arXiv subjects

Ben S. Southworth

Publications and source records attributed to Ben S. Southworth.

61 records · Page 4Linked to original sources

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.

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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.

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arXiv2019

Multilevel convergence analysis of multigrid-reduction-in-time

This paper presents a multilevel convergence framework for multigrid-reduction-in-time (MGRIT) as a generalization of previous two-grid estimates. The framework provides a priori upper bounds on the convergence of MGRIT V- and F-cycles, with different relaxation schemes, by deriving the respective residual and error propagation operators. The residual and error operators are functions of the time stepping operator, analyzed directly and bounded in norm, both numerically and analytically. We present various upper bounds of different computational cost and varying sharpness. These upper bounds are complemented by proposing analytic formulae for the approximate convergence factor of V-cycle algorithms that take the number of fine grid time points, the temporal coarsening factors, and the eigenvalues of the time stepping operator as parameters. The paper concludes with supporting numerical investigations of parabolic (anisotropic diffusion) and hyperbolic (wave equation) model problems. We assess the sharpness of the bounds and the quality of the approximate convergence factors. Observations from these numerical investigations demonstrate the value of the proposed multilevel convergence framework for estimating MGRIT convergence a priori and for the design of a convergent algorithm. We further highlight that observations in the literature are captured by the theory, including that two-level Parareal and multilevel MGRIT with F-relaxation do not yield scalable algorithms and the benefit of a stronger relaxation scheme. An important observation is that with increasing numbers of levels MGRIT convergence deteriorates for the hyperbolic model problem, while constant convergence factors can be achieved for the diffusion equation. The theory also indicates that L-stable Runge-Kutta schemes are more amendable to multilevel parallel-in-time integration with MGRIT than A-stable Runge-Kutta schemes.

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arXiv2019

Necessary Conditions and Tight Two-level Convergence Bounds for Parareal and Multigrid Reduction in Time

Parareal and multigrid reduction in time (MGRiT) are two of the most popular parallel-in-time methods. The idea is to treat time integration in a parallel context by using a multigrid method in time. If $Φ$ is a (fine-grid) time-stepping scheme, let $Ψ$ denote a "coarse-grid" time-stepping scheme chosen to approximate $k$ steps of $Φ$, $k\geq 1$. In particular, $Ψ$ defines the coarse-grid correction, and evaluating $Ψ$ should be (significantly) cheaper than evaluating $Φ^k$. A number of papers have studied the convergence of Parareal and MGRiT. However, there have yet to be general conditions developed on the convergence of Parareal or MGRiT that answer simple questions such as, (i) for a given $Φ$ and $k$, what is the best $Ψ$, or (ii) can Parareal/MGRiT converge for my problem? This work derives necessary and sufficient conditions for the convergence of Parareal and MGRiT applied to linear problems, along with tight two-level convergence bounds. Results rest on the introduction of a "temporal approximation property" (TAP) that indicates how $Φ^k$ must approximate the action of $Ψ$ on different vectors. Loosely, for unitarily diagonalizable operators, the TAP indicates that fine-grid and coarse-grid time integration schemes must integrate geometrically smooth spatial components similarly, and less so for geometrically high frequency. In the (non-unitarily) diagonalizable setting, the conditioning of each eigenvector, $\mathbf{v}_i$, must also be reflected in how well $Ψ\mathbf{v}_i \simΦ^k\mathbf{v}_i$. In general, worst-case convergence bounds are exactly given by $\min φ< 1$ such that an inequality along the lines of $\|(Ψ-Φ^k)\mathbf{v}\| \leqφ\|(I - Ψ)\mathbf{v}\|$ holds for all $\mathbf{v}$. Such inequalities are formalized as different realizations of the TAP, and form the basis for convergence of MGRiT and Parareal.

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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.

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arXiv2019

The Role of Energy Minimization in Algebraic Multigrid Interpolation

Algebraic multigrid (AMG) methods are powerful solvers with linear or near-linear computational complexity for certain classes of linear systems, Ax=b. Broadening the scope of problems that AMG can effectively solve requires the development of improved interpolation operators. Such development is often based on AMG convergence theory. However, convergence theory in AMG tends to have a disconnect with AMG in practice due to the practical constraints of (i) maintaining matrix sparsity in transfer and coarse-grid operators, and (ii) retaining linear complexity in the setup and solve phase. This paper presents a review of fundamental results in AMG convergence theory, followed by a discussion on how these results can be used to motivate interpolation operators in practice. A general weighted energy minimization functional is then proposed to form interpolation operators, and a novel `diagonal' preconditioner for Sylvester- or Lyapunov-type equations developed simultaneously. Although results based on the weighted energy minimization typically underperform compared to a fully constrained energy minimization, numerical results provide new insight into the role of energy minimization and constraint vectors in AMG interpolation.

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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.

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