Extending the applicability of multigrid methods

Brannick, James; Brezina, Marian; Falgout, Robert D.; Manteuffel, Thomas A.; McCormick, Steve; Ruge, John W.; Sheehan, Brendan; Xu, Jinchao; Zikatanov, Ludmil T.

doi:10.1088/1742-6596/46/1/061

Cited by 6 publications

(5 citation statements)

References 27 publications

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“…Figure 3.3 compares the effectiveness of eigCG in approximating many eigenpairs to those of the competing GMRESDR and RecycledCG methods (the latter implementing the RMINRES ideas on CG). The experiments are run on a Wilson Fermion lattice of size 12 × 12 4 with periodic boundary conditions and quark mass equal to the critical mass. We use odd-even preconditioning (which yields a matrix of half the size) and focus on the symmetric normal equations.…”

Section: Convergence and Comparison With Other Methodsmentioning

confidence: 99%

“…However, such results are often obtained from heavier quark masses, where the problem is not as difficult or interesting. In our 12 × 12 4 problem, and with the same accuracy and computational platform, GMRESDR (55,34) took 1000 iterations and 1120 seconds. GMRESDR(85,60) improved convergence to 585 iterations, but its execution time was still 1001 seconds.…”

Section: Convergence and Comparison With Other Methodsmentioning

confidence: 99%

“…As eigenvalue density increases with volume, more eigenvalues need to be deflated to achieve a constant condition number. Traditionally, scalability across volumes is the topic of multigrid methods [1,2,6,3,4]. Although, our methods are very effective in that direction too, they only guarantee constant time across different masses.…”

Section: Numerical Experimentsmentioning

confidence: 96%

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Computing and Deflating Eigenvalues While Solving Multiple Right-Hand Side Linear Systems with an Application to Quantum Chromodynamics

Stathopoulos¹,

Orginos²

2010

SIAM J. Sci. Comput.

112

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We present a new algorithm that computes eigenvalues and eigenvectors of a Hermitian positive definite matrix while solving a linear system of equations with conjugate gradient (CG). Traditionally, all the CG iteration vectors could be saved and recombined through the eigenvectors of the tridiagonal projection matrix, which is equivalent theoretically to unrestarted Lanczos. Our algorithm capitalizes on the iteration vectors produced by CG to update only a small window of vectors that approximate the eigenvectors. While this window is restarted in a locally optimal way, the CG algorithm for the linear system is unaffected. Yet, in all our experiments, if the window has more than a properly chosen but small number of vectors, it converges to the required eigenvectors at a rate identical to unrestarted Lanczos. After the solution of the linear system, eigenvectors that have not accurately converged can be improved in an incremental fashion by solving additional linear systems. In this case, eigenvectors identified in earlier systems can be used to deflate, and thus accelerate, the convergence of subsequent systems. We have used this algorithm with excellent results in lattice quantum chromodynamics applications, where hundreds of right-hand sides may be needed. Specifically, about 70 eigenvectors are obtained to full accuracy after solving 24 right-hand sides. Deflating these from the large number of subsequent right-hand sides removes the dreaded critical slowdown, where the conditioning of the matrix increases as the quark mass reaches a critical value. Our experiments show almost a constant number of iterations for our method, regardless of quark mass, and speedups of 8 over original CG for light quark masses. Introduction.The numerical solution of linear systems of equations of large, sparse matrices is central to many scientific and engineering applications. One of the most computationally demanding applications is lattice quantum chromodynamics (QCD) because not only does it involve very large matrix sizes but it also requires the solution of several linear systems with the same matrix but different right-hand sides. Direct methods, although attractive for multiple right-hand sides, cannot be used because of the size of the matrix. Iterative methods provide the only means for solving these problems.QCD is the theory of the fundamental force known as strong interaction, which describes the interactions among quarks, one of the constituents of matter. Lattice QCD is the tool for nonperturbative numerical calculations of these interactions on a Euclidean space-time lattice [58]. The heart of the computations is the solution of the lattice-Dirac equation, which translates to a linear system of equations M x = b, often for a large number of right-hand sides [20]. The Dirac operator M is

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Section: Convergence and Comparison With Other Methodsmentioning

confidence: 99%

Section: Convergence and Comparison With Other Methodsmentioning

confidence: 99%

Section: Numerical Experimentsmentioning

confidence: 96%

See 1 more Smart Citation

Computing and Deflating Eigenvalues While Solving Multiple Right-Hand Side Linear Systems with an Application to Quantum Chromodynamics

Stathopoulos¹,

Orginos²

2010

SIAM J. Sci. Comput.

112

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“…In particular, parallel algorithm designs and implementations of AMG have received considerable attention in the past 15 years as a means of scaling up to massively parallel machines, see, for example . Overall, AMG plays an increasingly important role in large‐scale simulation for practical applications …”

Section: Introductionmentioning

confidence: 99%

Algebraic interface‐based coarsening AMG preconditioner for multi‐scale sparse matrices with applications to radiation hydrodynamics computation

2016

Numerical Linear Algebra App

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Coarsening is a crucial component of algebraic multigrid (AMG) methods for iteratively solving sparse linear systems arising from scientific and engineering applications. Its application largely determines the complexity of the AMG iteration operator. Usually, high operator complexities lead to fast convergence of the AMG method; however, they require additional memory and as such do not scale as well in parallel computation. In contrast, although low operator complexities improve parallel scalability, they often lead to deterioration in convergence. This study introduces a new type of coarsening strategy called algebraic interface-based coarsening that yields a better balance between convergence and complexity for a class of multi-scale sparse matrices. Numerical results for various model-type problems and a radiation hydrodynamics practical application are provided to show the effectiveness of the proposed AMG solver. KEYWORDS algebraic multigrid (AMG), coarsening, preconditioning, parallel computation, multiscale sparse matrices, radiation hydrodynamics | INTRODUCTIONMany challenging computational problems in science and engineering need the solutions of large-scale sparse linear systems, that is,where A = (a ij ) n × n ∈ R n × n is a sparse matrix with entries a ij and u , f ∈ R n are the unknown and right-side vectors, respectively. The solver for these linear systems is a crucial component of simulation codes in diverse realistic applications and usually dominates the time consumption of simulations. Algebraic multigrid (AMG) 1-3 is one of the most efficient methods for iteratively solving linear systems arising from discretization of partial differential equations. In the past decades, challenges in real applications have sparked great interest in and further development of AMG methods. Many algorithm and theory variants have been developed to handle various problems (e.g., ). In particular, parallel algorithm designs and implementations of AMG have received considerable attention in the past 15 years as a means of scaling up to massively parallel machines, see, for example. [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47] Overall, AMG plays an increasingly important role in large-scale simulation for practical applications. 23,[48][49][50][51][52][53][54] The operator complexity (C op ) is a key performance measure for AMG methods. It is used as an important indicator of the computational cost of each iteration and storage requirements and is defined as follows:( 1:2) where A l denotes a matrix of level l and |A l | denotes the number of nonzero entries in A l . A 0 = A is the matrix of the finest * These authors contributed equally to this work.level (l = 0). The coarse-level matrix A l (l >0) usually comes from an application of the Galerkin approach (see Section 2). For large-scale computations on modern supercomputers, both low operator complexity and fast convergence are required. The classical AMG methods 3,32,39 are designed for fast convergence; however, they often...

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“…We consider the problem of an efficient solution of large‐scale sparse linear systems that arise from the discretizations of PDEs. The multigrid methods, including geometric multigrid ( GMG ) and algebraic multigrid ( AMG ), are often the most efficient and effective methods for solving such linear systems . Traditionally, these methods utilize stationary iterative methods (such as Jacobi, Gauss–Seidel, or successive over‐relaxation/under‐relaxation) to smooth out high‐frequency errors and accelerate the convergence of the solution by transferring the residual and correction vectors across different resolutions (or levels) via the so‐called prolongation and restriction operators.…”

Section: Introductionmentioning

confidence: 99%

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

Jiao

Missirlis

2014

Numerical Linear Algebra App

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SUMMARYWe propose a multigrid method for solving large‐scale sparse linear systems arising from discretizations of partial differential equations, such as those from finite element and generalized finite difference methods. Our proposed method has the following two characteristics. First, we introduce a hybrid geometric+algebraic multigrid method, or HyGA, to leverage the rigor, accuracy, and efficiency of geometric multigrid (GMG) for hierarchical unstructured meshes, with the flexibility of algebraic multigrid (AMG). Second, we introduce efficient smoothers based on the Chebyshev–Jacobi method for both symmetric and asymmetric matrices. We also introduce a unified derivation of restriction and prolongation operators for weighted‐residual formulations over unstructured hierarchical meshes and apply it to both finite element and generalized finite difference methods. We present numerical results of our method for Poisson equations in both 2‐D and 3‐D and compare our method against the classical GMG and AMG as well as Krylov subspace methods. Copyright © 2014 John Wiley & Sons, Ltd.

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Extending the applicability of multigrid methods

Cited by 6 publications

References 27 publications

Computing and Deflating Eigenvalues While Solving Multiple Right-Hand Side Linear Systems with an Application to Quantum Chromodynamics

Computing and Deflating Eigenvalues While Solving Multiple Right-Hand Side Linear Systems with an Application to Quantum Chromodynamics

Algebraic interface‐based coarsening AMG preconditioner for multi‐scale sparse matrices with applications to radiation hydrodynamics computation

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

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