Adaptive Smoothed Aggregation ($\alpha$SA) Multigrid

Brezina, Marian; Falgout, Robert D.; MacLachlan, Scott; Manteuffel, Thomas A.; McCormick, Steve; Ruge, John W.

doi:10.1137/050626272

Cited by 128 publications

(149 citation statements)

References 14 publications

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“…SA generally has two strategies for generating B. There is the purely adaptive approach [39], which can unfortunately be expensive. The other approach uses the null-space of the unrestricted PDE, which can be inaccurate, especially near the boundaries.…”

Section: Discussionmentioning

confidence: 99%

“…An important component in representing B is the wavenumber , but the described by the PDE (1) is inaccurate for the low-energy modes of the system. For example, we present in Figure 2(a), the near null-space mode generated by the adaptive-SA [39] next to the mode based on the natural wavenumber of the PDE. The adaptive mode is identified with a different wavenumber and this difference becomes more pronounced for less resolved problems, which highlights the effect of discretization error.…”

Section: Near Null-space Selectionmentioning

confidence: 99%

“…The difference is also likely related to satisfying the boundary conditions. Adaptive-SA [39] is an approach that assumes no a priori knowledge of the near null-space of the operator. Columns in B are generated by a process that begins with a random initial guess, which is then refined through a strategy that resembles multigrid cycling until the guess becomes a suitable approximation of a low-energy mode not yet captured by the coarse space.…”

Section: Near Null-space Selectionmentioning

confidence: 99%

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Smoothed aggregation for Helmholtz problems

Olson

Schroder

2010

Numerical Linear Algebra App

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SUMMARYWe outline a smoothed aggregation algebraic multigrid method for 1D and 2D scalar Helmholtz problems with exterior radiation boundary conditions. We consider standard 1D finite difference discretizations and 2D discontinuous Galerkin discretizations. The scalar Helmholtz problem is particularly difficult for algebraic multigrid solvers. Not only can the discrete operator be complex-valued, indefinite, and non-self-adjoint, but it also allows for oscillatory error components that yield relatively small residuals. These oscillatory error components are not effectively handled by either standard relaxation or standard coarsening procedures. We address these difficulties through modifications of SA and by providing the SA setup phase with appropriate wave-like near null-space candidates. Much is known a priori about the character of the near null-space, and our method uses this knowledge in an adaptive fashion to find appropriate candidate vectors. Our results for GMRES preconditioned with the proposed SA method exhibit consistent performance for fixed points-per-wavelength and decreasing mesh size.

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Section: Discussionmentioning

confidence: 99%

Section: Near Null-space Selectionmentioning

confidence: 99%

Section: Near Null-space Selectionmentioning

confidence: 99%

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Smoothed aggregation for Helmholtz problems

Olson

Schroder

2010

Numerical Linear Algebra App

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“…In this section, we apply adaptive SA to solve matrix equation (31), which appears in Step 4 of Algorithm 4 [13]. The resulting linear system clearly contains complex entries.…”

Section: Numerical Experimentsmentioning

confidence: 99%

Finite element methods for quantum electrodynamics using a Helmholtz decomposition of the gauge field

Ketelsen

Manteuffel

McCormick

et al. 2010

Numerical Linear Algebra App

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SUMMARYThe Dirac equation of quantum electrodynamics describes the interaction between electrons and photons. Large-scale numerical simulations of the theory require repeated solution of the two-dimensional Dirac equation, a system of two first-order partial differential equations coupled to a background U(1) gauge field. Traditional discretizations of this system are sparse and highly structured, but contain random complex entries introduced by the background field. For even mildly disordered gauge fields, the near kernel components of the system are highly oscillatory, rendering standard multilevel methods ineffective. We consider an alternate formulation of the governing equations obtained by a transformation of the continuum operator that decouples the system into separate scalar diffusion-like equations. We discretize the transformed system using least-squares finite elements and use adaptive smoothed aggregation multigrid to solve the resulting linear system. We present numerical results and discuss implications of the transformed formulation in terms of the physical theory.

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“…For this purpose, we use the following strategy (similar to the one in [8]): we choose an integer parameter µ 0 (1 ≤ µ 0 ≤ µ and typically µ 0 µ); we run µ iterations with a random A -normalized initial iterate (i.e., x 0 A = 1) without further normalizing the consecutive iterates; we monitor (see Section 2.2.3)…”

Section: The Prototypementioning

confidence: 99%

Adaptive Algebraic Multigrid for Finite Element Elliptic Equations with Random Coefficients

Kalchev¹

2012

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This thesis presents a two-grid algorithm based on Smoothed Aggregation Spectral Element Agglomeration Algebraic Multigrid (SA-ρAMGe) combined with adaptation. The aim is to build an efficient solver for the linear systems arising from discretization of second-order elliptic partial differential equations (PDEs) with stochastic coefficients. Examples include PDEs that model subsurface flow with random permeability field. During a Markov Chain Monte Carlo (MCMC) simulation process, that draws PDE coefficient samples from a certain distribution, the PDE coefficients change, hence the resulting linear systems to be solved change. At every such step the system (discretized PDE) needs to be solved and the computed solution used to evaluate some functional(s) of interest that then determine if the coefficient sample is acceptable or not. The MCMC process is hence computationally intensive and requires the solvers used to be efficient and fast. This fact that at every step of MCMC the resulting linear system changes, makes an already existing solver built for the old problem perhaps not as efficient for the problem corresponding to the new sampled coefficient. This motivates the main goal of our study, namely, to adapt an already existing solver to handle the problem (with changed coefficient) with the objective to achieve this goal to be faster and more efficient than building a completely new solver from scratch. Our approach utilizes the local element matrices (for the problem with changed coefficients) to build local problems associated with constructed by the method agglomerated elements (a set of subdomains that cover the given computational domain). We solve a generalized eigenproblem for each set in a subspace spanned by the previous local coarse space (used for the old solver) and a vector, component of the error, that the old solver cannot handle. A portion of the spectrum of these local eigenproblems (corresponding to eigenvalues close to zero) form the coarse basis used to define the new two-level method of our interest. We illustrate the performance of this adaptive two-level procedure with a large set of numerical experiments that demonstrate its efficiency over building the solvers from scratch.

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Adaptive Smoothed Aggregation ($\alpha$SA) Multigrid

Cited by 128 publications

References 14 publications

Smoothed aggregation for Helmholtz problems

Smoothed aggregation for Helmholtz problems

Finite element methods for quantum electrodynamics using a Helmholtz decomposition of the gauge field

Adaptive Algebraic Multigrid for Finite Element Elliptic Equations with Random Coefficients

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