Non-Galerkin Multigrid Based on Sparsified Smoothed Aggregation

Treister, Eran; Yavneh, Irad

doi:10.1137/140952570

Cited by 25 publications

(25 citation statements)

References 45 publications

(65 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the discrete setting, small entries that arise in matrix operations are often not aligned with the characteristic and are more of a numerical effect, suggesting that some can be eliminated without degrading convergence. Numerical results confirm this; in particular, removing entries in the case of SPD matrices is a delicate process [9,24,58], but Section 6 shows that entries can be removed from discretizations of steady state transport aggressively, without a degradation in convergence. For some drop-tolerance ϕ, elements {a ij | j = i, |a ij | ≤ ϕ|a ii |} are eliminated (that is, set to zero) for each row i of matrix A .…”

Section: Amg Componentsmentioning

confidence: 86%

Nonsymmetric Reduction-Based Algebraic Multigrid

Manteuffel¹,

Münzenmaier²,

Ruge³

et al. 2019

SIAM J. Sci. Comput.

104

View full text Add to dashboard Cite

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 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 reductiontype 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 (nAIR). nAIR can be seen as a variation of local AIR ( AIR) introduced in previous work, specifically targeting matrices with triangular structure. Although less versatile than AIR, setup times for nAIR can be substantially faster for problems with high connectivity. nAIR 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. nAIR is also shown to be effective on several classes of "nearly triangular" matrices, resulting from curvilinear finite elements and artificial diffusion.

show abstract

Section: Amg Componentsmentioning

confidence: 86%

Nonsymmetric Reduction-Based Algebraic Multigrid

Manteuffel¹,

Münzenmaier²,

Ruge³

et al. 2019

SIAM J. Sci. Comput.

104

View full text Add to dashboard Cite

show abstract

“…were eliminated in row i that were smaller than 0.001 · max j |a ij |. It is known that such an approach is typically not effective on diffusive matrices, prompting research into more advanced techniques for reducing the number of matrix nonzeros [6,17,54]. Here, we use a technique similar to the elimination used in [33], but instead of actually eliminating entries, we add them to the diagonal in order to preserve the row sum.…”

Section: Filtering and Lumpingmentioning

confidence: 99%

Nonsymmetric Algebraic Multigrid Based on Local Approximate Ideal Restriction ($\ell$AIR)

Manteuffel¹,

Ruge²,

Southworth³

2018

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

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 AIR), 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. AIR 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.

show abstract

“…For 2D examples, the efficient semi-coarsening multigrid solver with one V-cycle [34] is utilized to approximately solve the sublinear system on each subdomain. However, any other efficient iterative solvers, such as algebraic multigrid methods [43], are also applicable.…”

Section: Numerical Examplesmentioning

confidence: 99%

Efficient time domain decomposition algorithms for parabolic PDE-constrained optimization problems

Wang

2018

Computers & Mathematics with Applications

View full text Add to dashboard Cite

Optimization with time-dependent partial differential equations (PDEs) as constraints appears in many science and engineering applications. The associated first-order necessary optimality system consists of one forward and one backward time-dependent PDE coupled with optimality conditions. An optimization process by using the one-shot method determines the optimal control, state and adjoint state at once, with the cost of solving a large scale, fully discrete optimality system. Hence, such a one-shot method could easily become computationally prohibitive when the time span is long or time step is small. To overcome this difficulty, we propose several time domain decomposition algorithms for improving the computational efficiency of the one-shot method. In these algorithms, the optimality system is split into many small subsystems over a much smaller time interval, which are coupled by appropriate continuity matching conditions. Both one-level and two-level multiplicative and additive Schwarz algorithms are developed for iteratively solving the decomposed subsystems in parallel. In particular, the convergence of the one-level, non-overlapping algorithms is proved.The effectiveness of our proposed algorithms is demonstrated by both 1D and 2D numerical experiments, where the developed two-level algorithms show convergence rates that are scalable with respect to the number of subdomains.

show abstract

Non-Galerkin Multigrid Based on Sparsified Smoothed Aggregation

Cited by 25 publications

References 45 publications

Nonsymmetric Reduction-Based Algebraic Multigrid

Nonsymmetric Reduction-Based Algebraic Multigrid

Nonsymmetric Algebraic Multigrid Based on Local Approximate Ideal Restriction ($\ell$AIR)

Efficient time domain decomposition algorithms for parabolic PDE-constrained optimization problems

Contact Info

Product

Resources

About