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mod_lorenz63_inc4dvar.f90
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mod_lorenz63_inc4dvar.f90
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module mod_lorenz63_inc4dvar
!$$$ program documentation block
! . . .
! program name: mod_lorenz63_3dvar
! programmer: da,cheng org: umd date: 2015-Jan-11
!
! purpose:
! incremental 4D-Var system with multi-outerloop
! CVT is used
!
! revision history:
! 2015-Jan-11 da - creator
!
! file dependencies:
!
! attributes:
! language: fortran 90
! machine :
!
!
!$$$ end documentation block
use mod_type, only : rdef, rdp
!use mod_math, only : inv=>invmtx, sqrtm=>sqrtmtx
use mod_math, only : inv, sqrtm=>sqrtmtx
use mod_optimization, only : run_lbfgs, destroy_lbfgs
implicit none
private
public :: lorenz63_inc4dvar
!real(rdp),parameter :: epsl = 1.d-7
real(rdp),parameter :: epsl = 1.d-5
contains
!
! Incremental 4D-Var with control variable transform
!
subroutine lorenz63_inc4dvar( nx, nyo, nda, xb, B, lyo, yo, erro, xa, kitermax_in, kitermax_out, dt, ndt, para)
use mod_lorenz63_fwd, only : lorenz63_rk4, tl_lorenz63_rk4, adj_lorenz63_rk4
implicit none
! passed args
integer,intent(in ) :: nx
integer,intent(in ) :: nyo
integer,intent(in ) :: nda
real(8),intent(in ) :: xb(nx)
real(8),intent(in ) :: B(nx,nx)
logical,intent(in ) :: lyo(nyo,nda)
real(8),intent(in ) :: yo(nyo,nda)
real(8),intent(in ) :: erro(nyo,nda)
real(8),intent( out) :: xa(nx)
integer,intent(in ) :: kitermax_in
integer,intent(in ) :: kitermax_out
real(8),intent(in ) :: dt
integer,intent(in ) :: ndt
real(8),intent(in ) :: para(nx)
! local vars
real(8) :: invB(nx,nx)
real(8) :: invR(nyo,nda)
real(8) :: Jc, Jo, Jb
real(8) :: dJc(nx), dJo(nx), dJb(nx)
real(8) :: ygt(nyo) ! H(xg)
real(8) :: dyg(nyo,nda) ! yobs - h(xg)
real(8) :: dy(nyo,nda) ! yobs- h(xg)-Hdx
real(8) :: xg(nx) ! reference state at first step
real(8) :: xgt0(nx,nda) ! reference state at all time slots
real(8) :: dx(nx) ! x-xg
real(8) :: dxt0(nx,nda)
real(8) :: tmpx(nx)
real(8) :: forcing0(nx)
real(8) :: tmpforcing0(nx)
real(8) :: dub(nx)
real(8) :: du(nx), dv(nx)
real(8) :: L(nx,nx), invL(nx,nx)
real(8) :: gnorm, xnorm
real(8) :: t
integer :: ierr
integer :: i, j, m, n
logical :: lfound
call sqrtm(nx,B,L)
!call inv(nx,L,invL,ierr)
call inv(nx,L,invL)
do j = 1, nda
do i = 1, nyo
invR(i,j) = 1.d0/( erro(i,j)*erro(i,j) )
enddo
enddo
! we transform with Ldu = dx
xg = xb
outer_loop: do m = 1, kitermax_out
print*, "outerloop=", m
! advance NWP & calculate nonlinear H(x)
xgt0(1:nx,1) = xg
do j = 1, nda
if ( j>1 ) then
!call step( xgt0(:,j-1), t, dt, xgt0(:,j) )
call lorenz63_rk4( xgt0(:,j-1), xgt0(:,j), para, ndt, dt )
endif
do i = 1, nyo
! <FIXME>: y=h(xg)
ygt(i) = xgt0(i,j)
dyg(i,j) = yo(i,j) - ygt(i) ! (yo_i - h_i(M(xg)))
enddo
enddo
! calculate dub = L^-1 * dxb = L^-1 * (xb - xg )
dub = matmul(invL,xb-xg)
! innter loop to get increment on the reference state
lfound = .false.
du = 0.d0
inner_loop: do n = 1, kitermax_in
! calculate Jb = 0.5* (du-dub)^t * (du-dub)
! du = L^-1 * dx
Jb = 0.d0
do i = 1, nx
Jb = Jb + 0.5d0*(du(i)-dub(i))**2
enddo
! calculate Jo = sum { 0.5 * ( yo_i-h_i(x_i) ) * R^-1 * ( yo_i-h_i(x_i) ) }
! ~= sum { 0.5 * ( yo_i-h_i(N(xg))- HMdx ) * R^-1 * ( yo_i-h_i(N(xg))- HMdx ) }
! ~= sum { 0.5 * ( yo_i-h_i(N(xg))- HMLdu ) * R^-1 * ( yo_i-h_i(N(xg))- HMLdu ) } used here
dx = matmul(L,du)
Jo = 0.d0
dxt0(1:nx,1) = dx
do j = 1, nda
! advance perturb with TLM
if (j>1) then
!call tl_step(dxt0(:,j-1),xgt0(:,j-1),t,dt,dxt0(:,j) )
call tl_lorenz63_rk4( xin9 = xgt0(:,j-1), &
xin = dxt0(:,j-1), &
xout9 = tmpx(:), &
xout = dxt0(:,j), &
para = para, &
nt = ndt, &
dt = dt )
endif
! HMdx
do i = 1, nyo
if ( lyo(i,j) ) then
! ----------
! add Hd(x) here
! ----------
dy(i,j) = dyg(i,j) - dxt0(i,j) ! modify here
Jo = Jo + 0.5d0*dy(i,j)*dy(i,j)*invR(i,j)
endif
enddo
enddo
! calculate total cost function Jc = Jb + Jo
Jc = Jb + Jo
! calculate gradient dJb = I^-1 * (du-dub)
dJb = 0.0d0
do i = 1, nx
dJb(i) = du(i) - dub(i)
enddo
! calculate gradient dJo = -L^t * sum { M^t * H^t * R^-1 * ( yo - h(xg+dx) ) }
! ~= -L^t * sum { M^t * H^t * R^-1 * ( yo - h(xg) - Hdx ) }
! ~= -L^t * sum { M^t * H^t * R^-1 * ( yo - h(xg) - HLdu ) } used here
dJo = 0.d0
dv = 0.d0 ! dv = sum { M^t * H^t * R^-1 * ( yo - h(xg) - HLdu ) }
! dJo = -L^t * dv
! do j = nda, 1, -1
!
! forcing0 = 0.d0
! do i = 1, nyo
! if ( lyo(i,j) ) then
! forcing0(i) = dy(i,j)*invR(i,j)
! endif
! enddo
! if (j>1) then
! do i = j, 2, -1
! call ad_step(forcing0, xgt0(:,i-1),t,dt,forcing0)
! enddo
! endif
! dv = dv + forcing0(1:nx)
!
! enddo
! add obs misfit advected from previous step with adjoint model.
! this approach saves computational cost
forcing0 = 0.d0
do j = nda, 1, -1
do i = 1, nyo
if ( lyo(i,j) ) then
forcing0(i) = forcing0(i)+dy(i,j)*invR(i,j)
endif
enddo
if (j>1) then
!call ad_step(forcing0, xgt0(:,j-1),t,dt,forcing0)
call adj_lorenz63_rk4( xin9 = xgt0(:,j-1), &
xin = tmpforcing0, &
xout9 = tmpx(:), &
xout = forcing0, &
para = para, &
nt = ndt, &
dt = dt)
forcing0 = tmpforcing0
endif
enddo
dv = dv + forcing0(1:nx)
do i = 1, nx
do j = 1, nx
dJo(i) = dJo(i) + L(j,i)*dv(j) ! L(j,i)*dv(j) = L^t(i,j)*dv(j)
enddo
enddo
dJo(:) = -dJo(:)
! calculate total gradient
dJc = dJo + dJb
call run_lbfgs( nx, du, Jc, dJc, epsl, ierr )
!gnorm = sqrt(dot_product(dJc,dJc))
!xnorm = max(sqrt(dot_product(du,du)),1.d0)
!if (gnorm/xnorm<epsl) then
! print*, "[msg] gnorm/xnorm<epsl, skip this outerloop: g/x,epsl=", gnorm, 1.d-10
! ierr = 0
!endif
if ( ierr < 0 ) Stop "[err] inc4dvar: fail to run lbfgs."
if ( ierr == 0 ) then
call destroy_lbfgs()
lfound = .true.
exit inner_loop
endif
enddo inner_loop
if (.not.lfound) then
print*, "[warning] inc4dvar: reach the maximum iteration=", kitermax_in
call destroy_lbfgs()
endif
xg = xg + matmul(L,du)
enddo outer_loop
xa = xg
print*, "finished"
endsubroutine
endmodule