Searcharxiv⌕ Search

arXiv subjects

Thomas A. Manteuffel

Publications and source records attributed to Thomas A. Manteuffel.

7 recordsLinked to original sources
arXiv2024

On Compatible Transfer Operators in Nonsymmetric Algebraic Multigrid

The standard goal for an effective algebraic multigrid (AMG) algorithm is to develop relaxation and coarse-grid correction schemes that attenuate complementary error modes. In the nonsymmetric setting, coarse-grid correction $Π$ will almost certainly be nonorthogonal (and divergent) in any known standard product, meaning $\|Π\| > 1$. This introduces a new consideration, that one wants coarse-grid correction to be as close to orthogonal as possible, in an appropriate norm. In addition, due to non-orthogonality, $Π$ may actually amplify certain error modes that are in the range of interpolation. Relaxation must then not only be complementary to interpolation, but also rapidly eliminate any error amplified by the non-orthogonal correction, or the algorithm may diverge. This paper develops analytic formulae on how to construct ``compatible'' transfer operators in nonsymmetric AMG such that $\|Π\| = 1$ in some standard matrix-induced norm. Discussion is provided on different options for the norm in the nonsymmetric setting, the relation between ``ideal'' transfer operators in different norms, and insight into the convergence of nonsymmetric reduction-based AMG.

math.NA↗
arXiv2020

Mixed $(\mathcal{L}\mathcal{L}^*)^{-1}$ and $\mathcal{L}\mathcal{L}^*$ least-squares finite element methods with application to linear hyperbolic problems

In this paper, a few dual least-squares finite element methods and their application to scalar linear hyperbolic problems are studied. The purpose is to obtain $L^2$-norm approximations on finite element spaces of the exact solutions to hyperbolic partial differential equations of interest. This is approached by approximating the generally infeasible quadratic minimization, that defines the $L^2$-orthogonal projection of the exact solution, by feasible least-squares principles using the ideas of the original $\mathcal{L}\mathcal{L}^*$ method proposed in the context of elliptic equations. All methods in this paper are founded upon and extend the $\mathcal{L}\mathcal{L}^*$ approach which is rather general and applicable beyond the setting of elliptic problems. Error bounds are shown that point to the factors affecting the convergence and provide conditions that guarantee optimal rates. Furthermore, the preconditioning of the resulting linear systems is discussed. Numerical results are provided to illustrate the behavior of the methods on common finite element spaces.

math.NA↗
arXiv2020

A Least-Squares Finite Element Method Based on the Helmholtz Decomposition for Hyperbolic Balance Laws

In this paper, a least-squares finite element method for scalar nonlinear hyperbolic balance laws is proposed and studied. The approach is based on a formulation that utilizes an appropriate Helmholtz decomposition of the flux vector and is related to the standard notion of a weak solution. This relationship, together with a corresponding connection to negative-norm least-squares, is described in detail. As a consequence, an important numerical conservation theorem is obtained, similar to the famous Lax-Wendroff theorem. The numerical conservation properties of the method in this paper do not fall precisely in the framework introduced by Lax and Wendroff, but they are similar in spirit as they guarantee that when $L^2$ convergence holds, the resulting approximations approach a weak solution to the hyperbolic problem. The least-squares functional is continuous and coercive in an $H^{-1}$-type norm, but not $L^2$-coercive. Nevertheless, the $L^2$ convergence properties of the method are discussed. Convergence can be obtained either by an explicit regularization of the functional, that provides control of the $L^2$ norm, or by properly choosing the finite element spaces, providing implicit control of the $L^2$ norm. Numerical results for the inviscid Burgers equation with discontinuous source terms are shown, demonstrating the $L^2$ convergence of the obtained approximations to the physically admissible solution. The numerical method utilizes a least-squares functional, minimized on finite element spaces, and a Gauss-Newton technique with nested iteration. We believe that the linear systems encountered with this formulation are amenable to multigrid techniques and combining the method with adaptive mesh refinement would make this approach an efficient tool for solving balance laws (this is the focus of a future study).

math.NA↗
arXiv2019

Nonsymmetric Reduction-based Algebraic Multigrid

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, and sufficient conditions derived for $\ell^2$-convergence of error and residual. In particular, classical multigrid approximation properties are connected with reduction-based measures to develop a robust framework for nonsymmetric, reduction-based AMG. Matrices with block-triangular structure are then recognized as being amenable to reduction-type algorithms, and a reduction-based AMG method is developed for upwind discretizations of hyperbolic PDEs, based on the concept of a Neumann approximation to ideal restriction ($n$AIR). $n$AIR can be seen as a variation of local AIR ($\ell$AIR) introduced in previous work, specifically targeting matrices with triangular structure. Although less versatile than $\ell$AIR, setup times for $n$AIR can be substantially faster for problems with high connectivity. $n$AIR is shown to be an effective and scalable solver of steady state transport for discontinuous, upwind discretizations, with unstructured meshes, and up to 6th-order finite elements, offering a significant improvement over existing AMG methods. $n$AIR is also shown to be effective on several classes of `nearly triangular' matrices, resulting from curvilinear finite elements and artificial diffusion.

math.NA↗
arXiv2019

Convergence in Norm of Nonsymmetric Algebraic Multigrid

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 operators, $R$ and $P$, respectively, let $Π:= P(RAP)^{-1}RA$ denote the projection corresponding to coarse-grid correction in AMG. It is invariably the case in the nonsymmetric setting that $\|Π\| > 1$ in any known norm. This causes an interesting dichotomy: coarse-grid correction is fundamental to AMG achieving fast convergence, but, in this case, can actually increase the error. Here, we present a detailed analysis of nonsymmetric AMG, discussing why SPD theory breaks down in the nonsymmetric setting, and developing a general framework for convergence of NS-AMG. Classical multigrid weak and strong approximation properties are generalized to a \textit{fractional approximation property}. Conditions are then developed on $R$ and $P$ to ensure that $\|Π\|_{\sqrt{A^*A}}$ is nicely bounded, independent of problem size. This is followed by the development of conditions for two-grid and multilevel W-cycle convergence in the $\sqrt{A^*A}$-norm.

math.NA↗
arXiv2019

Nonsymmetric Algebraic Multigrid Based on Local Approximate Ideal Restriction (lAIR)

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 symmetric positive definite operator. This paper presents a new variation on classical AMG for nonsymmetric matrices (denoted lAIR), based on a local approximation to the ideal restriction operator, coupled with F-relaxation. A new block decomposition of the AMG error-propagation operator is used for a spectral analysis of convergence, and the efficacy of the algorithm is demonstrated on systems arising from the discrete form of the advection-diffusion-reaction equation. lAIR is shown to be a robust solver for various discretizations of the advection-diffusion-reaction equation, including time-dependent and steady-state, from purely advective to purely diffusive. Convergence is robust for discretizations on unstructured meshes and using higher-order finite elements, and is particularly effective on upwind discontinuous Galerkin discretizations. Although the implementation used here is not parallel, each part of the algorithm is highly parallelizable, avoiding common multigrid adjustments for strong advection such as line-relaxation and K- or W-cycles that can be effective in serial, but suffer from high communication costs in parallel, limiting their scalability.

math.NA↗
arXiv2018

A Root-Node Based Algebraic Multigrid Method

This paper provides a unified and detailed presentation of root-node style algebraic multigrid (AMG). Algebraic multigrid is a popular and effective iterative method for solving large, sparse linear systems that arise from discretizing partial differential equations. However, while AMG is designed for symmetric positive definite matrices (SPD), certain SPD problems, such as anisotropic diffusion, are still not adequately addressed by existing methods. Non-SPD problems pose an even greater challenge, and in practice AMG is often not considered as a solver for such problems. The focus of this paper is on so-called root-node AMG, which can be viewed as a combination of classical and aggregation-based multigrid. An algorithm for root-node is outlined and a filtering strategy is developed, which is able to control the cost of using root-node AMG, particularly on difficult problems. New theoretical motivation is provided for root-node and energy-minimization as applied to symmetric as well non-symmetric systems. Numerical results are then presented demonstrating the robust ability of root-node to solve non-symmetric problems, systems-based problems, and difficult SPD problems, including strongly anisotropic diffusion, convection-diffusion, and upwind steady-state transport, in a scalable manner. New, detailed estimates of the computational cost of the setup and solve phase are given for each example, providing additional support for root-node AMG over alternative methods.

math.NA↗