Shift-and-invert parallel spectral transformation eigensolver: Massively parallel performance for density-functional...

Area of Science:

• Computational Physics and Chemistry
• Materials Science
• High-Performance Computing

Background:

• Solving large sparse eigenvalue problems is computationally intensive, crucial for electronic structure calculations in materials science.
• Existing methods often struggle with scalability on massively parallel architectures.
• Density-Functional Based Tight-Binding (DFTB) requires efficient diagonalization of large matrices.

Purpose of the Study:

• To develop and demonstrate the Shift-and-invert parallel spectral transformations (SIPs) for solving sparse eigenvalue problems.
• To showcase the parallel scalability and robustness of SIPs on massively parallel systems.
• To apply SIPs to real-world materials science problems, specifically DFTB calculations.

Main Methods:

• Implementation of Shift-and-invert parallel spectral transformations (SIPs) on massively parallel architectures.
• Diagonalization of Hamiltonian and overlap matrices for various carbon nanotube and diamond crystal structures.
• Analysis of weak and strong scaling performance, and comparison with other computational methods.
• Investigation of matrix ordering techniques to optimize factorization cost.
• Development of a parallel density matrix assembly from distributed eigenvectors.

Main Results:

• SIPs demonstrated exceptional parallel scalability and robustness for large-scale eigenvalue problems.
• Efficient computation of hundreds of thousands of eigenvalues and eigenfunctions for multi-thousand atom systems (e.g., 128,000 atoms) in seconds.
• Successful parallel implementation of density matrix assembly, crucial for iterative electronic structure calculations.

Conclusions:

• SIPs provide a powerful and scalable computational approach for tackling large sparse eigenvalue problems in materials science.
• The method's performance on massively parallel systems, like Blue Gene/Q, enables unprecedented simulation sizes.
• SIPs offer a significant advancement for electronic structure calculations using methods like DFTB.