BSE solvers overview

This tutorial explores the options available in Yambo for the numerical solution of the Bethe-Salpeter equation. Before running this tutorial, you should have carried out the tutorial Calculating optical spectra including excitonic effects: a step-by-step guide. We will use the same example system (bulk hBN). The various options correspond to different numerical methods for solving (large) eigenvalue problems. The numerical methods are not covered in this tutorial and the interested user is directed to the references.

After completion of this tutorial, you are expected to recognize which solver to use depending on the system, computational load and desired output. Also, for each of the solver, you should be able to choose the values for the key input parameters.

Three main solvers are available in Yambo:

Full diagonalization solver (-y d)

Diagonalization of the full Hamiltonian can be done in serial, using the standard LAPACK library. Also, it is possible to do full diagonalization asking only for a subset of eigenvalues of eigenvectors.

This solver outputs the macroscopic dielectric function over a range of energies chosen by the user. It also calculates all exciton energies and composition in terms of electron-hole pairs. Upon user's choice, the latter is stored on disk for use in postprocessing for exciton analysis and plotting as seen in the tutorial How to analyse excitons.

The computational cost of full diagonalization is cubic in the number of electron-hole pairs so this solver can be used for small system only (few light atoms per unit cell). However, thanks to modern libraries like scalapack and ELPA, it is nowadays possible to fully diagonalize very large BSE matrices exploiting MPI and GPUs.

In Lumen we have two working interfaces with Scalapack and one with ELPA. The modern one, which requires the interface with the ldiago library developed by M. Nalabothula, is controlled by

BSSmod="d"

The old one, which was recently revised to include memory distribution, but it is still significantly less efficient, is controlled by

BSSmod="o"

An example of a tutorial doing full diagonalization can be found here: Bethe-Salpeter solver: diagonalization

Lanczos-Haydock solver (-y h)

Subspace iterative | Lanczos algorithm which by-pass diagonalization with the Haydock approach.

This solver outputs the macroscopic dielectric function over a range of energies chosen by the user. It can be applied to medium-large systems as it is substantially cheaper than the full diagonalization (quadratic with the number of electron-hole pairs). However, this approach does not provide the individual exciton energies and composition in terms of electron-hole pairs.

This solver is controlled by

BSSmod="h"

An example of a tutorial using Haydock can be found here: Bethe-Salpeter solver: Lanczos-Haydock
This solver is also compatible with the double-grid, see tutorial: Fast convergence of dielectric constant with double grid.

SLEPc solver (-y s)

Subspace iterative algorithm using the SLEPc library (SLEPC solver).

This solver outputs the individual exciton energies and composition about an energy value chosen by the user. It can be applied to medium-large systems but does not provide the overall spectrum. This solver is typically used after obtaining the spectra from the Lanczos-Haydock solver to either analyze a specific peak or getting the energy and composition of a few low-lying excitons. It can be used for large-medium systems. The computational cost grows with the number of excitons one asks to compute.

This solver is controlled by

BSSmod="s"

Important: To exploit the slepc solver, you need to link the code with Slepc and Petsc library. See here for more info: [Install Lumen with external slepc and petsc libraries].

An example of tutorial using Slepc can be found here: Bethe-Salpeter solver: SLEPC

ChASE solver (-y c)

This solver is controlled by

BSSmod="c"

Important: To exploit the chase solver, you need to link the code with the ChASE library. See here for more info: [Install Lumen with external chase libraries].

TODO: add a tutorial and few more info

INVERSION solver (-y i)

The inversion solver was designed to invert the Dyson equation frequency by frequency. For each frequency the inversion (which can be done with Lapack) has roughly the same cost of a diagonalization. Needing to do so for every frequency, this solver easily becomes very expensive. It was designed to deal with frequency dependent kernel, and also extended with an old implementation of the double-grid see Ref.. However, it is not used much anymore.

This solver is controlled by

BSSmod="i"

Summary

A first summary about, the output (dielectric function, exciton energies and composition) and cost of the three solvers are summarized in this table.

Also some performances on the performance of solvers able to extract the eigenvectors on a matrix of size 61440 x 61440 (28,125 GB). Requested eigenvalues: 500. Runs on 100 CPU (4 nodesx25 cores, 500 GB), BLACS 10x10

Solver :    Slepc    -   Chase      - Ldiago(ELPA2) - Ldiago(ELPA1) - Ldiago(heevr) - Odiago(heevx)
25920  :  104.7396s  -  127.4608s   -    53.8467s   -   212.3914s   -   240.1606s   -   1045.s
61440  :  499.0757s  -  497.0590s   -   361.0130s   -     1269.s    -     2134.s    -  12660.s

As you can see, the new interface with diagonalization library (Ldiago) using ELPA2 solver on CPU (unfortunately ELPA2 is not GPU ported) is even more efficient that iterative solvers like Slepc and Chase. However, it has a larger memory footprint (peak memory is about 6 times the BSE matrix). The timing of the Haydock solver is not reported, but it remains the fastest method.

More info about the performances of different solvers can be found in this Merge Request: Performances of BSE solvers

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