! Copyright (c) 2004-2024 Lars Nerger ! ! This file is part of PDAF. ! ! PDAF is free software: you can redistribute it and/or modify ! it under the terms of the GNU Lesser General Public License ! as published by the Free Software Foundation, either version ! 3 of the License, or (at your option) any later version. ! ! PDAF is distributed in the hope that it will be useful, ! but WITHOUT ANY WARRANTY; without even the implied warranty of ! MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ! GNU Lesser General Public License for more details. ! ! You should have received a copy of the GNU Lesser General Public ! License along with PDAF. If not, see <http://www.gnu.org/licenses/>. ! !$Id$ !BOP ! ! !ROUTINE: PDAF_seik_analysis_trans --- SEIK analysis/ensemble transformation ! ! !INTERFACE: SUBROUTINE PDAF_seik_analysis_trans(step, dim_p, dim_obs_p, dim_ens, rank, & state_p, Uinv, ens_p, state_inc_p, forget, & U_init_dim_obs, U_obs_op, U_init_obs, U_init_obsvar, U_prodRinvA, & screen, incremental, type_forget, type_sqrt, flag) ! !DESCRIPTION: ! Analysis step of the SEIK filter with direct ! transformation of the forecast into the ! analysis ensemble. This variant does not ! compute the analysis state, but only the ! analysis ensemble, whose mean is the analysis ! state. ! Supported is also the adaptive forgetting factor. ! ! Variant for domain decomposed states. ! ! ! This is a core routine of PDAF and ! should not be changed by the user ! ! ! !REVISION HISTORY: ! 2009-07 - Lars Nerger - Initial code ! Later revisions - see svn log ! ! !USES: ! Include definitions for real type of different precision ! (Defines BLAS/LAPACK routines and MPI_REALTYPE) #include "typedefs.h" USE mpi USE PDAF_timer, & ONLY: PDAF_timeit USE PDAF_memcounting, & ONLY: PDAF_memcount USE PDAF_mod_filter, & ONLY: Nm1vsN, type_trans, filterstr, obs_member, observe_ens, debug USE PDAF_mod_filtermpi, & ONLY: mype, MPIerr, COMM_filter USE PDAFomi, & ONLY: omi_n_obstypes => n_obstypes, omi_omit_obs => omit_obs IMPLICIT NONE ! !ARGUMENTS: INTEGER, INTENT(in) :: step ! Current time step INTEGER, INTENT(in) :: dim_p ! PE-local dimension of model state INTEGER, INTENT(out) :: dim_obs_p ! PE-local dimension of observation vector INTEGER, INTENT(in) :: dim_ens ! Size of ensemble INTEGER, INTENT(in) :: rank ! Rank of initial covariance matrix REAL, INTENT(inout) :: state_p(dim_p) ! on exit: PE-local forecast mean state REAL, INTENT(inout) :: Uinv(rank, rank) ! Inverse of matrix U - temporary use only REAL, INTENT(inout) :: ens_p(dim_p, dim_ens) ! PE-local state ensemble REAL, INTENT(inout) :: state_inc_p(dim_p) ! PE-local state analysis increment REAL, INTENT(in) :: forget ! Forgetting factor INTEGER, INTENT(in) :: screen ! Verbosity flag INTEGER, INTENT(in) :: incremental ! Control incremental updating INTEGER, INTENT(in) :: type_forget ! Type of forgetting factor INTEGER, INTENT(in) :: type_sqrt ! Type of square-root of A ! (0): symmetric sqrt; (1): Cholesky decomposition INTEGER, INTENT(inout) :: flag ! Status flag ! ! External subroutines ! ! (PDAF-internal names, real names are defined in the call to PDAF) EXTERNAL :: U_init_dim_obs, & ! Initialize dimension of observation vector U_obs_op, & ! Observation operator U_init_obsvar, & ! Initialize mean observation error variance U_init_obs, & ! Initialize observation vector U_prodRinvA ! Provide product R^-1 A ! !CALLING SEQUENCE: ! Called by: PDAF_seek_update ! Calls: U_init_dim_obs ! Calls: U_obs_op ! Calls: U_init_obs ! Calls: U_prodRinvA ! Calls: PDAF_timeit ! Calls: PDAF_memcount ! Calls: PDAF_set_forget ! Calls: PDAF_seik_matrixT ! Calls: PDAF_seik_Uinv ! Calls: PDAF_seik_Omega ! Calls: PDAF_seik_TtimesA ! Calls: gemmTYPE (BLAS; dgemm or sgemm dependent on precision) ! Calls: gemvTYPE (BLAS; dgemv or sgemv dependent on precision) ! Calls: gesvTYPE (LAPACK; dgesv or sgesv dependent on precision) ! Calls: syevTYPE (LAPACK; dsyev or ssyev dependent on precision) ! Calls: potrfTYPE (LAPACK; dpotrf or spotrf dependent on precision) ! Calls: trtrsTYPE (LAPACK; dtrtrs or strtrs dependent on precision) ! Calls: MPI_allreduce (MPI) !EOP ! *** local variables *** INTEGER :: i, j, member, col, row ! counters INTEGER, SAVE :: allocflag = 0 ! Flag whether first time allocation is done INTEGER :: lib_info ! Status flag for LAPACK calls INTEGER :: ldwork ! Size of work array for syev INTEGER :: maxblksize, blkupper, blklower ! Variables for blocked ensemble update REAL :: invdimens ! Inverse global ensemble size REAL :: fac ! Temporary variable sqrt(dim_ens) or sqrt(rank) REAL :: forget_ana ! Forgetting factor used for analysis LOGICAL :: storeOmega = .FALSE. ! Store matrix Omega instead of recomputing it LOGICAL, SAVE :: firsttime = .TRUE.! Indicates first call to resampling REAL, ALLOCATABLE :: HL_p(:,:) ! Temporary matrices for analysis REAL, ALLOCATABLE :: RiHL_p(:,:) ! Temporary matrices for analysis REAL, ALLOCATABLE :: resid_p(:) ! PE-local observation residual REAL, ALLOCATABLE :: obs_p(:) ! PE-local observation vector REAL, ALLOCATABLE :: HXbar_p(:) ! PE-local observed state REAL, ALLOCATABLE :: RiHLd(:) ! Temporary vector for analysis REAL, ALLOCATABLE :: RiHLd_p(:) ! PE-local RiHLd REAL, ALLOCATABLE :: VRiHLd(:) ! Temporary vector for analysis REAL, ALLOCATABLE :: Usqrt(:, :) ! Square-root of matrix U REAL, ALLOCATABLE :: Uinv_p(:,:) ! Uinv for PE-local domain REAL, ALLOCATABLE :: tmp_Uinv(:,:) ! Temporary storage of Uinv REAL, ALLOCATABLE :: Omega(:,:) ! Orthogonal matrix Omega REAL, ALLOCATABLE :: OmegaT(:,:) ! Transpose of Omega REAL, SAVE, ALLOCATABLE :: OmegaTsave(:,:) ! Saved transpose of Omega REAL, ALLOCATABLE :: TA(:,:) ! Temporary matrix REAL, ALLOCATABLE :: ens_blk(:,:) ! Temporary blocked state ensemble REAL, ALLOCATABLE :: svals(:) ! Singular values of Uinv REAL, ALLOCATABLE :: work(:) ! Work array for SYEV INTEGER, ALLOCATABLE :: ipiv(:) ! vector of pivot indices for GESV INTEGER :: incremental_dummy ! Dummy variable to avoid compiler warning REAL :: state_inc_p_dummy(1) ! Dummy variable to avoid compiler warning ! ********************** ! *** INITIALIZATION *** ! ********************** CALL PDAF_timeit(51, 'new') IF (debug>0) & WRITE (*,*) '++ PDAF-debug: ', debug, 'PDAF_seik_analysis -- START' ! Initialize variable to prevent compiler warning incremental_dummy = incremental state_inc_p_dummy(1) = state_inc_p(1) IF (mype == 0 .AND. screen > 0) THEN WRITE (*, '(a, 1x, i7, 3x, a)') & 'PDAF', step, 'Assimilating observations - SEIK with ensemble transformation' END IF ! *********************************** ! *** Compute mean forecast state *** ! *********************************** CALL PDAF_timeit(11, 'new') state_p = 0.0 invdimens = 1.0 / REAL(dim_ens) DO member = 1, dim_ens DO row = 1, dim_p state_p(row) = state_p(row) + invdimens * ens_p(row, member) END DO END DO CALL PDAF_timeit(11, 'old') CALL PDAF_timeit(51, 'old') ! ********************************* ! *** Get observation dimension *** ! ********************************* IF (debug>0) THEN WRITE (*,*) '++ PDAF-debug PDAF_seik_analysis:', debug, ' dim_p', dim_p WRITE (*,*) '++ PDAF-debug: ', debug, 'PDAF_seik_analysis -- call init_dim_obs' END IF CALL PDAF_timeit(15, 'new') CALL U_init_dim_obs(step, dim_obs_p) CALL PDAF_timeit(15, 'old') IF (debug>0) & WRITE (*,*) '++ PDAF-debug PDAF_seik_analysis:', debug, ' dim_obs_p', dim_obs_p IF (screen > 2) THEN WRITE (*, '(a, 5x, a13, 1x, i6, 1x, a, i10)') & 'PDAF', '--- PE-domain', mype, 'dimension of observation vector', dim_obs_p END IF ! ************************ ! *** Compute residual *** ! *** d = y - H x *** ! ************************ CALL PDAF_timeit(12, 'new') haveobsB: IF (dim_obs_p > 0) THEN ! *** The residual only exists for domains with observations *** ALLOCATE(resid_p(dim_obs_p)) ALLOCATE(obs_p(dim_obs_p)) ALLOCATE(HXbar_p(dim_obs_p)) IF (allocflag == 0) CALL PDAF_memcount(3, 'r', 3 * dim_obs_p) ! Project state onto observation space IF (debug>0) & WRITE (*,*) '++ PDAF-debug: ', debug, 'PDAF_etkf_analysis -- observe_ens', observe_ens IF (.NOT.observe_ens) THEN IF (debug>0) & WRITE (*,*) '++ PDAF-debug: ', debug, & 'PDAF_etkf_analysis -- call obs_op for ensemble mean' obs_member = 0 ! Store member index (0 for central state) CALL PDAF_timeit(44, 'new') CALL U_obs_op(step, dim_p, dim_obs_p, state_p, HXbar_p) CALL PDAF_timeit(44, 'old') ELSE ! For nonlinear H: apply H to each ensemble state; then average ALLOCATE(HL_p(dim_obs_p, dim_ens)) IF (allocflag == 0) CALL PDAF_memcount(3, 'r', dim_obs_p * dim_ens) IF (debug>0) & WRITE (*,*) '++ PDAF-debug: ', debug, 'PDAF_seik_analysis -- call obs_op', dim_ens, 'times' CALL PDAF_timeit(44, 'new') ENS1: DO member = 1, dim_ens ! Store member index to make it accessible with PDAF_get_obsmemberid obs_member = member ! [Hx_1 ... Hx_(r+1)] CALL U_obs_op(step, dim_p, dim_obs_p, ens_p(:, member), HL_p(:, member)) END DO ENS1 CALL PDAF_timeit(44, 'old') CALL PDAF_timeit(51, 'new') HXbar_p = 0.0 DO member = 1, dim_ens DO row = 1, dim_obs_p HXbar_p(row) = HXbar_p(row) + invdimens * HL_p(row, member) END DO END DO CALL PDAF_timeit(51, 'old') END IF ! get observation vector IF (debug>0) & WRITE (*,*) '++ PDAF-debug: ', debug, 'PDAF_seik_analysis -- call init_obs' CALL PDAF_timeit(50, 'new') CALL U_init_obs(step, dim_obs_p, obs_p) CALL PDAF_timeit(50, 'old') ! get residual as difference of observation and ! projected state CALL PDAF_timeit(51, 'new') resid_p = obs_p - HXbar_p CALL PDAF_timeit(51, 'old') IF (debug>0) THEN WRITE (*,*) '++ PDAF-debug PDAF_seik_analysis:', debug, & 'innovation d(1:min(dim_obs_p,10))', resid_p(1:min(dim_obs_p,10)) WRITE (*,*) '++ PDAF-debug PDAF_seik_analysis:', debug, & 'MIN/MAX of innovation', MINVAL(resid_p), MAXVAL(resid_p) END IF ! Omit observations with too high innovation IF (omi_omit_obs) THEN CALL PDAF_timeit(51, 'new') CALL PDAFomi_omit_by_inno_cb(dim_obs_p, resid_p, obs_p) CALL PDAF_timeit(51, 'old') END IF END IF haveobsB CALL PDAF_timeit(12, 'old') ! ************************************************* ! *** Compute analyzed matrix Uinv *** ! *** *** ! *** -1 T T -1 *** ! *** U = forget*fac T T + (HL) R (HL) *** ! *** *** ! *** Here FAC is a scaling factor according *** ! *** to the definition of the ensemble *** ! *** covariance scaled by N^-1 (original SEIK) *** ! *** (N-1)^-1 (SEIK as ensemble KF) *** ! ************************************************* CALL PDAF_timeit(10, 'new') haveobsA: IF (dim_obs_p > 0) THEN ! *** The contribution of observation matrix ist only *** ! *** computed for domains with observations *** CALL PDAF_timeit(30, 'new') IF (.NOT.observe_ens) THEN ! This part is only required if H is applied to the ensemble mean before ! *** Compute HL = [Hx_1 ... Hx_N] T ALLOCATE(HL_p(dim_obs_p, dim_ens)) IF (allocflag == 0) CALL PDAF_memcount(3, 'r', dim_obs_p * dim_ens) IF (debug>0) & WRITE (*,*) '++ PDAF-debug: ', debug, 'PDAF_seik_analysis -- call obs_op', dim_ens, 'times' CALL PDAF_timeit(44, 'new') ENS: DO member = 1, dim_ens ! Store member index to make it accessible with PDAF_get_obsmemberid obs_member = member ! [Hx_1 ... Hx_N] CALL U_obs_op(step, dim_p, dim_obs_p, ens_p(:, member), HL_p(:, member)) END DO ENS CALL PDAF_timeit(44, 'old') END IF ! Set forgetting factor forget_ana = forget IF (type_forget == 1) THEN CALL PDAF_set_forget(step, filterstr, dim_obs_p, dim_ens, HL_p, & HXbar_p, obs_p, U_init_obsvar, forget, forget_ana) ENDIF DEALLOCATE(HXbar_p) ! Complete HL = [Hx_1 ... Hx_N] T CALL PDAF_timeit(51, 'new') CALL PDAF_seik_matrixT(dim_obs_p, dim_ens, HL_p) CALL PDAF_timeit(51, 'old') CALL PDAF_timeit(30, 'old') CALL PDAF_timeit(31, 'new') ! *** RiHL = Rinv HL *** ! *** this is implemented as a subroutine thus that *** ! *** Rinv does not need to be allocated explicitly *** IF (debug>0) & WRITE (*,*) '++ PDAF-debug: ', debug, 'PDAF_seik_analysis -- call prodRinvA_l' ALLOCATE(RiHL_p(dim_obs_p, rank)) IF (allocflag == 0) CALL PDAF_memcount(3, 'r', dim_obs_p * rank) CALL PDAF_timeit(48, 'new') CALL U_prodRinvA(step, dim_obs_p, rank, obs_p, HL_p, RiHL_p) CALL PDAF_timeit(48, 'old') DEALLOCATE(obs_p) CALL PDAF_timeit(51, 'new') ! *** Initialize Uinv = fac T^T T *** CALL PDAF_seik_Uinv(rank, Uinv) ! *** T *** ! *** Compute HL RiHL *** ALLOCATE(Uinv_p(rank, rank)) ALLOCATE(tmp_Uinv(rank, rank)) IF (allocflag == 0) CALL PDAF_memcount(3, 'r', 2 * rank**2) CALL gemmTYPE('t', 'n', rank, rank, dim_obs_p, & 1.0, HL_p, dim_obs_p, RiHL_p, dim_obs_p, & 0.0, Uinv_p, rank) DEALLOCATE(HL_p) ELSE haveobsA ! *** For domains with dim_obs_p=0 there is no *** ! *** direct observation-contribution to Uinv *** CALL PDAF_timeit(31, 'new') CALL PDAF_timeit(51, 'new') ! Set forgetting factor forget_ana = forget ! Initialize Uinv = fac T^T T CALL PDAF_seik_Uinv(rank, Uinv) ALLOCATE(Uinv_p(rank, rank)) ALLOCATE(tmp_Uinv(rank, rank)) IF (allocflag == 0) CALL PDAF_memcount(3, 'r', 2 * rank**2) ! No observation-contribution to Uinv from this domain Uinv_p = 0.0 ! For OMI we need to call observation operator also for dim_obs_p=0 ! in order to initialize pointer to observation type IF (omi_n_obstypes>0) THEN ALLOCATE(HL_p(1, 1)) obs_member = 1 ! [Hx_1 ... Hx_N] CALL U_obs_op(step, dim_p, dim_obs_p, ens_p(:, 1), HL_p(:, 1)) DEALLOCATE(HL_p) END IF END IF haveobsA ! get total sum on all filter PEs CALL MPI_allreduce(Uinv_p, tmp_Uinv, rank * rank, & MPI_REALTYPE, MPI_SUM, COMM_filter, MPIerr) ! *** Complete computation of Uinv *** ! *** -1 -1 T *** ! *** U = forget U + HL RiHL *** Uinv = forget_ana * Uinv + tmp_Uinv DEALLOCATE(Uinv_p) IF (debug>0) & WRITE (*,*) '++ PDAF-debug PDAF_seik_analysis:', debug, ' U^-1', Uinv CALL PDAF_timeit(51, 'old') CALL PDAF_timeit(31, 'old') CALL PDAF_timeit(10, 'old') ! *********************************************** ! *** Compute weight for model state update *** ! *** *** ! *** T f *** ! *** w = U RiHL d with d = (y - H x ) *** ! *********************************************** CALL PDAF_timeit(51, 'new') CALL PDAF_timeit(13, 'new') ! *** RiHLd = RiHL^T d *** ALLOCATE(RiHLd_p(rank)) ALLOCATE(RiHLd(rank)) IF (allocflag == 0) CALL PDAF_memcount(3, 'r', 2 * rank) haveobsC: IF (dim_obs_p > 0) THEN ! *** RiHLd_p/=0 only with observations *** ! local products (partial sum) CALL gemvTYPE('t', dim_obs_p, rank, 1.0, RiHL_p, & dim_obs_p, resid_p, 1, 0.0, RiHLd_p, 1) DEALLOCATE(RiHL_p, resid_p) ELSE haveobsC RiHLd_p = 0.0 END IF haveobsC ! get total sum on all filter PEs CALL MPI_allreduce(RiHLd_p, RiHLd, rank, & MPI_REALTYPE, MPI_SUM, COMM_filter, MPIerr) DEALLOCATE(RiHLd_p) ! *** Compute weight vector for state analysis: *** ! *** w = U RiHLd *** ! *** For this, two variants are implemented: *** ! *** 1. solve for w in: *** ! *** -1 *** ! *** U w = RiHLd *** ! *** We use the LAPACK routine GESV *** ! *** 2. Compute singular value decomposition *** ! *** of Uinv: Uinv = ASB^T *** ! *** Then: U = A S^(-1) B *** ! *** This is combined with a symmetric *** ! *** square-root for the ensemble transformation *** typeuinv1: IF (type_sqrt==1) THEN ! *** Variant 1: Solve Uinv w= RiHLd for w ALLOCATE(ipiv(rank)) IF (allocflag == 0) CALL PDAF_memcount(3, 'i', rank) ! save matrix Uinv tmp_Uinv = Uinv IF (debug>0) & WRITE (*,*) '++ PDAF-debug PDAF_seik_analysis:', debug, & ' Invert U^-1 using solver GESV' ! call solver (GESV - LU solver) CALL gesvTYPE(rank, 1, tmp_Uinv, rank, ipiv, & RiHLd, rank, lib_info) DEALLOCATE(ipiv) ELSE typeuinv1 ! *** Variant 2: Invert Uinv using SVD ALLOCATE(svals(rank)) ALLOCATE(work(3 * rank)) ldwork = 3 * rank IF (allocflag == 0) CALL PDAF_memcount(3, 'r', 3 * rank) IF (debug>0) & WRITE (*,*) '++ PDAF-debug PDAF_seik_analysis:', debug, & ' Compute eigenvalue decomposition of U^-1' ! Compute SVD of Uinv CALL syevTYPE('v', 'l', rank, Uinv, rank, svals, work, ldwork, lib_info) DEALLOCATE(work) ! Compute product RiHLd U IF (lib_info==0) THEN IF (debug>0) & WRITE (*,*) '++ PDAF-debug PDAF_seik_resample:', debug, ' eigenvalues', svals ALLOCATE(VRiHLd(rank)) IF (allocflag == 0) CALL PDAF_memcount(3, 'r', rank) CALL gemvTYPE('t', rank, rank, 1.0, Uinv, & rank, RiHLd, 1, 0.0, VRiHLd, 1) DO row = 1,rank VRiHLd(row) = VRiHLd(row) / svals(row) END DO CALL gemvTYPE('n', rank, rank, 1.0, Uinv, & rank, VRiHLd, 1, 0.0, RiHLd, 1) DEALLOCATE(VRiHLd) END IF END IF typeuinv1 CALL PDAF_timeit(13, 'old') ! *** check if solve was successful IF (lib_info == 0) THEN IF (debug>0) & WRITE (*,*) '++ PDAF-debug PDAF_seik_analysis:', debug, ' U(HL R^-1)^T d', RiHLd flag = 0 ELSE WRITE (*, '(/5x, a/)') 'PDAF-ERROR(1): Problem in computation of analysis weights!!!' flag = 1 END IF ! **************************************************************** ! *** Transform state ensemble *** ! *** a _f f *** ! *** X = X + X W *** ! *** with -T T *** ! *** W = T (RiHLd + sqrt(FAC) C Omega ) *** ! *** Here FAC depends on the use definition of the covariance *** ! *** matrix P using a factor (r+1)^-1 or r^-1. *** ! **************************************************************** ! *** Prepare weight matrix for ensemble transformation *** check1: IF (flag == 0) THEN IF (mype == 0 .AND. screen > 0) THEN WRITE (*, '(a, 5x, a)') 'PDAF', 'Transform state ensemble' IF (type_sqrt == 1) THEN WRITE (*, '(a, 5x, a)') 'PDAF', '--- use Cholesky square-root of U' ELSE WRITE (*, '(a, 5x, a)') 'PDAF', '--- use symmetric square-root of U' END IF END IF CALL PDAF_timeit(20, 'new') CALL PDAF_timeit(32, 'new') ! Usqrt is allocated with dim_ens cols, because this is ! required further below. Now only rank columns are used ALLOCATE(Usqrt(rank, dim_ens)) IF (allocflag == 0) CALL PDAF_memcount(3, 'r', dim_ens * rank) ! Part 1: square-root of U typeuinv2: IF (type_sqrt == 1) THEN ! Variant, if Uinv has been inverted above by solving ! *** Store Uinv for temporary use tmp_Uinv(:,:) = Uinv(:,:) ! Cholesky decomposition of tmp_Uinv = C C^T CALL potrfTYPE('l', rank, tmp_Uinv, rank, lib_info) ! check if Cholesky decomposition was successful CholeskyOK: IF (lib_info == 0) THEN ! Decomposition OK, continue flag = 0 ELSE ! Decomposition failed WRITE (*, '(/5x, a/)') & 'PDAF-ERROR(3): Problem with Cholesky decomposition of Uinv !!!' flag = 3 ENDIF CholeskyOK ELSE typeuinv2 ! Variant, if SVD inversion of Uinv has been performed DO col = 1, rank DO row = 1, rank Usqrt(row, col) = Uinv(row, col) / SQRT(svals(col)) END DO END DO CALL gemmTYPE('n', 't', rank, rank, rank, & 1.0, Usqrt, rank, Uinv, rank, & 0.0, tmp_Uinv, rank) DEALLOCATE(svals) ! Set flag flag = 0 END IF typeuinv2 CALL PDAF_timeit(32, 'old') END IF check1 ! *** Part 2: Initialize Omega *** check2: IF (flag == 0) THEN ! allocate fields ALLOCATE(OmegaT(rank, dim_ens)) IF (allocflag == 0) CALL PDAF_memcount(3, 'r', rank * dim_ens) IF (storeOmega .AND. allocflag == 0) THEN ALLOCATE(OmegaTsave(rank, dim_ens)) CALL PDAF_memcount(3, 'r', dim_ens * rank) END IF CALL PDAF_timeit(33, 'new') Omega_store: IF (storeOmega) THEN first: IF (firsttime) THEN ! *** At first call to SEIK_RESAMPLE initialize *** ! *** the matrix Omega in SEIK_Omega and store it *** ALLOCATE(Omega(dim_ens, rank)) IF (allocflag == 0) CALL PDAF_memcount(3, 'r', dim_ens * rank) ! *** Generate uniform orthogonal matrix OMEGA *** CALL PDAF_seik_Omega(rank, Omega, type_trans, screen) ! transpose Omega IF (type_sqrt == 1) THEN OmegaT = TRANSPOSE(Omega) ! store transposed Omega OmegaTsave = OmegaT ELSE Usqrt = TRANSPOSE(Omega) ! store transposed Omega OmegaTsave = Usqrt END IF firsttime = .FALSE. DEALLOCATE(Omega) ELSE first IF (mype == 0 .AND. screen > 0) & WRITE (*, '(a, 5x, a)') 'PDAF', '--- use stored Omega' IF (type_sqrt == 1) THEN OmegaT = OmegaTsave ELSE Usqrt = OmegaTsave END IF END IF first ELSE Omega_store ! *** Initialize the matrix Omega in SEIK_Omega *** ! *** each time SEIK_RESAMPLE is called *** ALLOCATE(Omega(dim_ens, rank)) IF (allocflag == 0) CALL PDAF_memcount(3, 'r', dim_ens * rank) ! *** Generate uniform orthogonal matrix OMEGA *** CALL PDAF_seik_Omega(rank, Omega, type_trans, screen) ! transpose Omega IF (type_sqrt == 1) THEN OmegaT = TRANSPOSE(Omega) ELSE Usqrt = TRANSPOSE(Omega) END IF DEALLOCATE(Omega) END IF Omega_store IF (debug>0) THEN IF (type_sqrt == 1) THEN WRITE (*,*) '++ PDAF-debug PDAF_seik_analysis:', debug, ' Omega^T', OmegaT ELSE WRITE (*,*) '++ PDAF-debug PDAF_seik_analysis:', debug, ' Omega^T', Usqrt END IF END IF CALL PDAF_timeit(33, 'old') ! *** Part 3: Product U^(1/2) Omega *** CALL PDAF_timeit(34, 'new') IF (type_sqrt == 1) THEN ! A = (Omega C^(-1)) by solving Ct A = OmegaT for A CALL trtrsTYPE('l', 't', 'n', rank, dim_ens, & tmp_Uinv, rank, OmegaT, rank, lib_info) ELSE ! TMP_UINV already contains matrix C (no more inversion) CALL gemmTYPE('n', 'n', rank, dim_ens, rank, & 1.0, tmp_Uinv, rank, Usqrt, rank, & 0.0, OmegaT, rank) END IF CALL PDAF_timeit(34, 'old') ! check if solve was successful solveOK: IF (lib_info == 0) THEN ! Solve for A OK, continue flag = 0 ELSE ! Solve for A failed WRITE (*, '(/5x, a/)') & 'PDAF-ERROR(2): Problem in computation of transformation matrix !!!' flag = 2 CALL PDAF_timeit(20, 'old') END IF solveOK DEALLOCATE(Usqrt) END IF check2 check3: IF (flag == 0) THEN ! *** Part 4: Add RiHLd and multiply by scaling factor CALL PDAF_timeit(35, 'new') IF (Nm1vsN == 1) THEN ! Use factor (N-1)^-1 fac = SQRT(REAL(dim_ens - 1)) ELSE ! Use factor N^-1 fac = SQRT(REAL(dim_ens)) END IF ! *** Add RiHLd to At DO j = 1, dim_ens DO i = 1, rank OmegaT(i,j) = fac * OmegaT(i,j) + RiHLd(i) END DO END DO DEALLOCATE(RiHLd) ! *** T A^T (A^T stored in OmegaT) *** ALLOCATE(TA(dim_ens, dim_ens)) IF (allocflag == 0) CALL PDAF_memcount(3, 'r', dim_ens**2) CALL PDAF_seik_TtimesA(rank, dim_ens, OmegaT, TA) IF (debug>0) & WRITE (*,*) '++ PDAF-debug PDAF_seik_resample:', debug, ' transform', TA CALL PDAF_timeit(35, 'old') CALL PDAF_timeit(20, 'old') ! *** Perform ensemble transformation *** ! Use block formulation for transformation maxblksize = 200 IF (mype == 0 .AND. screen > 0) & WRITE (*, '(a, 5x, a, i5)') & 'PDAF', '--- use blocking with size ', maxblksize ALLOCATE(ens_blk(maxblksize, dim_ens)) IF (allocflag == 0) CALL PDAF_memcount(3, 'r', maxblksize * dim_ens) blocking: DO blklower = 1, dim_p, maxblksize blkupper = MIN(blklower + maxblksize - 1, dim_p) ! Store old state ensemble CALL PDAF_timeit(21, 'new') DO col = 1, dim_ens ens_blk(1 : blkupper - blklower + 1, col) & = ens_p(blklower : blkupper, col) END DO DO col = 1,dim_ens ens_p(blklower : blkupper, col) = state_p(blklower : blkupper) END DO CALL PDAF_timeit(21, 'old') ! a _f f T ! Transform ensemble: X = X + X T(A ) CALL PDAF_timeit(22, 'new') CALL gemmTYPE('n', 'n', blkupper - blklower + 1, dim_ens, dim_ens, & 1.0, ens_blk(1, 1), maxblksize, TA(1, 1), dim_ens, & 1.0, ens_p(blklower, 1), dim_p) CALL PDAF_timeit(22, 'old') END DO blocking DEALLOCATE(ens_blk, TA) DEALLOCATE(OmegaT) END IF check3 CALL PDAF_timeit(51, 'old') ! ******************** ! *** Finishing up *** ! ******************** DEALLOCATE(tmp_Uinv) IF (allocflag == 0) allocflag = 1 IF (debug>0) & WRITE (*,*) '++ PDAF-debug: ', debug, 'PDAF_seik_analysis -- END' END SUBROUTINE PDAF_seik_analysis_trans