This package implements gradient-based probing for fast electronic structure calculations. It is written in C++, and supports CUDA, multi-threading, and MPI acceleration. Running on one or more GPUs is highly recommended. A modern GPU will typically accelerate this code by 100x.
Given a sparse Hamiltonian matrix H in a real-space, orthogonal basis, the
goal of FastKPM is to calculate elements of the density matrix D at a
computational cost that scales linearly with system size N. The density matrix
is defined as D = f(H) where f is the Fermi function. Although D might be
dense, typical applications require only a sparse set of matrix elements D_ij.
The Kernel Polynomial Method (KPM)
achieves linear scaling by employing two tricks: (1) Expanding the Fermi
function f up to some order M of Chebyshev polynomials, and (2) Using a
stochastic approximation to estimate D. This approximation is unbiased, and
can be controlled by a tuneable parameter S. The total computational cost to
estimate density matrix elements scales as O(N M S).
Saad and collaborators introduced a probing
technique to significantly reduce the
stochastic approximation error as a function of S. Their approach directly
leverages the fact that the density matrix elements D_ij decay with spatial
distance between orbitals i and j in a real-space basis. Our gradient-based
probing builds on prior work to achieve even
faster convergence.
The hardest system to simulate is a zero temperature metal, because the
electronic wavefunctions decay only polynomially. In this case, the
gradient-based probing error scales as S^{-(d+2)/(2d)}, where d is the
spatial dimension. This is much faster than in the original KPM method, for
which the stochastic error scales like sqrt(S).
Building is handled with CMake.
CMake requires the following libraries: Armadillo, Boost > 1.55.
CMake will use the following libraries, if it can find them: fftw, CUDA, TBB, MPI.
To tell CMake where it can find a dependency, use the command cmake -D CMAKE_PREFIX_PATH=...
CMake will compile a fastkpm library, which can then be linked into binaries. For usage examples, please see the Kondo package.
If you find FastKPM useful, please cite this paper:
@article{doi:10.1063/1.5017741,
author = {Wang, Zhentao and Chern, Gia-Wei and Batista, Cristian D. and Barros, Kipton},
title = {Gradient-based stochastic estimation of the density matrix},
journal = {J. Chem. Phys.},
volume = {148},
pages = {094107},
year = {2018},
}
The primary authors are Kipton Barros (LANL) and Zhentao Wang (ZJU).