Space-Filling Curves for Real-Space Electronic Structure Calculations

Area of Science:

• Computational Physics
• Materials Science
• High-Performance Computing

Background:

• Kohn-Sham calculations often involve large, sparse Hamiltonian matrices.
• Iterative eigensolvers require numerous matrix-vector multiplications, demanding efficient parallel algorithms.
• Real-space implementations face challenges with large matrix sizes and sparsity.

Purpose of the Study:

• To develop an efficient parallel sparse matrix-vector multiplication algorithm for large-scale Kohn-Sham calculations.
• To investigate the benefits of Hilbert space-filling curves (SFCs) for domain partitioning in real-space electronic structure methods.
• To improve the scalability and performance of iterative eigensolvers.

Main Methods:

• Utilizing Hilbert space-filling curves (SFCs) for grid-point partitioning to enhance data locality and communication balance.
• Implementing blockwise operations with SFCs to leverage vector-processing units on modern hardware.
• Applying the Chebyshev-filtered subspace iteration method for solving the Kohn-Sham equations.

Main Results:

• Hilbert SFCs improve locality, balance communication, and reduce overhead in parallel sparse matrix-vector multiplication.
• Blockwise operations coupled with SFCs exploit vectorization for speedup.
• Demonstrated scalability and speedup for solving Kohn-Sham equations for large silicon nanocrystals (up to 26,000 atoms).

Conclusions:

• Blockwise Hilbert SFCs offer a significant performance improvement for large-scale electronic structure calculations.
• The method enables the simulation of complex systems, revealing insights into material properties like the density of states.
• This approach advances the computational feasibility of studying quantum mechanical systems.