Block Tridiagonal Matrices in Electronic Structure Calculations
Research output: Book/Report › Ph.D. thesis › Research
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Block Tridiagonal Matrices in Electronic Structure Calculations. / Petersen, Dan Erik.
København : Department of Computer Science, University of Copenhagen, 2008. 247 p.Research output: Book/Report › Ph.D. thesis › Research
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TY - BOOK
T1 - Block Tridiagonal Matrices in Electronic Structure Calculations
AU - Petersen, Dan Erik
PY - 2008
Y1 - 2008
N2 - This thesis focuses on some of the numerical aspects of the treatmentof the electronic structure problem, in particular that of determiningthe ground state electronic density for the non–equilibriumGreen’s function formulation of two–probe systems and the calculationof transmission in the Landauer–Büttiker ballistic transportregime. These calculations concentrate on determining the so–called Green’s function matrix, or portions thereof, which is the inverseof a block tridiagonal general complex matrix.To this end, a sequential algorithm based on Gaussian eliminationnamed Sweeps is developed and compared to standard Gaussianelimination, where it is shown to be qualitatively quicker for thetask of determining the block tridiagonal portion of the Green’sfunction matrix. The Sweep algorithm is then parallelized via astraightforward approach in order to enable moderate speedup andmemory distribution.The well known block cyclic reduction algorithm first developed byGene Golub is then presented and analyzed for further expandingour parallel options, and finally a new hybrid method that combinesblock cyclic reduction and a form of Schur complement calculationis introduced.The parallel algorithms are then benchmarked and the new hybridmethod is shown to possess promising speedup characteristics forcommon cases of problems that need to be modeled.
AB - This thesis focuses on some of the numerical aspects of the treatmentof the electronic structure problem, in particular that of determiningthe ground state electronic density for the non–equilibriumGreen’s function formulation of two–probe systems and the calculationof transmission in the Landauer–Büttiker ballistic transportregime. These calculations concentrate on determining the so–called Green’s function matrix, or portions thereof, which is the inverseof a block tridiagonal general complex matrix.To this end, a sequential algorithm based on Gaussian eliminationnamed Sweeps is developed and compared to standard Gaussianelimination, where it is shown to be qualitatively quicker for thetask of determining the block tridiagonal portion of the Green’sfunction matrix. The Sweep algorithm is then parallelized via astraightforward approach in order to enable moderate speedup andmemory distribution.The well known block cyclic reduction algorithm first developed byGene Golub is then presented and analyzed for further expandingour parallel options, and finally a new hybrid method that combinesblock cyclic reduction and a form of Schur complement calculationis introduced.The parallel algorithms are then benchmarked and the new hybridmethod is shown to possess promising speedup characteristics forcommon cases of problems that need to be modeled.
M3 - Ph.D. thesis
BT - Block Tridiagonal Matrices in Electronic Structure Calculations
PB - Department of Computer Science, University of Copenhagen
CY - København
ER -
ID: 14772653