Reducing the bandwidth of sparse symmetric matrices

Cuthill, Elizabeth; McKee, James

doi:10.1145/800195.805928

Cited by 1,085 publications

(658 citation statements)

References 12 publications

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“…Reordering, therefore, can improve the quality of incomplete factorization. χMD uses either original, Minimum degree (Tinney and Walker, 1967) or Reverse Cuthill-McKee orderings (Cuthill and McKee, 1969;George, 1971).…”

Section: Brief Outline Of Theorymentioning

confidence: 99%

MODFLOW-NWT, a Newton formulation for MODFLOW-2005

Niswonger¹,

Panday²,

Ibaraki³

2011

Techniques and Methods

328

257

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For more information on the USGS-the Federal source for science about the Earth, its natural and living resources, natural hazards, and the environment, visit http://www.usgs.gov or call 1-888-ASK-USGS.For an overview of USGS information products, including maps, imagery, and publications, visit http://www.usgs.gov/pubprodTo order this and other USGS information products, visit http://store.usgs.gov Any use of trade, product, or firm names is for descriptive purposes only and does not imply endorsement by the U.S. Government.Although this report is in the public domain, permission must be secured from the individual copyright owners to reproduce any copyrighted materials contained within this report. Altitude, as used in this report, refers to distance above the vertical datum.*Transmissivity: The standard unit for transmissivity is cubic foot per day per square foot times foot of aquifer thickness [(ft 3 /d)/ft 2 ]ft. In this report, the mathematically reduced form, foot squared per day (ft 2 /d), is used for convenience. The UPW Package treats nonlinearities of cell drying and rewetting by use of a continuous function of groundwater head, rather than the discrete approach of drying and rewetting that is used by the BCF, LPF, and HUF Packages. This further enables application of the Newton formulation for unconfined groundwater-flow problems because conductance derivatives required by the Newton method are smooth over the full range of head for a model cell. TablesThe NWT linearization approach generates an asymmetric matrix, which is different from the standard MODFLOW formulation that generates a symmetric matrix. Because all linear solvers presently available for use with MODFLOW-2005 solve only symmetric matrices, MODFLOW-NWT includes two previously developed asymmetric matrix-solver options. The matrix-solver options include a generalized-minimum-residual (GMRES) Solver and an Orthomin / stabilized conjugate-gradient (CGSTAB) Solver. The GMRES Solver is documented in a previously published report, such that only a brief description and input instructions are provided in this report. However, the CGSTAB Solver (called χMD) is documented in this report.Flow-property input for the UPW Package is designed based on the LPF Package and material-property input is identical to that for the LPF Package except that the rewetting and vertical-conductance correction options of the LPF Package are not available with the UPW Package. Input files constructed for the LPF Package can be used with slight modification as input for the UPW Package. This report presents the theory and methods used by MODFLOW-NWT, including the UPW Package. Additionally, this report provides comparisons of the new methodology to analytical solutions of groundwater flow and to standard MODFLOW-2005 results by use of an unconfined aquifer MODFLOW example problem. The standard MODFLOW-2005 simulation uses the LPF Package with the wet/dry option active. A new example problem also is presented to demonstrate MODFLOW-NWT's ability to provide a sol...

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Section: Brief Outline Of Theorymentioning

confidence: 99%

MODFLOW-NWT, a Newton formulation for MODFLOW-2005

Niswonger¹,

Panday²,

Ibaraki³

2011

Techniques and Methods

328

257

View full text Add to dashboard Cite

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“…To optimize cache usage, it is possible to use, for example, the Reverse Cuthill McKee (RCMK) reordering heuristic [14,9]. As shown in Figure 3, the RCMK tries to compress data around the matrix-diagonal, which improves cache-hit when sequentially retrieving data.…”

Section: Reordering For Better Cache Optimalitymentioning

confidence: 99%

Concurrent number cruncher: a GPU implementation of a general sparse linear solver

Buatois

Caumon²,

Lévy

2009

International Journal of Parallel, Emergent and Distributed Sys

116

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A wide class of numerical methods needs to solve a linear system, where the matrix pattern of non-zero coefficients can be arbitrary. These problems can greatly benefit from highly multithreaded computational power and large memory bandwidth available on GPUs, especially since dedicated general purpose APIs such as CTM (AMD-ATI) and CUDA (NVIDIA) have appeared. CUDA even provides a BLAS implementation, but only for dense matrices (CuBLAS). Other existing linear solvers for the GPU are also limited by their internal matrix representation.This paper describes how to combine recent GPU programming techniques and new GPU dedicated APIs with high performance computing strategies (namely block compressed row storage, register blocking and vectorization), to implement a sparse general-purpose linear solver. Our implementation of the Jacobi-preconditioned Conjugate Gradient algorithm outperforms by up to a factor of 6.0x leading-edge CPU counterparts, making it attractive for applications which content with single precision.

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“…In our experiments we make a comparison between two common reordering methods. The first method is the symmetric reverse CuthillMcKee reordering (SYMRCM) [6] and the second method is the approximate minimum degree (AMD) reordering [7].…”

Section: Short Review On Ilu Preconditionersmentioning

confidence: 99%

Simulation of Large Interconnect Structures Using ILU-Type Preconditioner

Harutyunyan

Schoenmaker

Schilders

2010

Scientific Computing in Electrical Engineering SCEE 2008

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For a fast simulation of interconnect structures we consider preconditioned iterative solution methods for large complex valued linear systems. In many applications the discretized equations result in ill-conditioned matrices, and efficient preconditioners are indispensable to solve the linear systems accurately. We apply the dual threshold incomplete LU (ILUT) factorization as preconditioners for the BICGSTAB iterative solver. On complicated problems with a different range of frequencies we show that the BICGSTAB method with the ILUT preconditioner provides a very accurate solution of the linear systems.

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Reducing the bandwidth of sparse symmetric matrices

Cited by 1,085 publications

References 12 publications

MODFLOW-NWT, a Newton formulation for MODFLOW-2005

MODFLOW-NWT, a Newton formulation for MODFLOW-2005

Concurrent number cruncher: a GPU implementation of a general sparse linear solver

Simulation of Large Interconnect Structures Using ILU-Type Preconditioner

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