Algebraic muligrid (AMG) is a go-to solver for symmetric positive definite linear systems resulting from the discretization of elliptic PDEs, or the spatial discretization of parabolic PDEs. For diffusion-like problems, the time to solution typically scales linearly with the number of unknowns. However, convergence theory and most variations of AMG rely on $A$ being symmetric positive definite. 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 new reduction-based AMG method is developed for upwind discretizations of hyperbolic PDEs and other matrices with triangular structure, based on the concept of ideal restriction (AMGir). Theory is presented motivating the reduction-based approach and connecting it in a big-picture sense to other nonsymmetric AMG methods. The model problems considered are variations of a steady-state transport equation, with source-terms, discontinuities, and non-constant flow. AMGir is shown to be an effective and scalable solver for various discontinuous, upwind discretizations, with unstructured meshes, and up to 6th-order finite elements, expanding the applicability of AMG to a new class of problems. AMGir is implemented in PyAMG and available at this https URL

This paper develops new adaptive mesh refinement strategies for first-order system least squares (FOSLS) in conjunction with algebraic multigrid (AMG) methods in the context of nested iteration (NI). The goal is to reach a certain error tolerance with the least amount of computational cost and nearly uniform distribution of the error over all elements. To accomplish this, the refinement decisions at each refinement level are determined on the basis of minimizing the accuracy-per-computational-cost efficiency (ACE) measure that takes into account both error reduction and computational cost. The NI-FOSLS-AMG-ACE approach produces a sequence of refinement levels in which the error is equally distributed across elements on a relatively coarse grid. Once the solution is numerically resolved, refinement becomes nearly uniform. Accommodations of the ACE approach to massively distributed memory architectures involve a geometric binning strategy to reduce communication cost. Load balancing begins at very coarse levels. Elements and nodes are redistributed using parallel quadtree structures and a space-filling curve, which automatically ameliorates load balancing issues at finer levels. Numerical results show that the NI-FOSLS-AMG-pACE approach is able to provide highly accurate approximations to rapidly varying solutions at relatively low cost. Excellent weak and strong scalability are demonstrated on 4096 processors for problems with 15 million biquadratic elements.Copyright (C) 2012 John Wiley & Sons, Ltd.

SUMMARYBootstrap algebraic multigrid (BAMG) is a multigrid‐based solver for matrix equations of the form Ax = b. Its aim is to automatically determine the interpolation weights used in algebraic multigrid by locally fitting a set of test vectors that have been relaxed as solutions to the corresponding homogeneous equation, Ax = 0. This paper studies an improved form of BAMG, called relaxation‐corrected bootstrap algebraic multigrid (rBAMG), that involves adding scaled residuals of the test vectors to the least‐squares equations.The basic rBAMG scheme was introduced in an earlier paper [1] and analyzed on a simple model problem. The purpose of the current paper is to further develop this algorithm by incorporating several new critical components and to systematically study its performance on an interesting model problem from quantum chromodynamics. Whereas the earlier paper introduced a new least‐squares principle involving the residuals of the test vectors, a simple extrapolation scheme is developed here to accurately estimate the convergence factors of the evolving algebraic multigrid solver. Such a capability is essential to the effective development of a fast solver, and the approach introduced here is shown numerically to be much more effective than the conventional approach of just observing successive error reduction factors. Another component of the setup process developed here is an adaptive cycling process. This component assesses the effectiveness of the V‐cycle constructed in the initial rBAMG phase by applying it to the homogeneous equation. When poor convergence is observed, the set of test vectors is enhanced with the resulting error, enabling the subsequent least‐squares fit of interpolation to produce an improved V‐cycle. A related component is the scaling and recombination Ritz process that targets the so‐called weak approximation property in an attempt to reveal the important elements of these evolving error and test vector spaces.The aim of the numerical study documented here is to provide insight into the various design choices that arise in the development of an rBAMG algorithm. With this in mind, the results for quantum chromodynamics focus on the behavior of rBAMG in terms of the number of initial test vectors used, the number of relaxation sweeps applied to them, and the size of the target matrices. Copyright © 2012 John Wiley & Sons, Ltd.

Magnetohydrodynamics (MHD) is a model of plasma physics that treats the plasma as a charged fluid. As a result, the set of partial differential equations that describe this model are a time-dependent, nonlinear system of equations. Thus, the equations can be difficult to solve and efficient numerical algorithms are needed. This work shows the use of such an efficient algorithm on the incompressible, resistive MHD equations. A first-order systems least-squares (FOSLS) [1, 2] finite element discretization is used along with nested iteration and algebraic multigrid (AMG) [3, 4, 5, 6, 7, 8]. The main focus of this work is to show that if a nested iteration algorithm along with an efficiency-based adaptive mesh refinement (AMR) scheme is used, then a nonlinear system of equations, such as the MHD equations, can be solved in only a handful of work units per time step. Here, a work unit is defined as the equivalent of one relaxation sweep on the finest grid. In other words, the accuracy-per-computational-cost for solving the MHD equations can be maximized by the use of nested iteration and AMR. An island coalescence instability was able to be resolved in less than 10 work units per time step. Further details of this work can be found in several companion papers. In [9], the FOSLS method applied to MHD is described. The nested iteration algorithm has also been develoeped in [10] and [11] and the efficiency-based AMR method known as ACE, is discussed in [12, 13, 14, 15].

Applying smoothed aggregation (SA) multigrid to solve a nonsymmetric linear system, $A\mathbf{x} =\mathbf{b}$, is often impeded by the lack of a minimization principle that can be used as a basis for the coarse-grid correction process. This paper proposes a Petrov-Galerkin (PG) approach based on applying SA to either of two symmetric positive definite (SPD) matrices, $\sqrt{A^{t}A}$ or $\sqrt{AA^{t}}$. These matrices, however, are typically full and difficult to compute, so it is not computationally efficient to use them directly to form a coarse-grid correction. The proposed approach approximates these coarse-grid corrections by using SA to accurately approximate the right and left singular vectors of $A$ that correspond to the lowest singular value. These left and right singular vectors are used to construct the restriction and interpolation operators, respectively. A preliminary two-level convergence theory is presented, suggesting that more relaxation should be applied than for an SPD problem. Additionally, a nonsymmetric version of adaptive SA ($\alpha$SA) is given that automatically constructs SA multigrid hierarchies using a stationary relaxation process on all levels. Numerical results are reported for convection-diffusion problems in two dimensions with varying amounts of convection for constant, variable, and recirculating convection fields. The results suggest that the proposed approach is algorithmically scalable for problems coming from these nonsymmetric scalar PDEs (with the exception of recirculating flow). This paper serves as a first step for nonsymmetric $\alpha$SA. The long-term goal of this effort is to develop nonsymmetric $\alpha$SA for systems of PDEs, where the SA framework has proven to be well suited for adaptivity in SPD problems.

An algebraic multigrid (AMG) method is presented for the calculation of the stationary probability vector of an irreducible Markov chain. The method is based on standard AMG for nonsingular linear systems, but in a multiplicative, adaptive setting. A modified AMG interpolation formula is proposed that produces a nonnegative interpolation operator with unit row sums. We show how the adoption of a previously described lumping technique maintains the irreducible singular M-matrix character of the coarse-level operators on all levels. Together, these properties are sufficient to guarantee the well-posedness of the algorithm. Numerical results show how it leads to nearly optimal multigrid efficiency for a representative set of test problems.

Bootstrap algebraic multigrid (BAMG) is a multigrid‐based solver for matrix equations of the form Ax = b . Its aim is to automatically determine the interpolation weights used in algebraic multigrid (AMG) by locally fitting a set of test vectors that have been relaxed as solutions to the corresponding homogeneous equation, Ax =0, and are then possibly improved later using a multilevel eigensolver. This paper introduces a flexible variant of BAMG that determines the interpolation weights indirectly by ‘collapsing’ the unwanted connections in ‘operator interpolation’. Compared to BAMG, this indirect BAMG approach ( i BAMG) is more in the spirit of classical AMG, which collapses unwanted connections in operator interpolation based on the (restrictive) assumption that smooth error is locally constant. This paper studies the numerical performance of i BAMG and establishes an equivalence, under certain assumptions, between it and a slightly modified (standard) BAMG scheme. To focus on camparing BAMG and i BAMG and exposing their behavior as the problem size grows, the numerical experiments concentrate on Poisson‐like scalar problems. Copyright © 2010 John Wiley & Sons, Ltd.

The Dirac equation of quantum electrodynamics describes the interaction between electrons and photons. Large‐scale numerical simulations of the theory require repeated solution of the two‐dimensional Dirac equation, a system of two first‐order partial differential equations coupled to a background U(1) gauge field. Traditional discretizations of this system are sparse and highly structured, but contain random complex entries introduced by the background field. For even mildly disordered gauge fields, the near kernel components of the system are highly oscillatory, rendering standard multilevel methods ineffective. We consider an alternate formulation of the governing equations obtained by a transformation of the continuum operator that decouples the system into separate scalar diffusion‐like equations. We discretize the transformed system using least‐squares finite elements and use adaptive smoothed aggregation multigrid to solve the resulting linear system. We present numerical results and discuss implications of the transformed formulation in terms of the physical theory. Copyright © 2010 John Wiley & Sons, Ltd.

A smoothed aggregation multigrid method is presented for the numerical calculation of the stationary probability vector of an irreducible sparse Markov chain. It is shown how smoothing the interpolation and restriction operators can dramatically increase the efficiency of aggregation multigrid methods for Markov chains that have been proposed in the literature. The proposed smoothing approach is inspired by smoothed aggregation multigrid for linear systems, supplemented with a new lumping technique that assures well-posedness of the coarse-level problems: the coarse-level operators are singular M-matrices on all levels, resulting in strictly positive coarse-level corrections on all levels. Numerical results show how these methods lead to nearly optimal multigrid efficiency for an extensive set of test problems, both when geometric and algebraic aggregation strategies are used.

Two efficiency‐based grid refinement strategies are investigated for adaptive finite element solution of partial differential equations. In each refinement step, the elements are ordered in terms of decreasing local error, and the optimal fraction of elements to be refined is determined based on efficiency measures that take both error reduction and work into account. The goal is to reach a pre‐specified bound on the global error with minimal amount of work. Two efficiency measures are discussed, ‘work times error’ and ‘accuracy per computational cost’. The resulting refinement strategies are first compared for a one‐dimensional (1D) model problem that may have a singularity. Modified versions of the efficiency strategies are proposed for the singular case, and the resulting adaptive methods are compared with a threshold‐based refinement strategy. Next, the efficiency strategies are applied to the case of hp ‐refinement for the 1D model problem. The use of the efficiency‐based refinement strategies is then explored for problems with spatial dimension greater than one. The ‘work times error’ strategy is inefficient when the spatial dimension, d , is larger than the finite element order, p , but the ‘accuracy per computational cost’ strategy provides an efficient refinement mechanism for any combination of d and p . Copyright © 2008 John Wiley & Sons, Ltd.

Consider the linear system A x = b , where A is a large, sparse, real, symmetric, and positive‐definite matrix and b is a known vector. Solving this system for unknown vector x using a smoothed aggregation (SA) multigrid algorithm requires a characterization of the algebraically smooth error, meaning error that is poorly attenuated by the algorithm's relaxation process. For many common relaxation processes, algebraically smooth error corresponds to the near‐nullspace of A . Therefore, having a good approximation to a minimal eigenvector is useful to characterize the algebraically smooth error when forming a linear SA solver. We discuss the details of a generalized eigensolver based on smoothed aggregation (GES‐SA) that is designed to produce an approximation to a minimal eigenvector of A . GES‐SA may be applied as a stand‐alone eigensolver for applications that desire an approximate minimal eigenvector, but the primary purpose here is to apply an eigensolver to the specific application of forming robust, adaptive linear solvers. This paper reports the first stage in our study of incorporating eigensolvers into the existing adaptive SA framework. Copyright © 2008 John Wiley & Sons, Ltd.

A spatial multigrid algorithm for isotropic neutron transport is presented in x-y geometry. The linear system is obtained using discrete ordinates in angle and corner balance finite differencing in space. Spatial smoothing is accomplished by a four-color block-Jacobi relaxation, where the diagonal blocks correspond to 4-cell blocks on the spatial grid. A bilinear interpolation operator and its transpose are used for the grid transfer operators. Good convergence factors were observed for homogeneous material properties. Heterogeneous material properties prove more difficult, especially the case of a vacuum region surrounded by a thick, diffusive region. In this case, a small amount of absorption, or "effective absorption" in a time-dependent problem, restores good convergence. Numerical results are presented.

Algebraic multigrid (AMG) is an iterative method that is often optimal for solving the matrix equations that arise in a wide variety of applications, including discretized partial differential equations. It automatically constructs a sequence of increasingly smaller matrix problems that hopefully enables efficient resolution of all scales present in the solution. The methodology is based on measuring how a so‐called algebraically smooth error value at one point depends on its value at another. Such a concept of strength of connection is well understood for operators whose principal part is an M‐matrix; however, the strength concept for more general matrices is not yet clearly understood, and this lack of knowledge limits the scope of AMG applicability. The purpose of this paper is to motivate a general definition of strength of connection, discuss its implementation, and present the results of initial numerical experiments. Copyright © 2006 John Wiley & Sons, Ltd.

Multigrid methods are ideal for solving the increasingly large-scale problems that arise in numerical simulations of physical phenomena because of their potential for computational costs and memory requirements that scale linearly with the degrees of freedom. Unfortunately, they have been historically limited by their applicability to elliptic-type problems and the need for special handling in their implementation. In this paper, we present an overview of several recent theoretical and algorithmic advances made by the TOPS multigrid partners and their collaborators in extending applicability of multigrid methods. specific examples that are presented include quantum chromodynamics, radiation transport, and electromagnetics.

Efficient numerical simulation of physical processes is constrained by our ability to solve the resulting linear systems, prompting substantial research into the development of multiscale iterative methods capable of solving these linear systems with an optimal amount of effort. Overcoming the limitations of geometric multigrid methods to simple geometries and differential equations, algebraic multigrid methods construct the multigrid hierarchy based only on the given matrix. While this allows for efficient black-box solution of the linear systems associated with discretizations of many elliptic differential equations, it also results in a lack of robustness due to unsatisfied assumptions made on the near null spaces of these matrices. This paper introduces an extension to algebraic multigrid methods that removes the need to make such assumptions by utilizing an adaptive process. Emphasis is on the principles that guide the adaptivity and their application to algebraic multigrid solution of certain symmetric positive-definite linear systems.

Substantial effort has been focused over the last two decades on developing multilevel iterative methods capable of solving the large linear systems encountered in engineering practice. These systems often arise from discretizing partial differential equations over unstructured meshes, and the particular parameters or geometry of the physical problem being discretized may be unavailable to the solver. Algebraic multigrid (AMG) and multilevel domain decomposition methods of algebraic type have been of particular interest in this context because of their promises of optimal performance without the need for explicit knowledge of the problem geometry. These methods construct a hierarchy of coarse problems based on the linear system itself and on certain assumptions about the smooth components of the error. For smoothed aggregation (SA) multigrid methods applied to discretizations of elliptic problems, these assumptions typically consist of knowledge of the near-kernel or near-nullspace of the weak form. This paper introduces an extension of the SA method in which good convergence properties are achieved in situations where explicit knowledge of the near-kernel components is unavailable. This extension is accomplished in an adaptive process that uses the method itself to determine near-kernel components and adjusts the coarsening processes accordingly.

This paper develops new first-order system LL * (FOSLL * ) formulations for scalar elliptic partial differential equations. It extends the work of [Z. Cal et al., SIAM J. Numer. Anal. , 39 (2001), pp. 1418--1445], where the FOSLL * methodology was first introduced. One focus of that paper was to develop \FL\ formulations that allow the use of H 1 -conforming finite element spaces and optimal multigrid solution techniques to construct L 2 approximations of the dependent variables in the presence of discontinuous coefficients. The problems for which this goal was achieved were limited to those with no reaction term and with Dirichlet and Neumann boundaries that were individually connected; that is, each had at most one component. Here, new FOSLL * formulations are developed to achieve the same goals on a wider class of problems, including problems with reaction terms, Dirichlet and Neumann boundaries with multiple components, reentrant corners, and points at which Dirichlet and Neumann boundaries meet with an inner angle greater than $\pi/2$. The efficiency of the improved FOSLL * formulations is illustrated by a series of numerical examples.

We introduce spectral element-based algebraic multigrid ($\rho$AMGe), a new algebraic multigrid method for solving systems of algebraic equations that arise in Ritz-type finite element discretizations of partial differential equations. The method requires access to the element stiffness matrices, which enables accurate approximation of algebraically "smooth" vectors (i.e., error components that relaxation cannot effectively eliminate). Most other algebraic multigrid methods are based in some manner on predefined concepts of smoothness. Coarse-grid selection and prolongation, for example, are often defined assuming that smooth errors vary slowly in the direction of "strong" connections (relatively large coefficients in the operator matrix). One aim of $\rho$AMGe is to broaden the range of problems to which the method can be successfully applied by avoiding any implicit premise about the nature of the smooth error. $\rho$AMGe uses the spectral decomposition of small collections of element stiffness matrices to determine local representations of algebraically smooth error components. This provides a foundation for generating the coarse level and for defining effective interpolation. This paper presents a theoretical foundation for $\rho$AMGe along with numerical experiments demonstrating its robustness.

We introduce spectral AMGe (ρAMGe), a new algebraic multigrid method for solving systems of algebraic equations that arise in Ritz-type finite element discretizations of partial differential equations. The method requires access to the element stiffness matrices, which enables accurate approximation of algebraically “smooth” vectors (i.e., error components that relaxation cannot effectively eliminate). Most other algebraic multigrid methods are based in some manner on predefined concepts of smoothness. Coarse-grid selection and prolongation, for example, are often defined assuming that smooth errors vary slowly in the direction of “strong” connections (relatively large coefficients in the operator matrix). One aim of ρAMGe is to broaden the range of problems to which the method can be successfully applied by avoiding any implicit premise about the nature of the smooth error. ρAMGe uses the spectral decomposition of small collections of element stiffness matrices to determine local representations of algebraically smooth error components. This provides a foundation for generating the coarse level and for defining effective interpolation. This paper presents a theoretical foundation for ρAMGe along with numerical experiments demonstrating its robustness.