Adaptive Algebraic Multigrid Research Articles

Coarsening in algebraic multigrid using Gaussian processes

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 algebraically smooth error as an instance of a Gaussian process, we develop a new, data driven approach to construct adaptive algebraic multigrid methods. Based on Gaussian a priori distributions, kriging interpolation minimizes the mean squared error of the a posteriori distribution, given the data on the coarse grid. Going one step further, we exploit the quantification of uncertainty in the Gaussian process model in order to construct efficient variable splittings. Using a semivariogram fit of a suitable covariance model we demonstrate that our approach yields efficient methods using a single algebraically smooth vector.

Space‐time finite element adaptive AMG for multi‐term time fractional advection diffusion equations

In this study, we construct a space‐time finite element (FE) scheme and furnish cost‐efficient approximations for one‐dimensional multi‐term time fractional advection diffusion equations on a bounded domain Ω. Firstly, a fully discrete scheme is obtained by the linear FE method in both temporal and spatial directions, and many characterizations on the resulting matrix are established. Secondly, the associated condition number estimation is derived, where τn and h, respectively, are the current time and space step sizes. Next, we propose a lossless in robustness and adaptive algebraic multigrid (AMG) with computational cost and matrix‐free storage, in contrast to classical AMG with , where M is the number of spatial segments. Meanwhile, uniform convergence analyses on the two‐level V(0,1)‐cycle of classical and adaptive AMGs are provided. Finally, we demonstrate the optimal error order of space‐time FE approximations in the L2(Ω) norm sense, present theoretical confirmations and predictable behaviors of the proposed algorithm in numerical experiments.

Improving solve time of aggregation‐based adaptive AMG

SummaryThis paper proposes improving the solve time of a bootstrap algebraic multigrid (AMG) designed previously by the authors. This is achieved by incorporating the information, a set of algebraically smooth vectors, generated by the bootstrap algorithm, in a single hierarchy by using sufficiently large aggregates, and these aggregates are compositions of aggregates already built throughout the bootstrap algorithm. The modified AMG method has good convergence properties and shows significant reduction in both memory and solve time. These savings with respect to the original bootstrap AMG are illustrated on some difficult (for standard AMG) linear systems arising from discretization of scalar and vector function elliptic partial differential equations in both 2D and 3D.

An Adaptive Multigrid Method Based on Path Cover

We propose a path cover adaptive algebraic multigrid (PC-$\alpha$AMG) method for solving linear systems with weighted graph Laplacians that can also be applied to discretized second order elliptic ...

BootCMatch

This article has two main objectives: one is to describe some extensions of an adaptive Algebraic Multigrid (AMG) method of the form previously proposed by the first and third authors, and a second one is to present a new software framework, named BootCMatch , which implements all the components needed to build and apply the described adaptive AMG both as a stand-alone solver and as a preconditioner in a Krylov method. The adaptive AMG presented is meant to handle general symmetric and positive definite (SPD) sparse linear systems, without assuming any a priori information of the problem and its origin; the goal of adaptivity is to achieve a method with a prescribed convergence rate. The presented method exploits a general coarsening process based on aggregation of unknowns, obtained by a maximum weight matching in the adjacency graph of the system matrix. More specifically, a maximum product matching is employed to define an effective smoother subspace (complementary to the coarse space), a process referred to as compatible relaxation, at every level of the recursive two-level hierarchical AMG process. Results on a large variety of test cases and comparisons with related work demonstrate the reliability and efficiency of the method and of the software.

Multiple Galerkin Adaptive Algebraic Multigrid Algorithm for the Helmholtz Equations

Introduced in this paper is a multiple Galerkin adaptive algebraic multigrid algorithm for solving linear discrete equations arising from discretization of partial differential operators with a rich and diverse near-kernel. Standard multigrid methods struggle with solving such equations due to their inability to produce accurate coarse correction to all near-kernel components. The motivating model problem here is the indefinite Helmholtz operators for which the character of near-kernel components is, at least approximately, known. In the algorithm, some of these components are used to create multiple prolongation operators and consecutive coarse descriptions which collectively approximate the entire near-kernel. The algorithm consists of two parts, with multiple Galerkin corrections accelerating a standard multigrid V-cycle. Both parts largely employ standard multigrid techniques. The resulting algorithm is then used to precondition the GMRES. Numerical experiments in one and two dimensions are presented, demonstrating the applicability of the method for problems with different types of wave numbers. Further leads in improving and extending the proposed approach conclude the paper.

Adaptive Aggregation-Based Domain Decomposition Multigrid for the Lattice Wilson--Dirac Operator

In lattice quantum chromodynamics (QCD) computations a substantial amount of work is spent in solving discretized versions of the Dirac equation. Conventional Krylov solvers show critical slowing down for large system sizes and physically interesting parameter regions. We present a domain decomposition adaptive algebraic multigrid method used as a preconditioner to solve the “clover improved” Wilson discretization of the Dirac equation. This approach combines and improves two approaches, namely domain decomposition and adaptive algebraic multigrid, that have been used separately in lattice QCD before. We show in extensive numerical tests conducted with a parallel production code implementation that considerable speedup can be achieved compared to conventional Krylov subspace methods, domain decomposition methods, and other hierarchical approaches for realistic system sizes.

Adaptive AMG with coarsening based on compatible weighted matching

We introduce a new composite adaptive Algebraic Multigrid (composite \(\alpha \)AMG) method to solve systems of linear equations without a-priori knowledge or assumption on characteristics of near-null components of the AMG preconditioned problem referred to as algebraic smoothness. Our version of \(\alpha \)AMG is a composite solver built through a bootstrap strategy aimed to obtain a desired convergence rate. The coarsening process employed to build each new solver component relies on a pairwise aggregation scheme based on weighted matching in a graph, successfully exploited for reordering algorithms in sparse direct methods to enhance diagonal dominance, and compatible relaxation. The proposed compatible matching process replaces the commonly used characterization of strength of connection in both the coarse space selection and in the interpolation scheme. The goal is to design a method leading to scalable AMG for a wide class of problems that go beyond the standard elliptic Partial Differential Equations (PDEs). In the present work, we introduce the method and demonstrate its potential when applied to symmetric positive definite linear systems arising from finite element discretization of highly anisotropic elliptic PDEs on structured and unstructured meshes. We also report on some preliminary tests for 2D and 3D elasticity problems as well as on problems from the University of Florida Sparse Matrix Collection.

Two-Level Adaptive Algebraic Multigrid for a Sequence of Problems with Slowly Varying Random Coefficients

This paper proposes an adaptive strategy for reusing a previously constructed coarse space by algebraic multigrid to construct a two-level solver for a problem with nearby characteristics. A main target application is the solution of the linear problems that appear throughout a sequence of Markov chain Monte Carlo simulations of subsurface flow with uncertain permeability field. The efficacy of the method is demonstrated with extensive set of numerical experiments.

Robust and adaptive multigrid methods: comparing structured and algebraic approaches

SUMMARYAlthough there have been significant advances in robust algebraic multigrid (AMG) methods in recent years, numerical studies and emerging hardware architectures continue to favor structured‐grid approaches. Specifically, implementations of logically structured robust variational multigrid algorithms, such as the black box multigrid (BoxMG) solver, have been shown to be 10 times faster than AMG for three‐dimensional heterogeneous diffusion problems on structured grids. BoxMG offers important features such as operator‐induced interpolation for robustness, while taking advantage of direct data access and bounded complexity in the Galerkin coarse‐grid operator. Moreover, because BoxMG uses a variational framework, it can be used to explore advances of modern adaptive AMG approaches in a structured setting. In this paper, we show how to extend the adaptive multigrid methodology to the BoxMG setting. This extension not only retains the favorable properties of the adaptive framework but also sheds light on the relationship between BoxMG and AMG. In particular, we show how classical BoxMG can be viewed as a special case of classical AMG and how this viewpoint leads to a richer family of adaptive BoxMG approaches. We present numerical results that explore this family of adaptive methods and compare its robustness and efficiency to the classical BoxMG solver.Copyright © 2012 John Wiley & Sons, Ltd.

An Efficient Algebraic Multigrid Solver Strategy for Adaptive Implicit Methods in Oil-Reservoir Simulation

Summary We propose a new, efficient, adaptive algebraic multigrid (AMG) solver strategy for the discrete systems of partial-differential equations (PDEs) arising from structured or unstructured grid models in reservoir simulation. The proposed strategy has been particularly tailored to linear systems of equations arising in adaptive implicit methods (AIMs). The coarsening process of the AMG method designed automatically employs information on the physical structure of the models; as a smoother, an adaptive incomplete LU factorization with thresholding (ILUT) method is employed, taking care of an efficient solution of the hyperbolic parts while providing adequately smooth errors for the elliptic parts. To achieve a good compromise of high efficiency and robustness for a variety of problem classes—ranging from simple, small black-oil to challenging, large compositional models—an automatic, adaptive ILUT parameter and AMG solver switching strategy, α-SAMG. has been developed. Its efficiency is demonstrated for eight industrial benchmark cases by comparison against standard one-level and AMG solvers, including constraint pressure residual (CPR), as well as the pure one-level variant of the proposed new strategy. In addition, very promising results of first parallel runs are shown.

Least-Squares Finite Element Methods for Quantum Electrodynamics

A significant amount of the computational time in large Monte Carlo simulations of lattice field theory is spent inverting the discrete Dirac operator. Unfortunately, traditional covariant finite difference discretizations of the Dirac operator present serious challenges for standard iterative methods. For interesting physical parameters, the discretized operator is large and ill-conditioned and has random coefficients. More recently, adaptive algebraic multigrid (AMG) methods have been shown to be effective preconditioners for Wilson's discretization [J. Brannick et al., Lect. Notes Comput. Sci. Eng., 55 (2006), pp. 499–506], [J. Brannick et al., Phys. Rev. Lett., 100 (2008), pp. 041601–041604] of the Dirac equation. This paper presents an alternate discretization of the two-dimensional Dirac operator of quantum electrodynamics (QED) based on least-squares finite elements. The discretization is systematically developed and, physical properties of the resulting matrix system are discussed. Finally, numerical experiments are presented that demonstrate the effectiveness of adaptive smoothed aggregation ($\alpha$SA) multigrid as a preconditioner for the discrete field equations.

General Constrained Energy Minimization Interpolation Mappings for AMG

This report proposes a new class of interpolation procedures for use in algebraic multigrid (AMG). The procedure is general in that it applies to s.p.d. finite element discretization problems that include scalar and vector elliptic PDEs, and it can be adapted to the time-domain Maxwell (definite $H(\mathrm{curl})$) problems. It can also be viewed as an extension of previously proposed vector-fitting interpolation procedures and can be used in adaptive AMG cycles. We illustrate the performance of the new interpolation matrices on a number of test problems.

Multilevel Adaptive Aggregation for Markov Chains, with Application to Web Ranking

A multilevel adaptive aggregation method for calculating the stationary probability vector of an irreducible stochastic matrix is described. The method is a special case of the adaptive smoothed aggregation and adaptive algebraic multigrid methods for sparse linear systems and is also closely related to certain extensively studied iterative aggregation/disaggregation methods for Markov chains. In contrast to most existing approaches, our aggregation process does not employ any explicit advance knowledge of the topology of the Markov chain. Instead, adaptive agglomeration is proposed that is based on the strength of connection in a scaled problem matrix, in which the columns of the original problem matrix at each recursive fine level are scaled with the current probability vector iterate at that level. The strength of connection is determined as in the algebraic multigrid method, and the aggregation process is fully adaptive, with optimized aggregates chosen in each step of the iteration and at all recursive levels. The multilevel method is applied to a set of stochastic matrices that provide models for web page ranking. Numerical tests serve to illustrate for which types of stochastic matrices the multilevel adaptive method may provide significant speedup compared to standard iterative methods. The tests also provide more insight into why Google's PageRank model is a successful model for determining a ranking of web pages.

An adaptive algebraic multigrid for multigroup neutron diffusion reactor core calculations

An adaptive algebraic multigrid solver has been constructed for the multigroup neutron diffusion equation in reactor core calculations. Coefficients for the weak connection approximation are calculated adaptively using the current approximation of the main eigenfunction. It is especially efficient if the multigrid solver is used for the reactor criticity calculation, where main eigenfunction presents naturally. The robustness is achieved by means of pivot selection and row reordering under prolongation operator construction.