Iterative methods for cyclically reduced nonselfadjoint linear systems

Elman, Howard C.; Golub, Gene H.

doi:10.1090/s0025-5718-1990-1011442-x

Cited by 24 publications

(8 citation statements)

References 27 publications

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“…In the early 1990s Elman and Golub offered a thorough analysis of the spectral properties and the convergence behavior of linear systems arising from a procedure of one step of cyclic reduction 8–10. Using red‐black ordering and eliminating all the unknowns corresponding to one of the two colors yields a linear system associated with a diamond‐shaped 9‐point computational molecule.…”

Section: Introductionmentioning

confidence: 99%

Protein Contacts and Ligand Binding in the Inward‐Facing Model of Human P‐Glycoprotein

2013

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The primary aim of this work was to analyze the contacts between residues in the nucleotide binding domains (NBDs) and at the interface between the transmembrane domains (TMDs) and the NBDs in the inward-open homology model of human P-glycoprotein (P-gp). The analysis revealed communication nets through hydrogen bonding in the NBD and at the NBD-TMD interface of each half involving residues from the adenosine triphosphate (ATP) motifs and the coupling helices of the intracellular loops. Similar networks have been identified in P-gp conformations generated by molecular dynamics simulation. Differences have been recorded in the networking between both halves of P-gp. Many of the residue contacts have also been observed in the X-ray crystal structures of other ATP binding cassette (ABC) transporters, which confirms their validity. Next, possible binding pockets involving residues of importance for the TMD-NBD communication were identified. By studying these pockets, binding sites were suggested for rhodamine 123 (R-site) and prazosin (regulatory site) at the NBD-TMD interface that agreed with the experimental data on their location. Additionally, one more R-site in the protein cavity was proposed, in accordance with the available biochemical data. Together with the previously suggested Hoechst 33342 site (H-site), all sites were interpreted with respect to their effects on the protein ATPase activity, in correspondence with the experimental observations. Several residues involved in key contacts in the P-gp NBDs were proposed for further targeted mutagenesis experiments.

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Section: Introductionmentioning

confidence: 99%

Protein Contacts and Ligand Binding in the Inward‐Facing Model of Human P‐Glycoprotein

2013

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“…We explore the effect of k-line and k-plane orderings on the convergence of iterative solvers applied to (7) and (8). For convenience, let us define k-line and k-plane orderings within the context of general block grid orderings.…”

Section: Block Grid Orderingsmentioning

confidence: 99%

“…A bound on the spectral radius of the 2D cyclically reduced matrix associated with the difference equation (3), with a diagonal one-line ordering strategy, is given by [8] …”

Section: Comparison Of Convergence Rates (2d)mentioning

confidence: 99%

“…As a result, when comparing the iterative solution procedure for the original system to that for the cyclically reduced system, the overall performance of solvers depends almost exclusively on the spectral structure of the reduced versus the unreduced operators, and not on the cost of a single iteration. The analysis and numerical experiments in [8][9][10] show that iterative solvers for the reduced system converge faster, and hence it pays off to perform one step of cyclic reduction in the 2D case.…”

Section: Introductionmentioning

confidence: 98%

“…In the early 1990s Elman and Golub offered a thorough analysis of the spectral properties and the convergence behavior of linear systems arising from a procedure of one step of cyclic reduction [8][9][10]. Using red-black ordering and eliminating all the unknowns corresponding to one of the two colors yields a linear system associated with a diamond-shaped 9-point computational molecule.…”

Section: Introductionmentioning

confidence: 99%

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A box-shaped cyclically reduced operator

Greif

Hocking

2010

Numer. Linear Algebra Appl.

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We propose a new procedure of partial cyclic reduction, where we apply a 2 d -color ordering (with d = 2, 3 the dimension of the problem), and use different operators for different gridpoints according to their color. These operators are chosen so that the gridpoints can be readily decoupled, and we then eliminate all colors but one. This yields a smaller cartesian mesh and box-shaped 9-point (in 2D) or 27-point (in 3D) operators that are easy to analyze and implement. Multi-line and multi-plane orderings are considered, and we perform convergence analysis and numerical experiments that demonstrate the merits of our approach.

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On the convergence of line iterative methods for cyclically reduced non-symmetrizable linear systems

Elman

Golub

Starke³

1994

Numerische Mathematik

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We derive analytic bounds on the convergence factors associated with block relaxation methods for solving the discrete two-dimensional convection-diffusion equation. The analysis applies to the reduced systems derived when one step of block Gaussian elimination is performed on red-black ordered two-cyclic discretizations. We consider the case where centered finite difference discretization is used and one cell Reynolds number is less than one in absolute value and the other is greater than one. It is shown that line ordered relaxation exhibits very fast rates of convergence.

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Iterative methods for cyclically reduced nonselfadjoint linear systems

Cited by 24 publications

References 27 publications

Protein Contacts and Ligand Binding in the Inward‐Facing Model of Human P‐Glycoprotein

Protein Contacts and Ligand Binding in the Inward‐Facing Model of Human P‐Glycoprotein

A box-shaped cyclically reduced operator

On the convergence of line iterative methods for cyclically reduced non-symmetrizable linear systems

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