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 -