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|
!
! Copyright (C) 2003-2010 Quantum ESPRESSO 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 .
!
!----------------------------------------------------------------------------
MODULE bfgs_module
!----------------------------------------------------------------------------
!
! ... Ionic relaxation through the Newton-Raphson optimization scheme
! ... based on the Broyden-Fletcher-Goldfarb-Shanno algorithm for the
! ... estimate of the inverse Hessian matrix.
! ... The ionic relaxation is performed converting cartesian (and cell)
! ... positions into internal coordinates.
! ... The algorithm uses a "trust radius" line search based on Wolfe
! ... conditions. Steps are rejected until the first Wolfe condition
! ... (sufficient energy decrease) is satisfied. Updated step length
! ... is estimated from quadratic interpolation.
! ... When the step is accepted inverse hessian is updated according to
! ... BFGS scheme and a new search direction is obtained from NR or GDIIS
! ... method. The corresponding step length is limited by trust_radius_max
! ... and can't be larger than the previous step multiplied by a certain
! ... factor determined by Wolfe and other convergence conditions.
!
! ... Originally written ( 5/12/2003 ) and maintained ( 2003-2007 ) by
! ... Carlo Sbraccia
! ... Modified for variable-cell-shape relaxation ( 2007-2008 ) by
! ... Javier Antonio Montoya, Lorenzo Paulatto and Stefano de Gironcoli
! ... Re-analyzed by Stefano de Gironcoli ( 2010 )
!
! ... references :
!
! ... 1) Roger Fletcher, Practical Methods of Optimization, John Wiley and
! ... Sons, Chichester, 2nd edn, 1987.
! ... 2) Salomon R. Billeter, Alexander J. Turner, Walter Thiel,
! ... Phys. Chem. Chem. Phys. 2, 2177 (2000).
! ... 3) Salomon R. Billeter, Alessandro Curioni, Wanda Andreoni,
! ... Comput. Mat. Science 27, 437, (2003).
! ... 4) Ren Weiqing, PhD Thesis: Numerical Methods for the Study of Energy
! ... Landscapes and Rare Events.
!
!
USE kinds, ONLY : DP
USE io_files, ONLY : iunbfgs, prefix
USE constants, ONLY : eps16
USE cell_base, ONLY : iforceh
!
USE basic_algebra_routines
!
IMPLICIT NONE
!
PRIVATE
!
! ... public methods
!
PUBLIC :: bfgs, terminate_bfgs
!
! ... public variables
!
PUBLIC :: bfgs_ndim, &
trust_radius_ini, trust_radius_min, trust_radius_max, &
w_1, w_2
!
! ... global module variables
!
SAVE
!
CHARACTER (len=8) :: fname="energy" ! name of the function to be minimized
!
REAL(DP), ALLOCATABLE :: &
pos(:), &! positions + cell
grad(:), &! gradients + cell_force
pos_p(:), &! positions at the previous accepted iteration
grad_p(:), &! gradients at the previous accepted iteration
inv_hess(:,:), &! inverse hessian matrix (updated using BFGS formula)
metric(:,:), &
h_block(:,:), &
hinv_block(:,:), &
step(:), &! the (new) search direction (normalized NR step)
step_old(:), &! the previous search direction (normalized NR step)
pos_old(:,:), &! list of m old positions - used only by gdiis
grad_old(:,:), &! list of m old gradients - used only by gdiis
pos_best(:) ! best extrapolated positions - used only by gdiis
REAL(DP) :: &
nr_step_length, &! length of (new) Newton-Raphson step
nr_step_length_old,&! length of previous Newton-Raphson step
trust_radius, &! new displacement along the search direction
trust_radius_old, &! old displacement along the search direction
energy_p ! energy at previous accepted iteration
INTEGER :: &
scf_iter, &! number of scf iterations
bfgs_iter, &! number of bfgs iterations
gdiis_iter, &! number of gdiis iterations
tr_min_hit = 0 ! set to 1 if the trust_radius has already been
! set to the minimum value at the previous step
! set to 2 if trust_radius is reset again: exit
LOGICAL :: &
conv_bfgs ! .TRUE. when bfgs convergence has been achieved
!
! ... default values for the following variables are set in
! ... Modules/read_namelist.f90 (SUBROUTINE ions_defaults)
!
! ... Note that trust_radius_max, trust_radius_min, trust_radius_ini,
! ... w_1, w_2, bfgs_ndim have a default value, but can also be assigned
! ... in the input.
!
INTEGER :: &
bfgs_ndim ! dimension of the subspace for GDIIS
! fixed to 1 for standard BFGS algorithm
REAL(DP) :: &
trust_radius_ini, &! suggested initial displacement
trust_radius_min, &! minimum allowed displacement
trust_radius_max ! maximum allowed displacement
REAL(DP) :: &! parameters for Wolfe conditions
w_1, &! 1st Wolfe condition: sufficient energy decrease
w_2 ! 2nd Wolfe condition: sufficient gradient decrease
!
CONTAINS
!
!------------------------------------------------------------------------
SUBROUTINE bfgs( pos_in, h, energy, grad_in, fcell, fixion, scratch, stdout,&
energy_thr, grad_thr, cell_thr, energy_error, grad_error, &
cell_error, istep, nstep, step_accepted, stop_bfgs, lmovecell )
!------------------------------------------------------------------------
!
! ... list of input/output arguments :
!
! pos : vector containing 3N coordinates of the system ( x )
! energy : energy of the system ( V(x) )
! grad : vector containing 3N components of grad( V(x) )
! fixion : vector used to freeze a deg. of freedom
! scratch : scratch directory
! stdout : unit for standard output
! energy_thr : treshold on energy difference for BFGS convergence
! grad_thr : treshold on grad difference for BFGS convergence
! the largest component of grad( V(x) ) is considered
! energy_error : energy difference | V(x_i) - V(x_i-1) |
! grad_error : the largest component of
! | grad(V(x_i)) - grad(V(x_i-1)) |
! cell_error : the largest component of: omega*(stress-press*I)
! nstep : the maximun nuber of scf-steps
! step_accepted : .TRUE. if a new BFGS step is done
! stop_bfgs : .TRUE. if BFGS convergence has been achieved
!
IMPLICIT NONE
!
REAL(DP), INTENT(INOUT) :: pos_in(:)
REAL(DP), INTENT(INOUT) :: h(3,3)
REAL(DP), INTENT(INOUT) :: energy
REAL(DP), INTENT(INOUT) :: grad_in(:)
REAL(DP), INTENT(INOUT) :: fcell(3,3)
INTEGER, INTENT(IN) :: fixion(:)
CHARACTER(LEN=*), INTENT(IN) :: scratch
INTEGER, INTENT(IN) :: stdout
REAL(DP), INTENT(IN) :: energy_thr, grad_thr, cell_thr
INTEGER, INTENT(OUT) :: istep
INTEGER, INTENT(IN) :: nstep
REAL(DP), INTENT(OUT) :: energy_error, grad_error, cell_error
LOGICAL, INTENT(OUT) :: step_accepted, stop_bfgs
LOGICAL, INTENT(IN) :: lmovecell
!
INTEGER :: n, i, j, k, nat
LOGICAL :: lwolfe
REAL(DP) :: dE0s, den
! ... for scaled coordinates
REAL(DP) :: hinv(3,3),g(3,3),ginv(3,3),garbage, omega
!
!
lwolfe=.false.
n = SIZE( pos_in ) + 9
nat = size (pos_in) / 3
if (nat*3 /= size (pos_in)) call errore('bfgs',' strange dimension',1)
!
! ... work-space allocation
!
ALLOCATE( pos( n ) )
ALLOCATE( grad( n ) )
!
ALLOCATE( grad_old( n, bfgs_ndim ) )
ALLOCATE( pos_old( n, bfgs_ndim ) )
!
ALLOCATE( inv_hess( n, n ) )
!
ALLOCATE( pos_p( n ) )
ALLOCATE( grad_p( n ) )
ALLOCATE( step( n ) )
ALLOCATE( step_old( n ) )
ALLOCATE( pos_best( n ) )
! ... scaled coordinates work-space
ALLOCATE( hinv_block( n-9, n-9 ) )
! ... cell related work-space
ALLOCATE( metric( n , n ) )
!
! ... the BFGS file read (pos & grad) in scaled coordinates
!
call invmat(3, h, hinv, omega)
! volume is defined to be positve even for left-handed vector triplet
omega = abs(omega)
!
hinv_block = 0.d0
FORALL ( k=0:nat-1, i=1:3, j=1:3 ) hinv_block(i+3*k,j+3*k) = hinv(i,j)
!
! ... generate metric to work with scaled ionic coordinates
g = MATMUL(TRANSPOSE(h),h)
call invmat(3,g,ginv,garbage)
metric = 0.d0
FORALL ( k=0:nat-1, i=1:3, j=1:3 ) metric(i+3*k,j+3*k) = g(i,j)
FORALL ( k=nat:nat+2, i=1:3, j=1:3 ) metric(i+3*k,j+3*k) = 0.04 * omega * ginv(i,j)
!
! ... generate bfgs vectors for the degrees of freedom and their gradients
pos = 0.0
pos(1:n-9) = pos_in
if (lmovecell) FORALL( i=1:3, j=1:3) pos( n-9 + j+3*(i-1) ) = h(i,j)
grad = 0.0
grad(1:n-9) = grad_in
if (lmovecell) FORALL( i=1:3, j=1:3) grad( n-9 + j+3*(i-1) ) = fcell(i,j)*iforceh(i,j)
!
! if the cell moves the quantity to be minimized is the enthalpy
IF ( lmovecell ) fname="enthalpy"
!
CALL read_bfgs_file( pos, grad, fixion, energy, scratch, n, stdout )
!
scf_iter = scf_iter + 1
istep = scf_iter
!
! ... convergence is checked here
!
energy_error = ABS( energy_p - energy )
grad_error = MAXVAL( ABS( MATMUL( TRANSPOSE(hinv_block), grad(1:n-9)) ) )
conv_bfgs = energy_error < energy_thr
conv_bfgs = conv_bfgs .AND. ( grad_error < grad_thr )
!
IF( lmovecell) THEN
cell_error = MAXVAL( ABS( MATMUL ( TRANSPOSE ( RESHAPE( grad(n-8:n), (/ 3, 3 /) ) ),&
TRANSPOSE(h) ) ) ) / omega
conv_bfgs = conv_bfgs .AND. ( cell_error < cell_thr )
#undef DEBUG
#ifdef DEBUG
write (*,'(3f15.10)') TRANSPOSE ( RESHAPE( grad(n-8:n), (/ 3, 3 /) ) )
write (*,*)
write (*,'(3f15.10)') TRANSPOSE(h)
write (*,*)
write (*,'(3f15.10)') MATMUL (TRANSPOSE( RESHAPE( grad(n-8:n), (/ 3, 3 /) ) ),&
TRANSPOSE(h) ) / omega
write (*,*)
write (*,*) cell_error/cell_thr*0.5d0
#endif
END IF
!
! ... converged (or useless to go on): quick return
!
conv_bfgs = conv_bfgs .OR. ( tr_min_hit > 1 )
IF ( conv_bfgs ) GOTO 1000
!
! ... some output is written
!
WRITE( UNIT = stdout, &
& FMT = '(/,5X,"number of scf cycles",T30,"= ",I3)' ) scf_iter
WRITE( UNIT = stdout, &
& FMT = '(5X,"number of bfgs steps",T30,"= ",I3,/)' ) bfgs_iter
IF ( scf_iter > 1 ) WRITE( UNIT = stdout, &
& FMT = '(5X,A," old",T30,"= ",F18.10," Ry")' ) fname,energy_p
WRITE( UNIT = stdout, &
& FMT = '(5X,A," new",T30,"= ",F18.10," Ry",/)' ) fname,energy
!
! ... the bfgs algorithm starts here
!
IF ( .NOT. energy_wolfe_condition( energy ) .AND. (scf_iter > 1) ) THEN
!
! ... the previous step is rejected, line search goes on
!
step_accepted = .FALSE.
!
WRITE( UNIT = stdout, &
& FMT = '(5X,"CASE: ",A,"_new > ",A,"_old",/)' ) fname,fname
!
! ... the new trust radius is obtained by quadratic interpolation
!
! ... E(s) = a*s*s + b*s + c ( we use E(0), dE(0), E(s') )
!
! ... s_min = - 0.5*( dE(0)*s'*s' ) / ( E(s') - E(0) - dE(0)*s' )
!
if (abs(scnorm(step_old(:))-1._DP) > 1.d-10) call errore('bfgs', &
' step_old is NOT normalized ',1)
! (normalized) search direction is the same as in previous step
step(:) = step_old(:)
!
dE0s = ( grad_p(:) .dot. step(:) ) * trust_radius_old
IF (dE0s > 0._DP ) CALL errore( 'bfgs', &
'dE0s is positive which should never happen', 1 )
den = energy - energy_p - dE0s
!
! estimate new trust radius by interpolation
trust_radius = - 0.5_DP*dE0s*trust_radius_old / den
!
WRITE( UNIT = stdout, &
& FMT = '(5X,"new trust radius",T30,"= ",F18.10," bohr")' ) &
trust_radius
!
! ... values from the last succeseful bfgs step are restored
!
pos(:) = pos_p(:)
energy = energy_p
grad(:) = grad_p(:)
!
IF ( trust_radius < trust_radius_min ) THEN
!
! ... the history is reset ( the history can be reset at most two
! ... consecutive times )
!
WRITE( UNIT = stdout, &
FMT = '(/,5X,"trust_radius < trust_radius_min")' )
WRITE( UNIT = stdout, FMT = '(/,5X,"resetting bfgs history",/)' )
!
! ... if tr_min_hit=1 the history has already been reset at the
! ... previous step : something is going wrong
!
IF ( tr_min_hit == 1 ) THEN
CALL infomsg( 'bfgs', &
'history already reset at previous step: stopping' )
tr_min_hit = 2
ELSE
tr_min_hit = 1
END IF
!
CALL reset_bfgs( n )
!
step(:) = - ( inv_hess(:,:) .times. grad(:) )
! normalize step but remember its length
nr_step_length = scnorm(step)
step(:) = step(:) / nr_step_length
!
trust_radius = min(trust_radius_ini, nr_step_length)
!
ELSE
!
tr_min_hit = 0
!
END IF
!
ELSE
!
! ... a new bfgs step is done
!
bfgs_iter = bfgs_iter + 1
!
IF ( bfgs_iter == 1 ) THEN
!
! ... first iteration
!
step_accepted = .FALSE.
!
ELSE
!
step_accepted = .TRUE.
!
nr_step_length_old = nr_step_length
!
WRITE( UNIT = stdout, &
& FMT = '(5X,"CASE: ",A,"_new < ",A,"_old",/)' ) fname,fname
!
CALL check_wolfe_conditions( lwolfe, energy, grad )
!
CALL update_inverse_hessian( pos, grad, n, stdout )
!
END IF
! compute new search direction and store NR step length
IF ( bfgs_ndim > 1 ) THEN
!
! ... GDIIS extrapolation
!
CALL gdiis_step()
!
ELSE
!
! ... standard Newton-Raphson step
!
step(:) = - ( inv_hess(:,:) .times. grad(:) )
!
END IF
IF ( ( grad(:) .dot. step(:) ) > 0.0_DP ) THEN
!
WRITE( UNIT = stdout, &
FMT = '(5X,"uphill step: resetting bfgs history",/)' )
!
CALL reset_bfgs( n )
step(:) = - ( inv_hess(:,:) .times. grad(:) )
!
END IF
!
! normalize the step and save the step length
nr_step_length = scnorm(step)
step(:) = step(:) / nr_step_length
!
! ... the new trust radius is computed
!
IF ( bfgs_iter == 1 ) THEN
!
trust_radius = min(trust_radius_ini, nr_step_length)
tr_min_hit = 0
!
ELSE
!
CALL compute_trust_radius( lwolfe, energy, grad, n, stdout )
!
END IF
!
WRITE( UNIT = stdout, &
& FMT = '(5X,"new trust radius",T30,"= ",F18.10," bohr")' ) &
trust_radius
!
END IF
!
! ... step along the bfgs direction
!
IF ( nr_step_length < eps16 ) &
CALL errore( 'bfgs', 'NR step-length unreasonably short', 1 )
!
! ... information required by next iteration is saved here ( this must
! ... be done before positions are updated )
!
CALL write_bfgs_file( pos, energy, grad, scratch )
!
! ... positions and cell are updated
!
pos(:) = pos(:) + trust_radius * step(:)
!
1000 stop_bfgs = conv_bfgs .OR. ( scf_iter >= nstep )
! ... input ions+cell variables
IF ( lmovecell ) FORALL( i=1:3, j=1:3) h(i,j) = pos( n-9 + j+3*(i-1) )
pos_in = pos(1:n-9)
! ... update forces
grad_in = grad(1:n-9)
!
! ... work-space deallocation
!
DEALLOCATE( pos )
DEALLOCATE( grad )
DEALLOCATE( pos_p )
DEALLOCATE( grad_p )
DEALLOCATE( pos_old )
DEALLOCATE( grad_old )
DEALLOCATE( inv_hess )
DEALLOCATE( step )
DEALLOCATE( step_old )
DEALLOCATE( pos_best )
DEALLOCATE( hinv_block )
DEALLOCATE( metric )
!
RETURN
!
CONTAINS
!
!--------------------------------------------------------------------
SUBROUTINE gdiis_step()
!--------------------------------------------------------------------
USE basic_algebra_routines
IMPLICIT NONE
!
REAL(DP), ALLOCATABLE :: res(:,:), overlap(:,:), work(:)
INTEGER, ALLOCATABLE :: iwork(:)
INTEGER :: k, k_m, info
REAL(DP) :: gamma0
!
!
gdiis_iter = gdiis_iter + 1
!
k = MIN( gdiis_iter, bfgs_ndim )
k_m = k + 1
!
ALLOCATE( res( n, k ) )
ALLOCATE( overlap( k_m, k_m ) )
ALLOCATE( work( k_m ), iwork( k_m ) )
!
work(:) = 0.0_DP
iwork(:) = 0
!
! ... the new direction is added to the workspace
!
DO i = bfgs_ndim, 2, -1
!
pos_old(:,i) = pos_old(:,i-1)
grad_old(:,i) = grad_old(:,i-1)
!
END DO
!
pos_old(:,1) = pos(:)
grad_old(:,1) = grad(:)
!
! ... |res_i> = H^-1 \times |g_i>
!
CALL DGEMM( 'N', 'N', n, k, n, 1.0_DP, &
inv_hess, n, grad_old, n, 0.0_DP, res, n )
!
! ... overlap_ij = <grad_i|res_j>
!
CALL DGEMM( 'T', 'N', k, k, n, 1.0_DP, &
res, n, res, n, 0.0_DP, overlap, k_m )
!
overlap( :, k_m) = 1.0_DP
overlap(k_m, : ) = 1.0_DP
overlap(k_m,k_m) = 0.0_DP
!
! ... overlap is inverted via Bunch-Kaufman diagonal pivoting method
!
CALL DSYTRF( 'U', k_m, overlap, k_m, iwork, work, k_m, info )
CALL DSYTRI( 'U', k_m, overlap, k_m, iwork, work, info )
CALL errore( 'gdiis_step', 'error in Bunch-Kaufman inversion', info )
!
! ... overlap is symmetrised
!
FORALL( i = 1:k_m, j = 1:k_m, j > i ) overlap(j,i) = overlap(i,j)
!
pos_best(:) = 0.0_DP
step(:) = 0.0_DP
!
DO i = 1, k
!
gamma0 = overlap(k_m,i)
!
pos_best(:) = pos_best(:) + gamma0*pos_old(:,i)
!
step(:) = step(:) - gamma0*res(:,i)
!
END DO
!
! ... the step must be consistent with the last positions
!
step(:) = step(:) + ( pos_best(:) - pos(:) )
!
IF ( ( grad(:) .dot. step(:) ) > 0.0_DP ) THEN
!
! ... if the extrapolated direction is uphill use only the
! ... last gradient and reset gdiis history
!
step(:) = - ( inv_hess(:,:) .times. grad(:) )
!
gdiis_iter = 0
!
END IF
!
DEALLOCATE( res, overlap, work, iwork )
!
END SUBROUTINE gdiis_step
!
END SUBROUTINE bfgs
!
!------------------------------------------------------------------------
SUBROUTINE reset_bfgs( n )
!------------------------------------------------------------------------
! ... inv_hess in re-initalized to the initial guess
! ... defined as the inverse metric
!
INTEGER, INTENT(IN) :: n
!
REAL(DP) :: garbage
!
call invmat(n, metric, inv_hess, garbage)
!
gdiis_iter = 0
!
END SUBROUTINE reset_bfgs
!
!------------------------------------------------------------------------
SUBROUTINE read_bfgs_file( pos, grad, fixion, energy, scratch, n, stdout )
!------------------------------------------------------------------------
!
IMPLICIT NONE
!
REAL(DP), INTENT(INOUT) :: pos(:)
REAL(DP), INTENT(INOUT) :: grad(:)
INTEGER, INTENT(IN) :: fixion(:)
CHARACTER(LEN=*), INTENT(IN) :: scratch
INTEGER, INTENT(IN) :: n
INTEGER, INTENT(IN) :: stdout
REAL(DP), INTENT(INOUT) :: energy
!
CHARACTER(LEN=256) :: bfgs_file
LOGICAL :: file_exists
REAL(DP) :: garbage
!
!
bfgs_file = TRIM( scratch ) // TRIM( prefix ) // '.bfgs'
!
INQUIRE( FILE = TRIM( bfgs_file ) , EXIST = file_exists )
!
IF ( file_exists ) THEN
!
! ... bfgs is restarted from file
!
OPEN( UNIT = iunbfgs, FILE = TRIM( bfgs_file ), &
STATUS = 'UNKNOWN', ACTION = 'READ' )
!
READ( iunbfgs, * ) pos_p
READ( iunbfgs, * ) grad_p
READ( iunbfgs, * ) scf_iter
READ( iunbfgs, * ) bfgs_iter
READ( iunbfgs, * ) gdiis_iter
READ( iunbfgs, * ) energy_p
READ( iunbfgs, * ) pos_old
READ( iunbfgs, * ) grad_old
READ( iunbfgs, * ) inv_hess
READ( iunbfgs, * ) tr_min_hit
READ( iunbfgs, * ) nr_step_length
!
CLOSE( UNIT = iunbfgs )
!
step_old = ( pos(:) - pos_p(:) )
trust_radius_old = scnorm( step_old )
step_old = step_old / trust_radius_old
!
ELSE
!
! ... bfgs initialization
!
WRITE( UNIT = stdout, FMT = '(/,5X,"BFGS Geometry Optimization")' )
!
! initialize the inv_hess to the inverse of the metric
call invmat(n, metric, inv_hess, garbage)
!
pos_p = 0.0_DP
grad_p = 0.0_DP
scf_iter = 0
bfgs_iter = 0
gdiis_iter = 0
energy_p = energy
step_old = 0.0_DP
nr_step_length = 0.0_DP
!
trust_radius_old = trust_radius_ini
!
pos_old(:,:) = 0.0_DP
grad_old(:,:) = 0.0_DP
!
tr_min_hit = 0
!
END IF
!
END SUBROUTINE read_bfgs_file
!
!------------------------------------------------------------------------
SUBROUTINE write_bfgs_file( pos, energy, grad, scratch )
!------------------------------------------------------------------------
!
IMPLICIT NONE
!
REAL(DP), INTENT(IN) :: pos(:)
REAL(DP), INTENT(IN) :: energy
REAL(DP), INTENT(IN) :: grad(:)
CHARACTER(LEN=*), INTENT(IN) :: scratch
!
!
OPEN( UNIT = iunbfgs, FILE = TRIM( scratch )//TRIM( prefix )//'.bfgs', &
STATUS = 'UNKNOWN', ACTION = 'WRITE' )
!
WRITE( iunbfgs, * ) pos
WRITE( iunbfgs, * ) grad
WRITE( iunbfgs, * ) scf_iter
WRITE( iunbfgs, * ) bfgs_iter
WRITE( iunbfgs, * ) gdiis_iter
WRITE( iunbfgs, * ) energy
WRITE( iunbfgs, * ) pos_old
WRITE( iunbfgs, * ) grad_old
WRITE( iunbfgs, * ) inv_hess
WRITE( iunbfgs, * ) tr_min_hit
WRITE( iunbfgs, * ) nr_step_length
!
CLOSE( UNIT = iunbfgs )
!
END SUBROUTINE write_bfgs_file
!
!------------------------------------------------------------------------
SUBROUTINE update_inverse_hessian( pos, grad, n, stdout )
!------------------------------------------------------------------------
!
IMPLICIT NONE
!
REAL(DP), INTENT(IN) :: pos(:)
REAL(DP), INTENT(IN) :: grad(:)
INTEGER, INTENT(IN) :: n
INTEGER, INTENT(IN) :: stdout
INTEGER :: info
!
REAL(DP), ALLOCATABLE :: y(:), s(:)
REAL(DP), ALLOCATABLE :: Hy(:), yH(:)
REAL(DP) :: sdoty, sBs, Theta
REAL(DP), ALLOCATABLE :: B(:,:)
!
ALLOCATE( y( n ), s( n ), Hy( n ), yH( n ) )
!
s(:) = pos(:) - pos_p(:)
y(:) = grad(:) - grad_p(:)
!
sdoty = ( s(:) .dot. y(:) )
!
IF ( ABS( sdoty ) < eps16 ) THEN
!
! ... the history is reset
!
WRITE( stdout, '(/,5X,"WARNING: unexpected ", &
& "behaviour in update_inverse_hessian")' )
WRITE( stdout, '( 5X," resetting bfgs history",/)' )
!
CALL reset_bfgs( n )
!
RETURN
!
ELSE
! Conventional Curvature Trap here
! See section 18.2 (p538-539 ) of Nocedal and Wright "Numerical
! Optimization"for instance
! LDM Addition, April 2011
!
! While with the Wolfe conditions the Hessian in most cases
! remains positive definite, if one is far from the minimum
! and/or "bonds" are being made/broken the curvature condition
! Hy = s ; or s = By
! cannot be satisfied if s.y < 0. In addition, if s.y is small
! compared to s.B.s too greedy a step is taken.
!
! The trap below is conventional and "OK", and has been around
! for ~ 30 years but, unfortunately, is rarely mentioned in
! introductory texts and hence often neglected.
!
! First, solve for inv_hess*t = s ; i.e. t = B*s
! Use yH as workspace here
ALLOCATE (B(n,n) )
B = inv_hess
yH= s
call DPOSV('U',n,1,B,n, yH, n, info)
! Info .ne. 0 should be trapped ...
if(info .ne. 0)write( stdout, '(/,5X,"WARNING: info=",i3," for Hessian")' )info
DEALLOCATE ( B )
!
! Calculate s.B.s
sBs = ( s(:) .dot. yH(:) )
!
! Now the trap itself
if ( sdoty < 0.20D0*sBs ) then
! Conventional damping
Theta = 0.8D0*sBs/(sBs-sdoty)
WRITE( stdout, '(/,5X,"WARNING: bfgs curvature condition ", &
& "failed, Theta=",F6.3)' )theta
y = Theta*y + (1.D0 - Theta)*yH
endif
END IF
!
Hy(:) = ( inv_hess .times. y(:) )
yH(:) = ( y(:) .times. inv_hess )
!
! ... BFGS update
!
inv_hess = inv_hess + 1.0_DP / sdoty * &
( ( 1.0_DP + ( y .dot. Hy ) / sdoty ) * matrix( s, s ) - &
( matrix( s, yH ) + matrix( Hy, s ) ) )
!
DEALLOCATE( y, s, Hy, yH )
!
RETURN
!
END SUBROUTINE update_inverse_hessian
!
!------------------------------------------------------------------------
SUBROUTINE check_wolfe_conditions( lwolfe, energy, grad )
!------------------------------------------------------------------------
IMPLICIT NONE
REAL(DP), INTENT(IN) :: energy
REAL(DP), INTENT(IN) :: grad(:)
LOGICAL, INTENT(OUT) :: lwolfe
!
lwolfe = energy_wolfe_condition ( energy ) .AND. &
gradient_wolfe_condition ( grad )
!
END SUBROUTINE check_wolfe_conditions
!
!------------------------------------------------------------------------
LOGICAL FUNCTION energy_wolfe_condition ( energy )
!------------------------------------------------------------------------
IMPLICIT NONE
REAL(DP), INTENT(IN) :: energy
!
energy_wolfe_condition = &
( energy-energy_p ) < w_1 * ( grad_p.dot.step_old ) * trust_radius_old
!
END FUNCTION energy_wolfe_condition
!
!------------------------------------------------------------------------
LOGICAL FUNCTION gradient_wolfe_condition ( grad )
!------------------------------------------------------------------------
IMPLICIT NONE
REAL(DP), INTENT(IN) :: grad(:)
!
gradient_wolfe_condition = &
ABS( grad .dot. step_old ) < - w_2 * ( grad_p .dot. step_old )
!
END FUNCTION gradient_wolfe_condition
!
!------------------------------------------------------------------------
SUBROUTINE compute_trust_radius( lwolfe, energy, grad, n, stdout )
!------------------------------------------------------------------------
!
IMPLICIT NONE
!
LOGICAL, INTENT(IN) :: lwolfe
REAL(DP), INTENT(IN) :: energy
REAL(DP), INTENT(IN) :: grad(:)
INTEGER, INTENT(IN) :: n
INTEGER, INTENT(IN) :: stdout
!
REAL(DP) :: a
LOGICAL :: ltest
!
ltest = ( energy - energy_p ) < w_1 * ( grad_p .dot. step_old ) * trust_radius_old
ltest = ltest .AND. ( nr_step_length_old > trust_radius_old )
!
IF ( ltest ) THEN
a = 1.5_DP
ELSE
a = 1.1_DP
END IF
IF ( lwolfe ) a = 2._DP * a
!
trust_radius = MIN( trust_radius_max, a*trust_radius_old, nr_step_length )
!
IF ( trust_radius < trust_radius_min ) THEN
!
! ... the history is reset
!
! ... if tr_min_hit the history has already been reset at the
! ... previous step : something is going wrong
!
IF ( tr_min_hit == 1 ) THEN
CALL infomsg( 'bfgs', &
'history already reset at previous step: stopping' )
tr_min_hit = 2
ELSE
tr_min_hit = 1
END IF
!
WRITE( UNIT = stdout, &
FMT = '(5X,"small trust_radius: resetting bfgs history",/)' )
!
CALL reset_bfgs( n )
step(:) = - ( inv_hess(:,:) .times. grad(:) )
!
nr_step_length = scnorm(step)
step(:) = step(:) / nr_step_length
!
trust_radius = min(trust_radius_min, nr_step_length )
!
ELSE
!
tr_min_hit = 0
!
END IF
!
END SUBROUTINE compute_trust_radius
!
!-----------------------------------------------------------------------
REAL(DP) FUNCTION scnorm1( vect )
!-----------------------------------------------------------------------
IMPLICIT NONE
REAL(DP), INTENT(IN) :: vect(:)
!
scnorm1 = SQRT( DOT_PRODUCT( vect , MATMUL( metric, vect ) ) )
!
END FUNCTION scnorm1
!
!-----------------------------------------------------------------------
REAL(DP) FUNCTION scnorm( vect )
!-----------------------------------------------------------------------
IMPLICIT NONE
REAL(DP), INTENT(IN) :: vect(:)
REAL(DP) :: ss
INTEGER :: i,k,l,n
!
scnorm = 0._DP
n = SIZE (vect) / 3
do i=1,n
ss = 0._DP
do k=1,3
do l=1,3
ss = ss + &
vect(k+(i-1)*3)*metric(k+(i-1)*3,l+(i-1)*3)*vect(l+(i-1)*3)
end do
end do
scnorm = MAX (scnorm, SQRT (ss) )
end do
!
END FUNCTION scnorm
!
!------------------------------------------------------------------------
SUBROUTINE terminate_bfgs( energy, energy_thr, grad_thr, cell_thr, &
lmovecell, stdout, scratch )
!------------------------------------------------------------------------
!
USE io_files, ONLY : prefix, delete_if_present
!
IMPLICIT NONE
REAL(DP), INTENT(IN) :: energy, energy_thr, grad_thr, cell_thr
LOGICAL, INTENT(IN) :: lmovecell
INTEGER, INTENT(IN) :: stdout
CHARACTER(LEN=*), INTENT(IN) :: scratch
!
IF ( conv_bfgs ) THEN
!
WRITE( UNIT = stdout, &
& FMT = '(/,5X,"bfgs converged in ",I3," scf cycles and ", &
& I3," bfgs steps")' ) scf_iter, bfgs_iter
IF ( lmovecell ) THEN
WRITE( UNIT = stdout, &
& FMT = '(5X,"(criteria: energy < ",ES8.1,", force < ",ES8.1, &
& ", cell < ",ES8.1,")")') energy_thr, grad_thr, cell_thr
ELSE
WRITE( UNIT = stdout, &
& FMT = '(5X,"(criteria: energy < ",ES8.1,", force < ",ES8.1, &
& ")")') energy_thr, grad_thr
END IF
WRITE( UNIT = stdout, &
& FMT = '(/,5X,"End of BFGS Geometry Optimization")' )
WRITE( UNIT = stdout, &
& FMT = '(/,5X,"Final ",A," = ",F18.10," Ry")' ) fname, energy
!
CALL delete_if_present( TRIM( scratch ) // TRIM( prefix ) // '.bfgs' )
!
ELSE
!
WRITE( UNIT = stdout, &
FMT = '(/,5X,"The maximum number of steps has been reached.")' )
WRITE( UNIT = stdout, &
FMT = '(/,5X,"End of BFGS Geometry Optimization")' )
!
END IF
!
END SUBROUTINE terminate_bfgs
!
END MODULE bfgs_module
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