A hybrid geometric + algebraic multigrid method with semi‐iterative smoothers

Lu, Chenguang; Jiao, Xiangmin; Missirlis, Nikolaos M.

doi:10.1002/nla.1925

Cited by 23 publications

(19 citation statements)

References 29 publications

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“…In this paper, we did not consider GMG methods. GMG often suffices as a standalone solver, and it typically outperforms KSP methods significantly if applicable (see, e.g., the work of Lu et al for comparisons of GMG, AMG, and a hybrid multigrid method). If GMG is not robust enough as a standalone solver, our recommendations regarding AMG preconditioners also hold to GMG as right preconditioners.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

A comparison of preconditioned Krylov subspace methods for large‐scale nonsymmetric linear systems

Ghai

Jiao

2018

Numerical Linear Algebra App

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Preconditioned Krylov subspace (KSP) methods are widely used for solving large-scale sparse linear systems arising from numerical solutions of partial differential equations (PDEs). These linear systems are often nonsymmetric due to the nature of the PDEs, boundary or jump conditions, or discretization methods. While implementations of preconditioned KSP methods are usually readily available, it is unclear to users which methods are the best for different classes of problems. In this work, we present a comparison of some KSP methods, including GMRES, TFQMR, BiCGSTAB, and QMRCGSTAB, coupled with three classes of preconditioners, namely, Gauss-Seidel, incomplete LU factorization (including ILUT, ILUTP, and multilevel ILU), and algebraic multigrid (including BoomerAMG and ML). Theoretically, we compare the mathematical formulations and operation counts of these methods. Empirically, we compare the convergence and serial performance for a range of benchmark problems from numerical PDEs in two and three dimensions with up to millions of unknowns and also assess the asymptotic complexity of the methods as the number of unknowns increases. Our results show that GMRES tends to deliver better performance when coupled with an effective multigrid preconditioner, but it is less competitive with an ineffective preconditioner due to restarts.BoomerAMG with a proper choice of coarsening and interpolation techniques typically converges faster than ML, but both may fail for ill-conditioned or saddle-point problems, whereas multilevel ILU tends to succeed. We also show that right preconditioning is more desirable. This study helps establish some practical guidelines for choosing preconditioned KSP methods and motivates the development of more effective preconditioners.

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Section: Conclusion and Discussionmentioning

confidence: 99%

A comparison of preconditioned Krylov subspace methods for large‐scale nonsymmetric linear systems

Ghai

Jiao

2018

Numerical Linear Algebra App

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“…For these shortcomings, the hybridization of algebraic and geometric multigrid is well-trodden ground. [19][20][21][22] Most hybrid approaches use algebraic multigrid where sole geometric operators fail, but employ geometric operators wherever possible. As material parameters change "infrequently" on fine meshes and diffusion dominates in many setups, it is indeed possible to rely on a geometric operator construction for the majority of the equation systems.…”

Section: Related Work and Influencing Ideasmentioning

confidence: 99%

Delayed approximate matrix assembly in multigrid with dynamic precisions

Murray

Weinzierl

2020

Concurrency and Computation

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The accurate assembly of the system matrix is an important step in any code that solves partial differential equations on a mesh. We either explicitly set up a matrix, or we work in a matrix-free environment where we have to be able to quickly return matrix entries upon demand. Either way, the construction can become costly due to nontrivial material parameters entering the equations, multigrid codes requiring cascades of matrices that depend upon each other, or dynamic adaptive mesh refinement that necessitates the recomputation of matrix entries or the whole equation system throughout the solve. We propose that these constructions can be performed concurrently with the multigrid cycles. Initial geometric matrices and low accuracy integrations kickstart the multigrid iterations, while improved assembly data is fed to the solver as and when it becomes available. The time to solution is improved as we eliminate an expensive preparation phase traditionally delaying the actual computation. We eliminate algorithmic latency. Furthermore, we desynchronize the assembly from the solution process. This anarchic increase in the concurrency level improves the scalability. Assembly routines are notoriously memory-and bandwidth-demanding. As we work with iteratively improving operator accuracies, we finally propose the use of a hierarchical, lossy compression scheme such that the memory footprint is brought down aggressively where the system matrix entries carry little information or are not yet available with high accuracy. K E Y W O R D S algebraic-geometric multigrid, asynchronous multigrid, delayed operator computation, dynamically adaptive Cartesian grids, finite element assembly, mixed precision computing 1 INTRODUCTION Multigrid algorithms are among the fastest solvers known for elliptic partial differential equations (PDEs) of the type −∇ (∇) u = f (1) on a d-dimensional, well-shaped domain Ω. An approximation of the function u : Ω  → R is what we are searching for with : Ω  → R + as a material parameter, and f : Ω  → R constituting the right-hand side. The system is closed by appropriate boundary conditions. We restrict ourselves to This paper is an extended version of C.D. Murray and T. Weinzierl: Lazy stencil integration in multigrid algorithms as introduced and published at the PPAM'19 Conference. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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“…If the problem is ill-suited for our code due to a lack of robustness, our approach however may act as building block applied on the finer mesh levels while coarser problems are solved algebraically [28,35,43,49]. We reiterate that such a level max can be determined automatically.…”

Section: Observation 18mentioning

confidence: 99%

“…On coarse meshes, one can employ algebraic multigrid, alternative iterative schemes or a direct solver. Such an approach [28,35,43,49] is robust and fast at modest memory requirements as long as the finest algebraic problem with a matrix setup remains reasonably coarse within the grid hierarchy. Finally, we may also decide to tailor all operators to the problem manually and to hard-code it into the solver software.…”

Section: Introductionmentioning

confidence: 99%

Quasi-matrix-free Hybrid Multigrid on Dynamically Adaptive Cartesian Grids

Weinzierl

2018

ACM Trans. Math. Softw.

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We present a family of spacetree-based multigrid realizations using the tree's multiscale nature to derive coarse grids. They align with matrix-free geometric multigrid solvers as they never assemble the system matrices which is cumbersome for dynamically adaptive grids and full multigrid. The most sophisticated realizations use BoxMG to construct operator-dependent prolongation and restriction in combination with Galerkin/Petrov-Galerkin coarse-grid operators. This yields robust solvers for nontrivial elliptic problems. We embed the algebraic, problem-and grid-dependent multigrid operators as stencils into the grid and evaluate all matrix-vector products in-situ throughout the grid traversals. While such an approach is not literally matrix-free-the grid carries the matrix-we propose to switch to a hierarchical representation of all operators. Only differences of algebraic operators to their geometric counterparts are held. These hierarchical differences can be stored and exchanged with small memory footprint. Our realizations support arbitrary dynamically adaptive grids while they vertically integrate the multilevel operations through spacetree linearization. This yields good memory access characteristics, while standard colouring of mesh entities with domain decomposition allows us to use parallel manycore clusters. All realization ingredients are detailed such that they can be used by other codes.

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A hybrid geometric + algebraic multigrid method with semi‐iterative smoothers

Cited by 23 publications

References 29 publications

A comparison of preconditioned Krylov subspace methods for large‐scale nonsymmetric linear systems

A comparison of preconditioned Krylov subspace methods for large‐scale nonsymmetric linear systems

Delayed approximate matrix assembly in multigrid with dynamic precisions

Quasi-matrix-free Hybrid Multigrid on Dynamically Adaptive Cartesian Grids

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