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quantum-espresso/KS_Solvers/ParO/pcg_k.f90

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! Copyright (C) 2015-2016 Aihui Zhou's group
!
! This file is distributed under the terms of the
! GNU General Public License. See the file `License'
! in the root directory of the present distribution,
! or http://www.gnu.org/copyleft/gpl.txt .
!
!
!-------------------------------------------------------------------------------
!
! We propose some parallel orbital updating based plane wave basis methods
! for electronic structure calculations, which aims to the solution of the corresponding eigenvalue
! problems. Compared to the traditional plane wave methods, our methods have the feature of two level
! parallelization, which make them have great advantage in large-scale parallelization.
!
! The approach following Algorithm is the parallel orbital updating algorithm:
! 1. Choose initial $E_{\mathrm{cut}}^{(0)}$ and then obtain $V_{N_G^{0}}$, use the SCF method to solve
! the Kohn-Sham equation in $V_{G_0}$ and get the initial $(\lambda_i^{0},u_i^{0}), i=1, \cdots, N$
! and let $n=0$.
! 2. For $i=1,2,\ldots,N$, find $e_i^{n+1/2}\in V_{G_n}$ satisfying
! $$a(\rho_{in}^{n}; e_i^{n+1/2}, v) = -[(a(\rho_{in}^{n}; u_i^{n}, v) - \lambda_i^{n} (u_i^{n}, v))] $$
! in parallel , where $\rho_{in}^{n}$ is the input charge density obtained by the orbits obtained in the
! $n$-th iteration or the former iterations.
! 3. Find $\{\lambda_i^{n+1},u_i^{n+1}\} \in \mathbf{R}\times \tilde{V}_N$ satisfying
! $$a(\tilde{\rho}; u_i^{n+1}, v) = ( \lambda_i^{n+1}u_i^{n+1}, v) \quad \forall v \in \tilde{V}_N$$
! where $\tilde{V}_N = \mathrm{span}\{e_1^{n+1/2},\ldots,e_N^{n+1/2},u_1^{n},\ldots,u_N^{n}\}$,
! $\tilde{\rho}(x)$ is the input charge density obtained from the previous orbits.
! 4. Convergence check: if not converged, set $n=n+1$, go to step 2; else, stop.
!
! You can see the detailed information through
! X. Dai, X. Gong, A. Zhou, J. Zhu,
! A parallel orbital-updating approach for electronic structure calculations, arXiv:1405.0260 (2014).
! X. Dai, Z. Liu, X. Zhang, A. Zhou,
! A Parallel Orbital-updating Based Optimization Method for Electronic Structure Calculations,
! arXiv:1510.07230 (2015).
! Yan Pan, Xiaoying Dai, Xingao Gong, Stefano de Gironcoli, Gian-Marco Rignanese, and Aihui Zhou,
! A Parallel Orbital-updating Based Plane Wave Basis Method. J. Comp. Phys. 348, 482-492 (2017).
!
! The file is written mainly by Stefano de Gironcoli and Yan Pan.
!
! The following file is for solving step 2 of the parallel orbital updating algorithm.
!
#define ZERO ( 0.D0, 0.D0 )
#define ONE ( 1.D0, 0.D0 )
!
!----------------------------------------------------------------------------
SUBROUTINE pcg_k( hs_1psi, g_1psi, psi0, spsi0, npw, npwx, nbnd, npol, psi, ethr, iter, e, nhpsi )
!----------------------------------------------------------------------------
!
! ... solve the linear system
!
! [ H - e S + lambda Pv ]|\tilde\psi> = [e S - H ] |psi>
! Pc [ H - e S ]|\tilde\psi> = Pc [ e S - H ] |psi>
!
! the solution is sought until the residual norm is a fixed fraction of the RHS norm
! in this way the more accurate is the original problem the more accuratly the correction is computed
!
USE util_param, ONLY : DP, stdout
USE mp_bands_util, ONLY : intra_bgrp_comm
USE mp, ONLY : mp_sum
!
IMPLICIT NONE
!
! Following varibales are temporary
COMPLEX(DP),INTENT(IN) :: psi0(npwx*npol,nbnd) ! psi0 needed to compute the Pv projection
COMPLEX(DP),INTENT(IN) :: spsi0(npwx*npol,nbnd) ! Spsi0 needed to compute the Pv projection
INTEGER, INTENT(IN) :: npw, npwx, nbnd, npol, iter ! input dimensions and iteration count
REAL(DP), INTENT(IN) :: ethr ! threshold for convergence.
REAL(DP), INTENT(IN) :: e ! current estimate of the target eigenvalue
COMPLEX(DP),INTENT(INOUT) :: psi(npwx*npol) ! input: the current estimate of the eigenvector,
! output: the estimated correction vector
INTEGER, INTENT(INOUT) :: nhpsi ! (updated) number of Hpsi evaluations
!
! ... LOCAL variables
!
INTEGER, PARAMETER :: maxter = 5 ! maximum number of CG iterations
!
COMPLEX(DP), ALLOCATABLE :: hpsi(:), & ! the product of H and psi
spsi(:), & ! the product of S and psi
b(:), & ! RHS for testing
r(:), p(:), sp(:), w(:),z(:) ! additional working vetors
COMPLEX(DP), ALLOCATABLE :: spsi0vec (:) ! the product of spsi0 and a trial vector
REAL(DP) :: g0, g1, g2, beta, alpha, gamma, ethr_cg, ff, ff0
INTEGER :: kdim, kdmx, cg_iter, ibnd
!
REAL(DP), EXTERNAL :: DDOT
COMPLEX(DP), EXTERNAL :: ZDOTC
EXTERNAL hs_1psi, g_1psi
! hs_1psi( npwx, npw, psi, hpsi, spsi )
!
CALL start_clock( 'pcg' )
!write (6,*) ' enter pcg' , e, 'npol = ', npol
!
kdim = npwx*(npol-1) + npw
kdmx = npwx* npol
!
ALLOCATE( hpsi( kdmx ), spsi( kdmx ), r( kdmx ), z( kdmx ), b( kdmx ) )
ALLOCATE( spsi0vec(nbnd) )
!
CALL start_clock( 'pcg:hs_1psi' )
CALL hs_1psi( npwx, npw, psi, hpsi, spsi ) ! apply H and S to a single wavefunction (no bgrp parallelization inside)
CALL stop_clock( 'pcg:hs_1psi' )
! define CG algorithm RHS and initial solution
r(:) = e * spsi(:) - hpsi(:) ! initial gradient
z(:) = r(:) ; call g_1psi(npwx,npw,z,e) ! initial preconditioned gradient
!- project on conduction bands
CALL start_clock( 'pcg:ortho' )
CALL ZGEMV( 'C', kdim, nbnd, ONE, spsi0, kdmx, z, 1, ZERO, spsi0vec, 1 )
CALL mp_sum( spsi0vec, intra_bgrp_comm )
CALL ZGEMV( 'N', kdim, nbnd, (-1.D0,0.D0), psi0, kdmx, spsi0vec, 1, ONE, z, 1 )
CALL stop_clock( 'pcg:ortho' )
!-
g0 = ZDOTC( kdim, z ,1 ,r ,1)
CALL mp_sum( g0, intra_bgrp_comm ) ! g0 = < initial z | initial r >
ff = 0.d0 ; ff0 = ff
!write (6,*) 0, g0, ff
ALLOCATE( p( kdmx ), sp( kdmx ), w( kdmx ) )
! ethr_cg = ethr ! CG convergence threshold could be set from input but it is not ...
ethr_cg = 1.0D-2 ! it makes more sense to fix the convergence of the CG solution to a
! fixed function of the RHS (see ethr_cg update later).
ethr_cg = max ( 0.01*ethr, ethr_cg * g0 ) ! here we set the convergence of the correction
! save RHS for later
b(:) = r(:)
! zero the trial solution: comment next line is you are looking for |\psi_new> = |\psi> + |\tilde \psi>
psi(:) = ZERO
! initial search direction
p(:) = z(:)
iterate: &
DO cg_iter = 1, maxter
CALL start_clock( 'pcg:hs_1psi' )
CALL hs_1psi( npwx, npw, p, w, sp ) ! apply H to a single wavefunction (no bgrp parallelization here!)
CALL stop_clock( 'pcg:hs_1psi' )
w = w - e* sp
gamma = ZDOTC( kdim, p ,1 ,w ,1)
CALL mp_sum( gamma, intra_bgrp_comm )
alpha = g0/gamma
psi(:) = psi(:) + alpha * p(:) ! updated solution
r(:) = r(:) - alpha * w(:) ! updated gradient
g2 = ZDOTC( kdim, z ,1 ,r ,1)
CALL mp_sum( g2, intra_bgrp_comm ) ! g2 = < old z | new r >
z(:) = r(:) ; call g_1psi(npwx,npw,z,e) ! updated preconditioned gradient
!- project on conduction bands
CALL start_clock( 'pcg:ortho' )
CALL ZGEMV( 'C', kdim, nbnd, ONE, spsi0, kdmx, z, 1, ZERO, spsi0vec, 1 )
CALL mp_sum( spsi0vec, intra_bgrp_comm )
CALL ZGEMV( 'N', kdim, nbnd, (-1.D0,0.D0), psi0, kdmx, spsi0vec, 1, ONE, z, 1 )
CALL stop_clock( 'pcg:ortho' )
!-
g1 = ZDOTC( kdim, z, 1, r ,1)
CALL mp_sum( g1, intra_bgrp_comm ) ! g1 = < new z | new r >
! evaluate the function
ff = - 0.5_DP * (ZDOTC( kdim, psi, 1, r ,1) + ZDOTC( kdim, psi, 1, b ,1) )
CALL mp_sum( ff, intra_bgrp_comm )
!write (6,*) cg_iter, g1, ff, gamma
if ( ff > ff0 .AND. ff0 < 0.d0 ) psi(:) = psi(:) - alpha * p(:) ! fallback solution if last iteration failed to improve the function... exit and hope next time it'll be better
IF ( ABS ( g1 ) < ethr_cg .OR. ( ff > ff0 ) ) EXIT iterate
beta = (g1-g2)/g0 ! Polak - Ribiere style update
g0 = g1 ! < new z | new r > -> < old z | old r >
p(:) = z(:) + beta * p(:) ! updated search direction
ff0 = ff ! updated minimum value reached by the function
END DO iterate
!write (6,*) ' exit pcg loop'
! orthogonalize to psi0 ...
! actually we are not doing that.. it would require both psi0 AND spsi0 to be computed and would
! remove an occupied orb contribution which is taken care of by the following rotate_wfc routine anyway
!if ( cg_iter == maxter.and. ABS(g1) > ethr_cg) &
! write (*,*) 'CG not converged maxter exceeded', cg_iter, g1, g0, ethr_cg
!IF ( ABS ( g1 ) < ethr_cg) write (*,*) 'CG correction converged ', cg_iter, g1, g0, ethr_cg
!IF ( ABS ( g1 ) > ethr_cg) write (*,*) 'CG not converged ', cg_iter, g1, g0, ethr_cg
nhpsi = nhpsi + cg_iter + 1
!
DEALLOCATE( spsi0vec )
DEALLOCATE( r, p, sp, w, z )
DEALLOCATE( hpsi, spsi )
!
CALL stop_clock( 'pcg' )
!
RETURN
!
END SUBROUTINE pcg_k
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