subroutine da_minimise_lz(grid, config_flags, & it, cv_size, xbx, be, iv, & j_grad_norm_target, xhat, qhat, cv, & re, y, j_cost, eignvec, eignval, neign) !------------------------------------------------------------------------- ! Purpose: Main Lanczos minimisation routine ! ! Here ! cv is updated in outer-loop. ! xhat is the control variable in inner-loop. ! ! Called from da_solve ! ! History: 07/30/2008 Creation (Tom Auligne) ! ! Reference: Golub and Van Loan 1996 (p493) ! !------------------------------------------------------------------------- implicit none type(domain), intent(inout) :: grid type(grid_config_rec_type), intent(inout) :: config_flags integer, intent(in) :: it ! external iteration. integer, intent(in) :: cv_size ! Total cv size type (xbx_type),intent(inout) :: xbx ! Header & non-gridded vars. type (be_type), intent(in) :: be ! background error structure. type (iv_type), intent(inout) :: iv ! ob. increment vector. real, intent(inout) :: j_grad_norm_target ! Target norm. real, intent(out) :: xhat(1:cv_size) ! control variable (local). real, intent(out) :: qhat(1:cv_size, 0:ntmax) ! cv copy. real, intent(inout) :: cv(1:cv_size) ! control variable (local). type (y_type), intent(inout) :: re ! residual (o-a) structure. type (y_type), intent(inout) :: y ! y = H(x_inc) structure. type (j_type), intent(out) :: j_cost ! cost function real*8, intent(out) :: eignvec(ntmax, ntmax) real*8, intent(out) :: eignval(ntmax) integer, intent(out) :: neign integer :: i, j, k, n, inst, iter #ifdef CLOUD_CV integer :: mz(11), info, nsstwrk #else integer :: mz(7), info, nsstwrk #endif integer :: npred, ipred integer :: jp_start, jp_end ! Start/end indices of Jp. real :: fhat(1:cv_size) ! cv copy. real :: ghat(1:cv_size) ! cv copy. real :: ghat0(1:cv_size) ! cv copy. real :: shat(1:cv_size) ! cv copy. real :: gdot, rho, mu, bndlm real :: alpha(ntmax), beta(0:ntmax) real*8 :: subdiag(ntmax) real :: c(cv_size), d(ntmax) real*8 :: sstwrk(2*ntmax-2) real :: bnds(ntmax) real :: adj_rhs, adj_sum_rhs ! < cv, cv > real :: j_total, j0_total real :: dfs, dfsmax if (trace_use) call da_trace_entry("da_minimise_lz") write(unit=stdout,fmt='(A)') 'Minimize cost function using Lanczos method' write(unit=stdout,fmt=*) ' ' !------------------------------------------------------------------------- ! [1.0] Initialization: !------------------------------------------------------------------------- #ifdef CLOUD_CV mz = (/ be%v1%mz, be%v2%mz, be%v3%mz, be%v4%mz, be%v5%mz, be%v6%mz, be%v7%mz, be%v8%mz, be%v9%mz, be%alpha%mz, be % ne /) #else mz = (/ be%v1%mz, be%v2%mz, be%v3%mz, be%v4%mz, be%v5%mz,be%alpha%mz, be % ne /) #endif jp_start = be % cv % size_jb + be % cv % size_je + 1 jp_end = be % cv % size_jb + be % cv % size_je + be % cv % size_jp call da_calculate_j(it, 0, cv_size, be%cv%size_jb, be%cv%size_je, be%cv%size_jp, & be%cv%size_jl, xbx, be, iv, xhat, cv, re, y, j_cost, ghat0, grid, config_flags) j0_total = j_cost%total if (j0_total == 0.0) return ghat = - ghat0 beta(0) = SQRT(da_dot_cv(cv_size, ghat, ghat, grid, mz, jp_start, jp_end)) qhat(:,0) = 0.0 j_grad_norm_target = beta(0) write(unit=stdout,fmt='("Starting outer iteration : ",i3)') it write(unit=stdout,fmt=11) j0_total, beta(0),eps(it) 11 format('Starting cost function: ' ,1PD15.8,', Gradient: ',1PD15.8,/,& 'For this outer iteration, the targeted reduction is: ',1PD15.8) write(unit=stdout,fmt='(A)') & '----------------------------------------------------------' write(unit=stdout,fmt='(A)') 'Iter Cost Function Info Content' !------------------------------------------------------------------------- ! [2.0] Iteratively solve for minimum of cost function: !------------------------------------------------------------------------- do iter=1, ntmax qhat(:,iter) = ghat / beta(iter-1) call da_calculate_gradj(it,iter,cv_size,be%cv%size_jb,be%cv%size_je,be%cv%size_jp, & be%cv%size_jl, xbx,be,iv,qhat(:,iter),y,fhat,grid,config_flags) ! Apply Lanczos recurrence and orthonormalize new gradient (using modified Gramm-Schmidt) !---------------------------------------------------------------------------------------- alpha(iter) = da_dot_cv(cv_size, qhat(:,iter), fhat, grid, mz, jp_start, jp_end) ghat = fhat - alpha(iter)*qhat(:,iter) - beta(iter-1)*qhat(:,iter-1) do i = iter, 1, -1 gdot = da_dot_cv(cv_size, ghat, qhat(:,i), grid, mz, jp_start, jp_end) ghat = ghat - gdot * qhat(:,i) end do beta(iter) = SQRT(da_dot_cv (cv_size, ghat, ghat, grid, mz, jp_start, jp_end)) ! Lanczos iteration !------------------ if (iter == 1) then d(1) = alpha(1) c = qhat(:,1) rho = beta(0) / alpha(1) xhat = rho * qhat(:,1) dfs = da_dot_cv(cv_size, xhat, xhat, grid, mz, jp_start, jp_end) dfsmax = dfs else mu = beta(iter-1) / d(iter-1) d(iter) = alpha(iter) - beta(iter-1)*mu c = qhat(:,iter) - mu*c rho = - mu*d(iter-1)*rho / d(iter) xhat = xhat + rho*c dfs = da_dot_cv(cv_size, rho*c, rho*c, grid, mz, jp_start, jp_end) if (dfs > dfsmax) dfsmax = dfs end if ! Determine eigenvalues and eigenvectors of the Lanczos tri-diagonal matrix !-------------------------------------------------------------------------- eignval(1:iter) = alpha(1:iter) subdiag(1:iter-1) = beta(1:iter-1) nsstwrk = MAX(2*iter-2,1) info = 0 call DSTEQR('I',iter,eignval(1:iter),subdiag(1:iter-1),eignvec(:,1:iter),ntmax,& sstwrk(1:nsstwrk),info) if (info /=0) write(stdout,*) 'Error in Lanczos minimization: SSTEQR returned ',info ! Count converged eigenpairs (using Arnoldi relation) !---------------------------------------------------- bndlm = eps(it) * eignval(iter) bnds(1:iter) = abs(beta(iter) * eignvec(iter,1:iter)) neign = COUNT(bnds(1:iter) <= bndlm) ! Print Cost Function and Infomation Content !----------------------------------- j_total = j0_total + 0.5 * da_dot_cv(cv_size,ghat0,xhat,grid,mz,jp_start,jp_end) write(unit=stdout,fmt=12)iter, j_total, dfs 12 format(i3,5x,1PD15.8,5x,1PD15.8) ! Stop minimization based on Information Content ! ---------------------------------------------- if (sqrt(dfs) < eps(it) * sqrt(dfsmax)) exit ! Option to calculate J Cost Function explicitly ! ---------------------------------------------- if (calculate_cg_cost_fn) & call da_calculate_j(it, iter, cv_size, be%cv%size_jb, be%cv%size_je, be%cv%size_jp, & be%cv%size_jl, xbx, be, iv, xhat, cv, re, y, j_cost, fhat, grid, config_flags) end do !------------------------------------------------------------------------- ! End of the minimization of cost function !------------------------------------------------------------------------- iter = MIN(iter, ntmax) write(unit=stdout,fmt='(A)') & '----------------------------------------------------------' write(unit=stdout,fmt='(A)') " " write(unit=stdout, fmt='("Inner iteration stopped after ",i4," iterations")') iter write(unit=stdout,fmt='(A)') " " call da_calculate_j(it, iter, cv_size, be%cv%size_jb, be%cv%size_je, be%cv%size_jp, & be%cv%size_jl, xbx, be, iv, xhat, cv, re, y, j_cost, ghat, grid, config_flags) write(unit=stdout,fmt=15) iter, j_cost%total , & SQRT(da_dot_cv(cv_size, ghat, ghat, grid, mz, jp_start, jp_end)) 15 format('Final: ',I3,' iter, J=',1PD15.8,', g=',1PD15.8) write(unit=stdout,fmt='(A)') & '----------------------------------------------------------' write(stdout,*) 'Ritz eigenvalues: ',eignval(iter:1:-1) write(stdout,*) 'Number of converged eigenpairs: ', neign neign = iter if (trace_use) call da_trace_exit("da_minimise_lz") end subroutine da_minimise_lz