A review of algebraic multigrid

Stüben, Klaus

doi:10.1016/b978-0-444-50616-0.50012-9

Cited by 167 publications

(183 citation statements)

References 30 publications

(1 reference statement)

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“…We then examine the interplay of the source model approaches with three FE solver methods: a Jacobi-CG, an incomplete Cholesky CG (e.g., [27]), and an algebraic multigrid preconditioned CG (AMG-CG), which has already shown to be especially suited for problems with discontinuous and anisotropic coefficients [22,32,21,9,38]. …”

Section: Introductionmentioning

confidence: 99%

Accuracy and run-time comparison for different potential approaches and iterative solvers in finite element method based EEG source analysis

Lew

Wolters

Dierkes

et al. 2009

Applied Numerical Mathematics

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Accuracy and run-time play an important role in medical diagnostics and research as well as in the field of neuroscience. In Electroencephalography (EEG) source reconstruction, a current distribution in the human brain is reconstructed noninvasively from measured potentials at the head surface (the EEG inverse problem). Numerical modeling techniques are used to simulate head surface potentials for dipolar current sources in the human cortex, the so-called EEG forward problem.In this paper, the efficiency of algebraic multigrid (AMG), incomplete Cholesky (IC) and Jacobi preconditioners for the conjugate gradient (CG) method are compared for iteratively solving the finite element (FE) method based EEG forward problem. The interplay of the three solvers with a full subtraction approach and two direct potential approaches, the Venant and the partial integration method for the treatment of the dipole singularity is examined. The examination is performed in a four-compartment sphere model with anisotropic skull layer, where quasi-analytical solutions allow for an exact quantification of computational speed versus numerical error. Specifically-tuned constrained Delaunay tetrahedralization (CDT) FE meshes lead to high accuracies for both the full subtraction and the direct potential approaches. Best accuracies are achieved by the full subtraction approach if the homogeneity condition is fulfilled. It is shown that the AMG-CG achieves an order of magnitude higher computational speed than the CG with the standard preconditioners with an increasing gain factor when decreasing mesh size. Our results should broaden the application of accurate and fast high-resolution FE volume conductor modeling in source analysis routine.

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Section: Introductionmentioning

confidence: 99%

Accuracy and run-time comparison for different potential approaches and iterative solvers in finite element method based EEG source analysis

Lew

Wolters

Dierkes

et al. 2009

Applied Numerical Mathematics

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“…Several tracer tests in heterogeneous aquifers [Adams and Gelhar, 1992;Sidle et al, 1998;Becker and Shapiro 2000;Haggerty et al, 2001] have indicated behavior that cannot be predicted assuming Gaussian spreading of the plume. Di Donato et al [2003] demonstrated using streamline-based simulation that advective movement with particles experiencing a wide range of velocities could be the physical origin of anomalous transport at the large scale, confirming the theoretical and numerical studies of Berkowitz and Scher [1995, 1998, 2001]. We will pursue this concept further by including rate-limited sorption in the model.…”

Section: Introductionmentioning

confidence: 88%

“…[12] From equations (8) - (11) a pressure equation can be derived that is solved on a regular Cartesian grid using an algebraic multigrid method [Stueben, 2001]. Using Darcy's law, the total velocity can be found at all cell edges.…”

Section: Streamline Methodsmentioning

confidence: 99%

“…Notice the huge range of timescales; while breakthrough can occur in the order 10 days (8.64 Â 10 5 s), significant amounts of tracer still remain in the system after 10,000 days (8.64 Â 10 8 s). This anomalous behavior has been explained and predicted by Berkowitz and Scher [1995, 1998, 2001, using a continuous time random walk (CTRW) formulation. Define ψ(s, t) as the probability per unit time for tracer movement between sites separated by a distance s in a time t. If, at large times, ψ(s, t) $ t À1Àl for 1 > l > 0, then at late times the concentration at a production well, C(t) $ t À(1+l) [Berkowitz et al, 2000].…”

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confidence: 99%

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Streamline‐based dual‐porosity simulation of reactive transport and flow in fractured reservoirs

Donato

Blunt

2004

Water Resources Research

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[1] We present a streamline-based formulation to model two processes: transport of a tracer undergoing rate-limited sorption and two-phase (water/oil or air/water) transport in fractured systems, using a dual-porosity approach. We show that these two processes can be simulated using mathematically equivalent formulations. In both cases the system conceptually has two components: a flowing fraction connected to stagnant regions with transfer between the two domains. Streamlines capture movement through the flowing fraction. Fluid transfer between flowing and stagnant regions enters as a source/sink term in the one-dimensional transport equations along a streamline. To model flow and transport in fractured systems, we develop a new formulation for the transfer function that matches experimental imbibition data. Then we illustrate the streamline approach with synthetic reservoir problems. We use a finely gridded (over one million grid blocks) threedimensional domain with a highly heterogeneous permeability field to study both fracture flow and tracer transport. We find breakthrough curves that are consistent with anomalous transport described by an exponent that characterizes the longtime tail of the transit time distribution. For fracture flow we demonstrate that the speed of fluid advance in the fractures is controlled by the imbibition rate. The run times for the simulations scale approximately linearly with system size, making the method appropriate for the simulation of large numerical models.

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“…[29,30]. Two prototypical variants of multigrid methods can be distinguished: the algebraic multigrid (AMG) variant is based on a purely algebraic problem formulation by means of a linear system of equations [31]. In contrast, the geometric multigrid (GMG) variant is based on the discretization of the underlying model equations on several grid refinement levels [32].…”

Section: Energy-efficient Techniques For Multigrid Methods On Distribmentioning

confidence: 99%

Energy-Aware High Performance Computing

Wlotzka¹,

Heuveline²,

Dolz³

et al. 2017

ICT - Energy Concepts for Energy Efficiency and Sustainability

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High performance computing centres consume substantial amounts of energy to power large-scale supercomputers and the necessary building and cooling infrastructure. Recently, considerable performance gains resulted predominantly from developments in multi-core, many-core and accelerator technology. Computing centres rapidly adopted this hardware to serve the increasing demand for computational power. However, further performance increases in large-scale computing systems are limited by the aggregate energy budget required to operate them. Power consumption has become a major cost factor for computing centres. Furthermore, energy consumption results in carbon dioxide emissions, a hazard for the environment and public health; and heat, which reduces the reliability and lifetime of hardware components. Energy efficiency is therefore crucial in high performance computing.This chapter addresses key issues of energy-aware high performance computing. We outline some numerical methods which are often used in scientific applications, and present an energy profiling and tracing technique suitable to analyse the power consumption of applications. The next section is devoted to the performance and energy characterization of the sparse matrix-vector product, a basic numerical building block. Finally, we discuss opportunities for saving energy in computations by means of two examples. First, we present energy-aware runtimes on shared memory multi-core platforms for the Conjugate Gradient method. Second, we present energy-efficient techniques for multigrid methods on distributed memory clusters.

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A review of algebraic multigrid

Cited by 167 publications

References 30 publications

Accuracy and run-time comparison for different potential approaches and iterative solvers in finite element method based EEG source analysis

Accuracy and run-time comparison for different potential approaches and iterative solvers in finite element method based EEG source analysis

Streamline‐based dual‐porosity simulation of reactive transport and flow in fractured reservoirs

Energy-Aware High Performance Computing

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