full Approximation Scheme. Numerical tests show that for certain test problems the multilevel method significantly outperforms standalone ALS when a high level of accuracy is required.
The aim of this paper is to develop an algebraic multigrid method to solve eigenvalue problems based on the combination of the multilevel correction scheme and the algebraic multigrid method for linear equations. Our approach uses the algebraic multigrid method setup procedure to construct the hierarchy and the intergrid transfer operators. In this algebraic multigrid scheme, a large scale eigenvalue problem is solved by some algebraic multigrid smoothing steps in the hierarchy and very small-dimensional eigenvalue problems. To emphasize the efficiency and flexibility of the proposed method, h…
Multigrid methods have proven to be an invaluable tool to efficiently solve large sparse linear systems arising in the discretization of partial differential equations (PDEs). Algebraic multigrid methods and in particular adaptive algebraic multigrid approaches have shown that multigrid efficiency can be obtained without having to resort to properties of the PDE. Yet the required setup of these methods poses a not negligible overhead cost. Methods from machine learning have attracted attention to streamline processes based on statistical models being trained on the available data. Interpreting…
A multigrid framework is described for multiphysics applications. The framework allows one to construct, adapt, and tailor a monolithic multigrid methodology to different linear systems coming from discretized partial differential equations. The main idea centers on developing multigrid components in a blocked fashion where each block corresponds to separate sets of physical unknowns and equations within the larger discretization matrix. Once defined, these components are ultimately assembled into a monolithic multigrid solver for the entire system. We demonstrate the potential of the framewor…
We construct an algebraic multigrid (AMG) based preconditioner for the reduced Hessian of a linear-quadratic optimization problem constrained by an elliptic partial differential equation. While the preconditioner generalizes a geometric multigrid preconditioner introduced in earlier works, its construction relies entirely on a standard AMG infrastructure built for solving the forward elliptic equation, thus allowing for it to be implemented using a variety of AMG methods and standard packages. Our analysis establishes a clear connection between the quality of the preconditioner and the AMG met…
Multigrid methods are well suited to large massively parallel computer architectures because they are mathematically optimal and display excellent parallelization properties. Since current architecture trends are favoring regular compute patterns to achieve high performance, the ability to express structure has become much more important. The hypre software library provides high-performance multigrid preconditioners and solvers through conceptual interfaces, including a semi-structured interface that describes matrices primarily in terms of stencils and logically structured grids. This paper p…
The paper focuses on developing and studying efficient block preconditioners based on classical algebraic multigrid for the large-scale sparse linear systems arising from the fully coupled and implicitly cell-centered finite volume discretization of multi-group radiation diffusion equations, whose coefficient matrices can be rearranged into the $(G+2)\times(G+2)$ block form, where $G$ is the number of energy groups. The preconditioning techniques are based on the monolithic classical algebraic multigrid method, physical-variable based coarsening two-level algorithm and two types of block Schur…
Nodal Coarsening and Sparse Ideal Interpolation for H(curl) Problems in Algebraic Multigrid
The Pennsylvania State University（宾夕法尼亚州立大学）；Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）；Bergische Universität Wuppertal（伍珀塔尔大学）；University of New Mexico（新墨西哥大学）
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 energy made up of edge and vertex contributions, we are able to develop coarsening criteria that guarantee two-level convergence even for systems of equations. This energy also allows us to construct prolongations with prescribed sparsity pattern that still preserve kernel vectors exactly. These allow for a straightforward optimization that simplifies paralleliz…
On the Ideal Interpolation Operator in Algebraic Multigrid Methods
LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
We present an efficient, robust and fully GPU-accelerated aggregation-based algebraic multigrid preconditioning technique for the solution of large sparse linear systems. These linear systems arise from the discretization of elliptic PDEs. The method involves two stages, setup and solve. In the setup stage, hierarchical coarse grids are constructed through aggregation of the fine grid nodes. These aggregations are obtained using a set of maximal independent nodes from the fine grid nodes. We use a ``fine-grain'' parallel algorithm for finding a maximal independent set from a graph of strong ne…
A self-learning algebraic multigrid method for dominant and minimal singular triplets and eigenpairs is described. The method consists of two multilevel phases. In the first, multiplicative phase (setup phase), tentative singular triplets are calculated along with a multigrid hierarchy of interpolation operators that approximately fit the tentative singular vectors in a collective and self-learning manner, using multiplicative update formulas. In the second, additive phase (solve phase), the tentative singular triplets are improved up to the desired accuracy by using an additive correction sch…
Problems arising in Earth's mantle convection involve finding the solution to Stokes systems with large viscosity contrasts. These systems contain localized features which, even with adaptive mesh refinement, result in linear systems that can be on the order of 10^9 or more unknowns. One common approach for preconditioning to the velocity block of these systems is to apply an Algebraic Multigrid (AMG) v-cycle (as is done in the ASPECT software, for example), however, with AMG, robustness can be difficult with respect to problem size and number of parallel processes. Additionally, we see an inc…
Efficient solution of 3D elasticity problems is an important part of many industrial and scientific applications. Smoothed aggregation algebraic multigrid using rigid body modes for the tentative prolongation operator construction is an efficient and robust choice for the solution of linear systems arising from the discretization of elasticity equations. The system matrices on every level of the multigrid hierarchy have block structure, so using block representation and block arithmetics should significantly improve the solver efficiency. However, the tentative prolongation operator constructi…
This paper establishes the first theoretical framework for analyzing the rounding-error effects on multigrid methods using mixed-precision iterative-refinement solvers. While motivated by the sparse symmetric positive definite (SPD) matrix equations that arise from discretizing linear elliptic PDEs, the framework is purely algebraic such that it applies to matrices that do not necessarily come from the continuum. Based on the so-called energy or $A$ norm, which is the natural norm for many problems involving SPD matrices, we provide a normwise forward error analysis, and introduce the notion o…
Algebraic multigrid (AMG) is one of the fastest numerical methods for solving large sparse linear systems. For SPD matrices, convergence of AMG is well motivated in the $A$-norm, and AMG has proven to be an effective solver for many applications. Recently, several AMG algorithms have been developed that are effective on nonsymmetric linear systems. Although motivation was provided in each case, the convergence of AMG for nonsymmetric linear systems is still not well understood, and algorithms are based largely on heuristics or incomplete theory. For multigrid restriction and interpolation oper…
University of Colorado at Boulder（科罗拉多大学博尔德分校）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）
Algebraic multigrid (AMG) is a widely used scalable solver and preconditioner for large-scale linear systems resulting from the discretization of a wide class of elliptic PDEs. While AMG has optimal computational complexity, the cost of communication has become a significant bottleneck that limits its scalability as processor counts continue to grow on modern machines. This paper examines the design, implementation, and parallel performance of a novel algorithm, Algebraic Multigrid Domain Decomposition (AMG-DD), designed specifically to limit communication. The goal of AMG-DD is to provide a l…
Automated Grammar-based Algebraic Multigrid Design With Evolutionary Algorithms
Friedrich-Alexander-Universität Erlangen-Nürnberg（埃朗根-纽伦堡弗里德里希-亚历山大大学）；Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）；CERFACS；Department of Applied Mathematics, VSB-Technical University of Ostrava（应用数学系，奥斯特拉瓦技术大学）
Multiphase flow is a critical process in a wide range of applications, including carbon sequestration, contaminant remediation, and groundwater management. Typically, this process is modeled by a nonlinear system of partial differential equations derived by considering the mass conservation of each phase (e.g., oil, water), along with constitutive laws for the relationship of phase velocity to phase pressure. In this study, we develop and study efficient solution algorithms for solving the algebraic systems of equations derived from a fully coupled and time-implicit treatment of models of mult…
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 sparse} systems is still an interesting open question [Brezina, Manteuffel, McCormick, Runge, and Sanders, SIAM J. Sci. Comput. (2010); Manteuffel and Southworth, SIAM J. Sci. Comput. (2019)]. This paper deals with the block-structured dense and Toeplitz-like-plus-Cross systems, including {\em nonsymmetric} indefinite, symmetric positive definite (SPD), arising …
This paper describes a massively parallel algebraic multigrid method based on non-smoothed aggregation. It is especially suited for solving heterogeneous elliptic problems as it uses a greedy heuristic algorithm for the aggregation that detects changes in the coefficients and prevents aggregation across them. Using decoupled aggregation on each process with data agglomeration onto fewer processes on the coarse level, it weakly scales well in terms of both total time to solution and time per iteration to nearly 300,000 cores. Because of simple piecewise constant interpolation between the levels…
Algebraic multigrid (AMG) is often an effective solver for symmetric positive definite (SPD) linear systems resulting from the discretization of general elliptic PDEs, or the spatial discretization of parabolic PDEs. However, convergence theory and most variations of AMG rely on $A$ being SPD. Hyperbolic PDEs, which arise often in large-scale scientific simulations, remain a challenge for AMG, as well as other fast linear solvers, in part because the resulting linear systems are often highly nonsymmetric. Here, a novel convergence framework is developed for nonsymmetric, reduction-based AMG, a…
Iterative solvers preconditioned with algebraic multigrid have been devised as an optimal technology to speed up the response of large sparse linear systems. In this work, this technique was implemented in the framework of the dual delineation approach. This involves a single groundwater flow solve and a pure advective transport solve with different right-hand sides. The new solver was compared with traditional preconditioned iterative methods and direct sparse solvers on several two- and three-dimensional benchmark problems spanning homogeneous and heterogeneous formations. For the groundwate…
This paper builds on the algebraic theory in the companion paper [Algebraic Error Analysis for Mixed-Precision Multigrid Solvers] to obtain discretization-error-accurate solutions for linear elliptic partial differential equations (PDEs) by mixed-precision multigrid solvers. It is often assumed that the achievable accuracy is limited by discretization or algebraic errors. On the contrary, we show that the quantization error incurred by simply storing the matrix in any fixed precision quickly begins to dominate the total error as the discretization is refined. We extend the existing theory to a…
