Blocked-Davidson algorithm: Difference between revisions

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* Move on to the next block <math>\{ \psi^1_k| k=n_1+1,..,2 n_1\}</math>.
* Move on to the next block <math>\{ \psi^1_k| k=n_1+1,..,2 n_1\}</math>.
* When {{TAG|LDIAG}}=.TRUE. (default), a Rayleigh-Ritz optimization in the complete subspace <math>\{ \psi_k| k=1,..,N_{\rm bands}\}</math> is performed after all orbitals have been optimized.
* When {{TAG|LDIAG}}=.TRUE. (default), a Rayleigh-Ritz optimization in the complete subspace <math>\{ \psi_k| k=1,..,N_{\rm bands}\}</math> is performed after all orbitals have been optimized.
The blocksize <math>n_1</math> used in the blocked-Davidson algorithm can be set by means of the {{TAG|NSIM}} tag.
In principle <math>n_1= 2\times</math> {{TAG|NSIM}}, but for technical reasons it needs to be dividable by an integer ''N'':
:<math>
n_1 = {\rm int}\left(\frac{2*{\rm NSIM} + N - 1}{N}\right) N
</math>
where <math>N</math> is the "number of band groups per k-point group":
:<math>
N = \frac{{\rm \#\; of\; MPI\; ranks}}{{\rm IMAGES}*{\rm KPAR}*{\rm NCORE}}
</math>
(see [[Parallelization#Basic_parallelization|the section on parallelization basics]]).
As mentioned before, the optimization of a block of orbitals is stopped when either the maximum iteration depth ({{TAG|NRMM}}), or a certain convergence threshold has been reached. The latter may be fine-tuned by means of the {{TAG|EBREAK}}, {{TAG|DEPER}}, and {{TAG|WEIMIN}} tags. Note: we do not recommend you to do so! Rather rely on the defaults instead.


The blocked-Davidson algorithm is approximately a factor of 1.5-2 slower than the [[RMM-DIIS]], but more robust.
The blocked-Davidson algorithm is approximately a factor of 1.5-2 slower than the [[RMM-DIIS]], but more robust.

Latest revision as of 12:31, 14 November 2023

The workflow of the blocked-Davidson iterative matrix diagonalization scheme implemented in VASP is as follows:[1][2]

  • Take a subset (block) of orbitals out of the total set of NBANDS orbitals:
.
  • Extend the subspace spanned by by adding the preconditioned residual vectors of :
  • Rayleigh-Ritz optimization ("subspace rotation") within the -dimensional space spanned by , to determine the lowest eigenvectors:
  • Extend the subspace with the residuals of :
  • Rayleigh-Ritz optimization ("subspace rotation") within the -dimensional space spanned by :
  • If need be the subspace may be extended by repetition of this cycle of adding residual vectors and Rayleigh-Ritz optimization of the resulting subspace:
Per default VASP will not iterate deeper than , though it may break off even sooner when certain criteria that measure the convergence of the orbitals have been met.
  • When the iteration is finished, store the optimized block of orbitals back into the set:
.
  • Move on to the next block .
  • When LDIAG=.TRUE. (default), a Rayleigh-Ritz optimization in the complete subspace is performed after all orbitals have been optimized.


The blocksize used in the blocked-Davidson algorithm can be set by means of the NSIM tag. In principle NSIM, but for technical reasons it needs to be dividable by an integer N:

where is the "number of band groups per k-point group":

(see the section on parallelization basics).

As mentioned before, the optimization of a block of orbitals is stopped when either the maximum iteration depth (NRMM), or a certain convergence threshold has been reached. The latter may be fine-tuned by means of the EBREAK, DEPER, and WEIMIN tags. Note: we do not recommend you to do so! Rather rely on the defaults instead.

The blocked-Davidson algorithm is approximately a factor of 1.5-2 slower than the RMM-DIIS, but more robust.

References