示例#1
0
def optcon_nse(problemname='drivencavity',
               N=10,
               Nts=10,
               nu=1e-2,
               clearprvveldata=False,
               ini_vel_stokes=False,
               stst_control=False,
               closed_loop=True,
               outernwtnstps=1,
               t0=None,
               tE=None,
               use_ric_ini_nu=None,
               alphau=1e-9,
               gamma=1e-3,
               spec_tip_dict=None,
               nwtn_adi_dict=None,
               linearized_nse=False,
               stokes_flow=False,
               ystar=None):

    tip = time_int_params(Nts, t0=t0, tE=tE)
    if spec_tip_dict is not None:
        tip.update(spec_tip_dict)
    if nwtn_adi_dict is not None:
        tip['nwtn_adi_dict'] = nwtn_adi_dict

    problemdict = dict(drivencavity=dnsps.drivcav_fems,
                       cylinderwake=dnsps.cyl_fems)

    problemfem = problemdict[problemname]
    femp = problemfem(N)

    # output
    ddir = 'data/'
    try:
        os.chdir(ddir)
    except OSError:
        raise Warning('need "' + ddir + '" subdir for storing the data')
    os.chdir('..')

    if linearized_nse and not outernwtnstps == 1:
        raise Warning('Linearized problem can have only one Newton step')

    if closed_loop:
        if stst_control:
            data_prfx = ddir + 'stst_' + problemname + '__'
        else:
            data_prfx = ddir + 'tdst_' + problemname + '__'

    else:
        data_prfx = ddir + problemname + '__'

    if stokes_flow:
        data_prfx = data_prfx + 'stokes__'

    # specify in what spatial direction Bu changes. The remaining is constant
    if problemname == 'drivencavity':
        uspacedep = 0
    elif problemname == 'cylinderwake':
        uspacedep = 1

    stokesmats = dts.get_stokessysmats(femp['V'], femp['Q'], nu)
    rhsd_vf = dts.setget_rhs(femp['V'], femp['Q'], femp['fv'], femp['fp'], t=0)

    # remove the freedom in the pressure
    stokesmats['J'] = stokesmats['J'][:-1, :][:, :]
    stokesmats['JT'] = stokesmats['JT'][:, :-1][:, :]
    rhsd_vf['fp'] = rhsd_vf['fp'][:-1, :]

    # reduce the matrices by resolving the BCs
    (stokesmatsc, rhsd_stbc, invinds, bcinds,
     bcvals) = dts.condense_sysmatsbybcs(stokesmats, femp['diribcs'])

    print 'Dimension of the div matrix: ', stokesmatsc['J'].shape
    # pressure freedom and dirichlet reduced rhs
    rhsd_vfrc = dict(fpr=rhsd_vf['fp'], fvc=rhsd_vf['fv'][invinds, ])

    # add the info on boundary and inner nodes
    bcdata = {'bcinds': bcinds, 'bcvals': bcvals, 'invinds': invinds}
    femp.update(bcdata)

    # casting some parameters
    NV = len(femp['invinds'])

    soldict = stokesmatsc  # containing A, J, JT
    soldict.update(femp)  # adding V, Q, invinds, diribcs
    # soldict.update(rhsd_vfrc)  # adding fvc, fpr
    soldict.update(fv=rhsd_stbc['fv'] + rhsd_vfrc['fvc'],
                   fp=rhsd_stbc['fp'] + rhsd_vfrc['fpr'],
                   N=N,
                   nu=nu,
                   trange=tip['tmesh'],
                   get_datastring=get_datastr,
                   data_prfx=data_prfx,
                   clearprvdata=clearprvveldata,
                   paraviewoutput=tip['ParaviewOutput'],
                   vfileprfx=tip['proutdir'] + 'vel_',
                   pfileprfx=tip['proutdir'] + 'p_')

    #
    # Prepare for control
    #

    contp = ContParams(femp['odcoo'], ystar=ystar, alphau=alphau, gamma=gamma)
    # casting some parameters
    NY, NU = contp.NY, contp.NU

    contsetupstr = problemname + '__NV{0}NU{1}NY{2}'.format(NV, NU, NY)

    # get the control and observation operators
    try:
        b_mat = dou.load_spa(ddir + contsetupstr + '__b_mat')
        u_masmat = dou.load_spa(ddir + contsetupstr + '__u_masmat')
        print 'loaded `b_mat`'
    except IOError:
        print 'computing `b_mat`...'
        b_mat, u_masmat = cou.get_inp_opa(cdcoo=femp['cdcoo'],
                                          V=femp['V'],
                                          NU=NU,
                                          xcomp=uspacedep)
        dou.save_spa(b_mat, ddir + contsetupstr + '__b_mat')
        dou.save_spa(u_masmat, ddir + contsetupstr + '__u_masmat')
    try:
        mc_mat = dou.load_spa(ddir + contsetupstr + '__mc_mat')
        y_masmat = dou.load_spa(ddir + contsetupstr + '__y_masmat')
        print 'loaded `c_mat`'
    except IOError:
        print 'computing `c_mat`...'
        mc_mat, y_masmat = cou.get_mout_opa(odcoo=femp['odcoo'],
                                            V=femp['V'],
                                            NY=NY)
        dou.save_spa(mc_mat, ddir + contsetupstr + '__mc_mat')
        dou.save_spa(y_masmat, ddir + contsetupstr + '__y_masmat')

    # restrict the operators to the inner nodes
    mc_mat = mc_mat[:, invinds][:, :]
    b_mat = b_mat[invinds, :][:, :]

    # for further use:
    c_mat = lau.apply_massinv(y_masmat, mc_mat, output='sparse')

    if contp.ystarx is None:
        c_mat = c_mat[NY:, :][:, :]  # TODO: Do this right
        mc_mat = mc_mat[NY:, :][:, :]  # TODO: Do this right
        y_masmat = y_masmat[:NY, :][:, :NY]  # TODO: Do this right

    mct_mat_reg = lau.app_prj_via_sadpnt(amat=stokesmatsc['M'],
                                         jmat=stokesmatsc['J'],
                                         rhsv=mc_mat.T,
                                         transposedprj=True)

    # set the weighing matrices
    contp.R = contp.alphau * u_masmat

    #
    # solve the differential-alg. Riccati eqn for the feedback gain X
    # via computing factors Z, such that X = -Z*Z.T
    #
    # at the same time we solve for the affine-linear correction w
    #

    # tilde B = BR^{-1/2}
    tb_mat = lau.apply_invsqrt_fromright(contp.R, b_mat, output='sparse')
    # tb_dense = np.array(tb_mat.todense())

    trct_mat = lau.apply_invsqrt_fromright(y_masmat,
                                           mct_mat_reg,
                                           output='dense')

    if closed_loop:
        cntpstr = 'NV{3}NY{0}NU{1}alphau{2}gamma{4}'.\
            format(contp.NU, contp.NY, contp.alphau, NV, contp.gamma)
    else:
        cntpstr = ''

    # we gonna use this quite often
    M, A = stokesmatsc['M'], stokesmatsc['A']
    datastrdict = dict(time=None, meshp=N, nu=nu, Nts=Nts, data_prfx=data_prfx)

    # compute the uncontrolled steady state (Navier-)Stokes solution
    # as initial value
    if ini_vel_stokes:
        # compute the uncontrolled steady state Stokes solution
        ini_vel, newtonnorms = snu.solve_steadystate_nse(vel_nwtn_stps=0,
                                                         vel_pcrd_stps=0,
                                                         **soldict)
        soldict.update(dict(iniv=ini_vel))
    else:
        ini_vel, newtonnorms = snu.solve_steadystate_nse(**soldict)
        soldict.update(dict(iniv=ini_vel))

    if closed_loop:
        if stst_control:
            if stokes_flow:
                convc_mat = sps.csr_matrix((NV, NV))
                rhs_con, rhsv_conbc = np.zeros((NV, 1)), np.zeros((NV, 1))
                lin_point = None
            else:
                lin_point, newtonnorms = snu.solve_steadystate_nse(**soldict)
                (convc_mat, rhs_con,
                 rhsv_conbc) = snu.get_v_conv_conts(prev_v=lin_point,
                                                    invinds=invinds,
                                                    V=femp['V'],
                                                    diribcs=femp['diribcs'])
            # infinite control horizon, steady target state
            cdatstr = get_datastr(time=None,
                                  meshp=N,
                                  nu=nu,
                                  Nts=None,
                                  data_prfx=data_prfx)

            try:
                Z = dou.load_npa(cdatstr + cntpstr + '__Z')
                print 'loaded ' + cdatstr + cntpstr + '__Z'
            except IOError:
                if use_ric_ini_nu is not None:
                    cdatstr = get_datastr(nwtn=None,
                                          time=None,
                                          meshp=N,
                                          nu=use_ric_ini_nu,
                                          Nts=None,
                                          data_prfx=data_prfx)
                    try:
                        zini = dou.load_npa(ddir + cdatstr + cntpstr + '__Z')
                        print 'Initialize Newton ADI by Z from ' + cdatstr
                    except IOError:
                        raise Warning('No data for initialization of '
                                      ' Newton ADI -- need ' + cdatstr + '__Z')
                    cdatstr = get_datastr(meshp=N, nu=nu, data_prfx=data_prfx)
                else:
                    zini = None

                parnadi = pru.proj_alg_ric_newtonadi
                Z = parnadi(mmat=M,
                            amat=-A - convc_mat,
                            jmat=stokesmatsc['J'],
                            bmat=tb_mat,
                            wmat=trct_mat,
                            z0=zini,
                            nwtn_adi_dict=tip['nwtn_adi_dict'])['zfac']

                dou.save_npa(Z, fstring=cdatstr + cntpstr + '__Z')
                print 'saved ' + cdatstr + cntpstr + '__Z'

                if tip['compress_z']:
                    Zc = pru.compress_Zsvd(Z,
                                           thresh=tip['comprz_thresh'],
                                           k=tip['comprz_maxc'])
                    Z = Zc

            fvnstst = rhs_con + rhsv_conbc + rhsd_stbc['fv'] + rhsd_vfrc['fvc']

            # X = -ZZ.T
            mtxtb_stst = -pru.get_mTzzTtb(M.T, Z, tb_mat)
            mtxfv_stst = -pru.get_mTzzTtb(M.T, Z, fvnstst)

            fl = mc_mat.T * contp.ystarvec(0)

            wft = lau.solve_sadpnt_smw(amat=A.T + convc_mat.T,
                                       jmat=stokesmatsc['J'],
                                       rhsv=fl + mtxfv_stst,
                                       umat=mtxtb_stst,
                                       vmat=tb_mat.T)[:NV]

            auxstrg = cdatstr + cntpstr
            dou.save_npa(wft, fstring=cdatstr + cntpstr + '__w')
            dou.save_npa(mtxtb_stst, fstring=cdatstr + cntpstr + '__mtxtb')
            feedbackthroughdict = {
                None: dict(w=auxstrg + '__w', mtxtb=auxstrg + '__mtxtb')
            }

            cns = 0
            soldict.update(data_prfx=data_prfx + '_cns{0}'.format(cns))
            if linearized_nse:
                soldict.update(vel_pcrd_stps=0,
                               vel_nwtn_stps=1,
                               lin_vel_point={None: lin_point})
            dictofvels = snu.\
                solve_nse(return_dictofvelstrs=True,
                          closed_loop=True,
                          static_feedback=True,
                          tb_mat=tb_mat,
                          stokes_flow=stokes_flow,
                          clearprvveldata=True,
                          feedbackthroughdict=feedbackthroughdict, **soldict)

        else:  # time dep closed loop

            cns_data_prfx = 'data/cnsvars'
            invd = init_nwtnstps_value_dict
            curnwtnsdict = invd(tmesh=tip['tmesh'], data_prfx=cns_data_prfx)
            # initialization: compute the forward solution
            if stokes_flow:
                dictofvels = None
            else:
                dictofvels = snu.solve_nse(return_dictofvelstrs=True,
                                           stokes_flow=stokes_flow,
                                           **soldict)

            # dbs.plot_vel_norms(tip['tmesh'], dictofvels)

            # function for the time depending parts
            # -- to be passed to the solver
            def get_tdpart(time=None,
                           dictofvalues=None,
                           feedback=False,
                           V=None,
                           invinds=None,
                           diribcs=None,
                           **kw):

                if stokes_flow:
                    convc_mat = sps.csr_matrix((NV, NV))
                    rhs_con, rhsv_conbc = np.zeros((NV, 1)), np.zeros((NV, 1))
                else:
                    curvel = dou.load_npa(dictofvalues[time])
                    convc_mat, rhs_con, rhsv_conbc = \
                        snu.get_v_conv_conts(prev_v=curvel, invinds=invinds,
                                             V=V, diribcs=diribcs)

                return convc_mat, rhsv_conbc + rhs_con

            gttdprtargs = dict(dictofvalues=dictofvels,
                               V=femp['V'],
                               diribcs=femp['diribcs'],
                               invinds=invinds)

            # old version rhs
            # ftilde = rhs_con + rhsv_conbc + rhsd_stbc['fv']
            for cns in range(outernwtnstps):

                datastrdict.update(data_prfx=data_prfx + cntpstr +
                                   '_cns{0}'.format(cns))
                soldict.update(data_prfx=data_prfx + cntpstr +
                               '_cns{0}'.format(cns))

                sfd = sdr.solve_flow_daeric
                feedbackthroughdict = \
                    sfd(mmat=M, amat=A, jmat=stokesmatsc['J'],
                        bmat=b_mat,
                        # cmat=ct_mat_reg.T,
                        mcmat=mct_mat_reg.T,
                        v_is_my=True, rmat=contp.alphau*u_masmat,
                        vmat=y_masmat, rhsv=rhsd_stbc['fv'],
                        gamma=contp.gamma,
                        rhsp=None,
                        tmesh=tip['tmesh'], ystarvec=contp.ystarvec,
                        nwtn_adi_dict=tip['nwtn_adi_dict'],
                        comprz_thresh=tip['comprz_thresh'],
                        comprz_maxc=tip['comprz_maxc'], save_full_z=False,
                        get_tdpart=get_tdpart, gttdprtargs=gttdprtargs,
                        curnwtnsdict=curnwtnsdict,
                        get_datastr=get_datastr, gtdtstrargs=datastrdict)

                # for t in tip['tmesh']:  # feedbackthroughdict.keys():
                #     curw = dou.load_npa(feedbackthroughdict[t]['mtxtb'])
                #     print cns, t, np.linalg.norm(curw)

                cdatstr = get_datastr(time='all',
                                      meshp=N,
                                      nu=nu,
                                      Nts=None,
                                      data_prfx=data_prfx)

                if linearized_nse:
                    dictofvels = snu.\
                        solve_nse(return_dictofvelstrs=True,
                                  closed_loop=True, tb_mat=tb_mat,
                                  lin_vel_point=dictofvels,
                                  feedbackthroughdict=feedbackthroughdict,
                                  vel_nwtn_stps=1,
                                  vel_pcrd_stps=0,
                                  **soldict)
                else:
                    dictofvels = snu.\
                        solve_nse(return_dictofvelstrs=True,
                                  closed_loop=True, tb_mat=tb_mat,
                                  stokes_flow=stokes_flow,
                                  feedbackthroughdict=feedbackthroughdict,
                                  vel_pcrd_stps=1,
                                  vel_nwtn_stps=2,
                                  **soldict)

                # for t in dictofvels.keys():
                #     curw = dou.load_npa(dictofvels[t])
                #     print cns, t, np.linalg.norm(curw)

                gttdprtargs.update(dictofvalues=dictofvels)
    else:
        # no control
        feedbackthroughdict = None
        tb_mat = None
        cdatstr = get_datastr(meshp=N,
                              nu=nu,
                              time='all',
                              Nts=Nts,
                              data_prfx=data_prfx)

        soldict.update(clearprvdata=True)
        dictofvels = snu.solve_nse(feedbackthroughdict=feedbackthroughdict,
                                   tb_mat=tb_mat,
                                   closed_loop=closed_loop,
                                   stokes_flow=stokes_flow,
                                   return_dictofvelstrs=True,
                                   static_feedback=stst_control,
                                   **soldict)

    (yscomplist, ystarlist) = dou.extract_output(dictofpaths=dictofvels,
                                                 tmesh=tip['tmesh'],
                                                 c_mat=c_mat,
                                                 ystarvec=contp.ystarvec)

    save_output_json(yscomplist,
                     tip['tmesh'].tolist(),
                     ystar=ystarlist,
                     fstring=cdatstr + cntpstr + '__sigout')

    costfunval = eval_costfunc(W=y_masmat,
                               V=contp.gamma * y_masmat,
                               R=None,
                               tbmat=tb_mat,
                               cmat=c_mat,
                               ystar=contp.ystarvec,
                               tmesh=tip['tmesh'],
                               veldict=dictofvels,
                               fbftdict=feedbackthroughdict)

    print 'Value of cost functional: ', costfunval

    costfunval = eval_costfunc(W=y_masmat,
                               V=contp.gamma * y_masmat,
                               R=None,
                               tbmat=tb_mat,
                               cmat=c_mat,
                               ystar=contp.ystarvec,
                               penau=False,
                               tmesh=tip['tmesh'],
                               veldict=dictofvels,
                               fbftdict=feedbackthroughdict)

    print 'Value of cost functional not considering `u`: ', costfunval

    print 'dim of v :', femp['V'].dim()
    charlene = .15 if problemname == 'cylinderwake' else 1.0
    print 'Re = charL / nu = {0}'.format(charlene / nu)
示例#2
0
def solve_flow_daeric(mmat=None, amat=None, jmat=None, bmat=None,
                      cmat=None, rhsv=None, rhsp=None,
                      mcmat=None, v_is_my=False,
                      rmat=None, vmat=None,
                      gamma=1.0,
                      tmesh=None, ystarvec=None,
                      nwtn_adi_dict=None,
                      curnwtnsdict=None,
                      comprz_thresh=None, comprz_maxc=None, save_full_z=False,
                      get_tdpart=None, gttdprtargs=None,
                      get_datastr=None, gtdtstrargs=None,
                      check_c_consist=True):

    """
    Routine for the solution of the DAE Riccati

    .. math::

        \\dot{MXM^T} + F^TXM + M^TXM + M^TXGXM + L(Y) = W \\\\
                JXM = 0 \\quad \\text{and} \\quad M^TXJ = 0 \\\\
                M^TX(T)M = W

    where :math:`F=A+N(t)`,
    where :math:`W:=C^T V C`, :math:`G:=B R^{-1} B^T`,
    and where :math:`L(Y)` is the Lagrange multiplier term.

    Simultaneously we solve for the feedforward term :math:`w`:

    .. math::

        \\dot{M^Tw} - [M^TXG+F^T]w - J^Tv = C^T V y^* + M^T[Xf + Yg] \\\\
                Jw = 0 \\\\
                M^Tw(T) = C^T V y^*(T)

    Note that :math:`V=M_y` if the norm of :math:`Y` is used
    in the cost function.


    Parameters
    ----------
    cmat : (NY, NV) array
        the (regularized aka projected) output matrix
    mcmat : (NY, NV) array
        output matrix times the mass matrix in the output space
    gamma : float, optional
        weighting parameter for penalization of the terminal value,
        TODO: rather provide the right weighting matrix V,
        defaults to `1.0`
    v_is_my : boolean
        whether the weighting matrix is the same as the mass matrix, \
                defaults to `False`
    get_tdpart : callable f(t)
        returns the `mattd, rhstd` -- time dependent coefficients matrices
        and right hand side at time `t`
    gtdtstrargs : dictionary
        **kwargs to the current data string
    gttdprtargs : dictionary
        `**kwargs` to get_tdpart

    Returns
    -------
    feedbackthroughdict : dictionary
        with time instances as keys and
        | `w` -- the current feedthrough value
        | `mtxtb` -- the current feedback gain part `(R.-1/2 * B.T * X * M).T`
        as values
    """

    if check_c_consist:
        if v_is_my and mcmat is not None:
            mic = lau.apply_massinv(mmat.T, mcmat.T)
            if np.linalg.norm(jmat*mic) > 1e-12:
                raise Warning('mcmat.T needs to be in the kernel of J*M.-1')
        elif cmat is not None:
            mic = lau.apply_massinv(mmat.T, cmat.T)
            if np.linalg.norm(jmat*mic) > 1e-12:
                raise Warning('cmat.T needs to be in the kernel of J*M.-1')

    MT, AT, NV = mmat.T, amat.T, amat.shape[0]

    gtdtstrargs.update(time=tmesh[-1])
    cdatstr = get_datastr(**gtdtstrargs)

    # set/compute the terminal values aka starting point
    if v_is_my and mcmat is not None:
        tct_mat = lau.apply_invsqrt_fromright(vmat, mcmat.T, output='dense')
    else:
        tct_mat = lau.apply_sqrt_fromright(vmat, cmat.T, output='dense')

    # TODO: good handling of bmat and umasmat
    tb_mat = lau.apply_invsqrt_fromright(rmat, bmat, output='sparse')
    # bmat_rpmo = bmat * np.linalg.inv(np.array(rmat.todense()))

    Zc = np.sqrt(gamma)*lau.apply_massinv(mmat, tct_mat)
    mtxtb = -pru.get_mTzzTtb(mmat.T, Zc, tb_mat)
    # mtxbrm = pru.get_mTzzTtb(mmat.T, Zc, bmat_rpmo)

    dou.save_npa(Zc, fstring=cdatstr + '__Z')
    dou.save_npa(mtxtb, fstring=cdatstr + '__mtxtb')

    if ystarvec is not None:
        wc = lau.apply_massinv(MT, gamma*np.dot(mcmat.T, ystarvec(tmesh[-1])))
        dou.save_npa(wc, fstring=cdatstr + '__w')
    else:
        wc = None

    feedbackthroughdict = {tmesh[-1]: dict(w=cdatstr + '__w',
                                           mtxtb=cdatstr + '__mtxtb')}

    # save the end values
    if curnwtnsdict is not None:
        dou.save_npa(wc, fstring=curnwtnsdict[tmesh[-1]]['w'])
        dou.save_npa(mtxtb, fstring=curnwtnsdict[tmesh[-1]]['mtxtb'])

    # time integration
    for tk, t in reversed(list(enumerate(tmesh[:-1]))):
        cts = tmesh[tk+1] - t

        print 'Time is {0}, timestep is {1}'.\
            format(t, cts)

        # get the previous time time-dep matrices
        gtdtstrargs.update(time=t)
        cdatstr = get_datastr(**gtdtstrargs)
        nmattd, rhsvtd = get_tdpart(time=t, **gttdprtargs)

        # get the feedback from the current newton step
        if curnwtnsdict is not None:
            try:
                cnsw = dou.load_npa(curnwtnsdict[t]['w'])
                cnsmtxtb = dou.load_npa(curnwtnsdict[t]['mtxtb'])
            except IOError:
                cnsw, cnsmtxtb = None, None
        else:
            cnsw, cnsmtxtb = None, None

        try:
            Zc = dou.load_npa(cdatstr + '__Z')
        except IOError:
            # coeffmat for nwtn adi
            ft_mat = -(0.5*MT + cts*(AT + nmattd.T))
            # rhs for nwtn adi
            w_mat = np.hstack([MT*Zc, np.sqrt(cts)*tct_mat])
            # feedback from a previous Newton step
            mtxb = np.sqrt(cts)*cnsmtxtb if cnsmtxtb is not None else None
            Zp = pru.proj_alg_ric_newtonadi(mmat=MT,
                                            amat=ft_mat, transposed=True,
                                            mtxoldb=mtxb,
                                            jmat=jmat,
                                            bmat=np.sqrt(cts)*tb_mat,
                                            wmat=w_mat, z0=Zc,
                                            nwtn_adi_dict=nwtn_adi_dict
                                            )['zfac']

            if comprz_maxc is not None or comprz_thresh is not None:
                Zc = pru.compress_Zsvd(Zp, thresh=comprz_thresh,
                                       k=comprz_maxc)
            else:
                Zc = Zp

            if save_full_z:
                dou.save_npa(Zp, fstring=cdatstr + '__Z')
            else:
                dou.save_npa(Zc, fstring=cdatstr + '__Z')

        # and the affine correction
        at_mat = MT + cts*(AT + nmattd.T)

        # current rhs
        ftilde = rhsvtd + rhsv

        # apply the feedback and through
        if cnsw is not None:
            ftilde = rhsvtd + rhsv + cnsw
        cnsmtxtb = cnsmtxtb + mtxtb if cnsmtxtb is not None else mtxtb

        mtxft = pru.get_mTzzTtb(MT, Zc, ftilde)

        fl1 = np.dot(mcmat.T, ystarvec(t))

        rhswc = MT*wc + cts*(fl1 - mtxft)

        mtxtb = -pru.get_mTzzTtb(MT, Zc, tb_mat)
        # mtxtbrm = pru.get_mTzzTtb(MT, Zc, bmat_rpmo)

        wc = lau.solve_sadpnt_smw(amat=at_mat, jmat=jmat,
                                  umat=cts*cnsmtxtb, vmat=tb_mat.T,
                                  rhsv=rhswc)[:NV]
        # wc = lau.solve_sadpnt_smw(amat=at_mat, jmat=jmat,
        #                           umat=-cts*mtxbrm, vmat=bmat.T,
        #                           rhsv=rhswc)[:NV]

        # update the feedback in Newton
        if curnwtnsdict is not None:
            cnsw = cnsw + wc if cnsw is not None else wc
            cnsmtxtb = cnsmtxtb + mtxtb if cnsmtxtb is not None else mtxtb
            dou.save_npa(cnsw, fstring=curnwtnsdict[t]['w'])
            dou.save_npa(cnsmtxtb, fstring=curnwtnsdict[t]['mtxtb'])

        dou.save_npa(wc, fstring=cdatstr + '__w')
        dou.save_npa(mtxtb, fstring=cdatstr + '__mtxtb')
        # dou.save_npa(mtxbrm, fstring=cdatstr + '__mtxbrm')
        feedbackthroughdict.update({t: dict(w=cdatstr + '__w',
                                            # mtxbrm=cdatstr + '__mtxbrm')})
                                            mtxtb=cdatstr + '__mtxtb')})

    return feedbackthroughdict
示例#3
0
def lqgbt(problemname='drivencavity',
          N=10, Re=1e2, plain_bt=False,
          use_ric_ini=None, t0=0.0, tE=1.0, Nts=11,
          NU=3, NY=3,
          bccontrol=True, palpha=1e-5,
          paraoutput=True,
          trunc_lqgbtcv=1e-6,
          nwtn_adi_dict=None,
          comp_freqresp=False, comp_stepresp='nonlinear',
          closed_loop=False, multiproc=False,
          perturbpara=1e-3):
    """Main routine for LQGBT

    Parameters
    ----------
    problemname : string, optional
        what problem to be solved, 'cylinderwake' or 'drivencavity'
    N : int, optional
        parameter for the dimension of the space discretization
    Re : real, optional
        Reynolds number, defaults to `1e2`
    plain_bt : boolean, optional
        whether to try simple *balanced truncation*, defaults to False
    use_ric_ini : real, optional
        use the solution with this Re number as stabilizing initial guess,
        defaults to `None`
    t0, tE, Nts : real, real, int, optional
        starting and endpoint of the considered time interval, number of
        time instancses, default to `0.0, 1.0, 11`
    bccontrol : boolean, optional
        whether to apply boundary control via penalized robin conditions,
        defaults to `False`
    NU, NY : int, optional
        dimensions of components of in and output space (will double because
        there are two components), default to `3, 3`
    comp_freqresp : boolean, optional
        whether to compute and compare the frequency responses,
        defaults to `False`
    comp_stepresp : {'nonlinear', False, None}
        whether to compute and compare the step responses

        | if False -> no step response
        | if == 'nonlinear' -> compare linear reduced to nonlinear full model
        | else -> linear reduced versus linear full model

        defaults to `False`

    trunc_lqgbtcv : real, optional
        threshold at what the lqgbt characteristiv values are truncated,
        defaults to `1e-6`
    closed_loop : {'full_state_fb', 'red_output_fb', False, None}
        how to do the closed loop simulation:

        | if False -> no simulation
        | if == 'full_state_fb' -> full state feedback
        | if == 'red_output_fb' -> reduced output feedback
        | else -> no control is applied

        defaults to `False`

    """

    typprb = 'BT' if plain_bt else 'LQG-BT'

    print '\n ### We solve the {0} problem for the {1} at Re={2} ###\n'.\
        format(typprb, problemname, Re)

    if nwtn_adi_dict is not None:
        nap = nwtn_adi_dict
    else:
        nap = nwtn_adi_params()['nwtn_adi_dict']
    # output
    ddir = 'data/'
    try:
        os.chdir(ddir)
    except OSError:
        raise Warning('need "' + ddir + '" subdir for storing the data')
    os.chdir('..')

    # stokesmats = dts.get_stokessysmats(femp['V'], femp['Q'], nu)

    # rhsd_vf = dts.setget_rhs(femp['V'], femp['Q'],
    #                          femp['fv'], femp['fp'], t=0)

    # # remove the freedom in the pressure
    # stokesmats['J'] = stokesmats['J'][:-1, :][:, :]
    # stokesmats['JT'] = stokesmats['JT'][:, :-1][:, :]
    # rhsd_vf['fp'] = rhsd_vf['fp'][:-1, :]

    # # reduce the matrices by resolving the BCs
    # (stokesmatsc,
    #  rhsd_stbc,
    #  invinds,
    #  bcinds,
    #  bcvals) = dts.condense_sysmatsbybcs(stokesmats,
    #                                      femp['diribcs'])

    # # pressure freedom and dirichlet reduced rhs
    # rhsd_vfrc = dict(fpr=rhsd_vf['fp'], fvc=rhsd_vf['fv'][invinds, ])

    # # add the info on boundary and inner nodes
    # bcdata = {'bcinds': bcinds,
    #           'bcvals': bcvals,
    #           'invinds': invinds}
    # femp.update(bcdata)

    femp, stokesmatsc, rhsd_vfrc, rhsd_stbc \
        = dnsps.get_sysmats(problem=problemname, N=N, Re=Re,
                            bccontrol=bccontrol, scheme='TH')

    nu = femp['charlen']/Re
    # specify in what spatial direction Bu changes. The remaining is constant
    uspacedep = femp['uspacedep']

    # casting some parameters
    invinds, NV = femp['invinds'], len(femp['invinds'])

    prbstr = '_bt' if plain_bt else '_lqgbt'
    # contsetupstr = 'NV{0}NU{1}NY{2}alphau{3}'.format(NV, NU, NY, alphau)
    if bccontrol:
        import scipy.sparse as sps
        contsetupstr = 'NV{0}_bcc_NY{1}'.format(NV, NY)
        stokesmatsc['A'] = stokesmatsc['A'] + 1./palpha*stokesmatsc['Arob']
        b_mat = 1./palpha*stokesmatsc['Brob']
        u_masmat = sps.eye(b_mat.shape[1], format='csr')
    else:
        contsetupstr = 'NV{0}NU{1}NY{2}'.format(NV, NU, NY)

    def get_fdstr(Re):
        return ddir + problemname + '_Re{0}_'.format(Re) + \
            contsetupstr + prbstr

    fdstr = get_fdstr(Re)

    soldict = stokesmatsc  # containing A, J, JT
    soldict.update(femp)  # adding V, Q, invinds, diribcs
    # soldict.update(rhsd_vfrc)  # adding fvc, fpr
    soldict.update(fv=rhsd_stbc['fv']+rhsd_vfrc['fvc'],
                   fp=rhsd_stbc['fp']+rhsd_vfrc['fpr'],
                   N=N, nu=nu, data_prfx=fdstr)

#
# Prepare for control
#

    # get the control and observation operators
    if not bccontrol:
        try:
            b_mat = dou.load_spa(ddir + contsetupstr + '__b_mat')
            u_masmat = dou.load_spa(ddir + contsetupstr + '__u_masmat')
            print 'loaded `b_mat`'
        except IOError:
            print 'computing `b_mat`...'
            b_mat, u_masmat = cou.get_inp_opa(cdcoo=femp['cdcoo'], V=femp['V'],
                                              NU=NU, xcomp=uspacedep)
            dou.save_spa(b_mat, ddir + contsetupstr + '__b_mat')
            dou.save_spa(u_masmat, ddir + contsetupstr + '__u_masmat')

        b_mat = b_mat[invinds, :][:, :]
        # tb_mat = 1./np.sqrt(alphau)

    try:
        mc_mat = dou.load_spa(ddir + contsetupstr + '__mc_mat')
        y_masmat = dou.load_spa(ddir + contsetupstr + '__y_masmat')
        print 'loaded `c_mat`'
    except IOError:
        print 'computing `c_mat`...'
        mc_mat, y_masmat = cou.get_mout_opa(odcoo=femp['odcoo'],
                                            V=femp['V'], NY=NY)
        dou.save_spa(mc_mat, ddir + contsetupstr + '__mc_mat')
        dou.save_spa(y_masmat, ddir + contsetupstr + '__y_masmat')

    c_mat = lau.apply_massinv(y_masmat, mc_mat, output='sparse')
    # restrict the operators to the inner nodes

    mc_mat = mc_mat[:, invinds][:, :]
    c_mat = c_mat[:, invinds][:, :]
    c_mat_reg = lau.app_prj_via_sadpnt(amat=stokesmatsc['M'],
                                       jmat=stokesmatsc['J'],
                                       rhsv=c_mat.T,
                                       transposedprj=True).T

    # c_mat_reg = np.array(c_mat.todense())

    # TODO: right choice of norms for y
    #       and necessity of regularization here
    #       by now, we go on number save
#
# setup the system for the correction
#
#
# compute the uncontrolled steady state Stokes solution
#
    v_ss_nse, list_norm_nwtnupd = snu.solve_steadystate_nse(**soldict)
    (convc_mat, rhs_con,
     rhsv_conbc) = snu.get_v_conv_conts(prev_v=v_ss_nse, invinds=invinds,
                                        V=femp['V'], diribcs=femp['diribcs'])

    f_mat = - stokesmatsc['A'] - convc_mat
    mmat = stokesmatsc['M']

    # ssv_rhs = rhsv_conbc + rhsv_conbc + rhsd_vfrc['fvc'] + rhsd_stbc['fv']

    if plain_bt:
        get_gramians = pru.solve_proj_lyap_stein
    else:
        get_gramians = pru.proj_alg_ric_newtonadi

    truncstr = '__lqgbtcv{0}'.format(trunc_lqgbtcv)
    try:
        tl = dou.load_npa(fdstr + '__tl' + truncstr)
        tr = dou.load_npa(fdstr + '__tr' + truncstr)
        print 'loaded the left and right transformations: \n' + \
            fdstr + '__tl/__tr' + truncstr
    except IOError:
        print 'computing the left and right transformations' + \
            ' and saving to: \n' + fdstr + '__tl/__tr' + truncstr

        try:
            zwc = dou.load_npa(fdstr + '__zwc')
            zwo = dou.load_npa(fdstr + '__zwo')
            print 'loaded factor of the Gramians: \n\t' + \
                fdstr + '__zwc/__zwo'
        except IOError:
            zinic, zinio = None, None
            if use_ric_ini is not None:
                fdstr = get_fdstr(use_ric_ini)
                try:
                    zinic = dou.load_npa(fdstr + '__zwc')
                    zinio = dou.load_npa(fdstr + '__zwo')
                    print 'Initialize Newton ADI by zwc/zwo from ' + fdstr
                except IOError:
                    raise UserWarning('No initial guess with Re={0}'.
                                      format(use_ric_ini))

            fdstr = get_fdstr(Re)
            print 'computing factors of Gramians: \n\t' + \
                fdstr + '__zwc/__zwo'

            def compobsg():
                try:
                    zwo = dou.load_npa(fdstr + '__zwo')
                    print 'at least __zwo is there'
                except IOError:
                    zwo = get_gramians(mmat=mmat.T, amat=f_mat.T,
                                       jmat=stokesmatsc['J'],
                                       bmat=c_mat_reg.T, wmat=b_mat,
                                       nwtn_adi_dict=nap,
                                       z0=zinio)['zfac']
                    dou.save_npa(zwo, fdstr + '__zwo')

            def compcong():
                try:
                    zwc = dou.load_npa(fdstr + '__zwc')
                    print 'at least __zwc is there'
                except IOError:
                    zwc = get_gramians(mmat=mmat, amat=f_mat,
                                       jmat=stokesmatsc['J'],
                                       bmat=b_mat, wmat=c_mat_reg.T,
                                       nwtn_adi_dict=nap,
                                       z0=zinic)['zfac']
                    dou.save_npa(zwc, fdstr + '__zwc')

            if multiproc:
                print '\n ### multithread start - ' +\
                    'output might be intermangled'
                p1 = multiprocessing.Process(target=compobsg)
                p2 = multiprocessing.Process(target=compcong)
                p1.start()
                p2.start()
                p1.join()
                p2.join()
                print '### multithread end'

            else:
                compobsg()
                compcong()

            zwc = dou.load_npa(fdstr + '__zwc')
            zwo = dou.load_npa(fdstr + '__zwo')

        print 'computing the left and right transformations' + \
            ' and saving to:\n' + fdstr + '__tr/__tl' + truncstr

        tl, tr = btu.\
            compute_lrbt_transfos(zfc=zwc, zfo=zwo,
                                  mmat=stokesmatsc['M'],
                                  trunck={'threshh': trunc_lqgbtcv})
        dou.save_npa(tl, fdstr + '__tl' + truncstr)
        dou.save_npa(tr, fdstr + '__tr' + truncstr)

    print 'NV = {0}, NP = {2}, k = {1}'.\
        format(tl.shape[0], tl.shape[1], stokesmatsc['J'].shape[0])

    if comp_freqresp:
        btu.compare_freqresp(mmat=stokesmatsc['M'], amat=f_mat,
                             jmat=stokesmatsc['J'], bmat=b_mat,
                             cmat=c_mat, tr=tr, tl=tl,
                             plot=True, datastr=fdstr + '__tl' + truncstr)

    if comp_stepresp is not False:
        if comp_stepresp == 'nonlinear':
            stp_rsp_nwtn = 3
            stp_rsp_dtpr = 'nonl_stepresp_'
        else:
            stp_rsp_nwtn = 1
            stp_rsp_dtpr = 'stepresp_'

        def fullstepresp_lnse(bcol=None, trange=None, ini_vel=None,
                              cmat=None, soldict=None):
            soldict.update(fv_stbc=rhsd_stbc['fv']+bcol,
                           vel_nwtn_stps=stp_rsp_nwtn, trange=trange,
                           iniv=ini_vel, lin_vel_point=ini_vel,
                           clearprvdata=True, data_prfx=stp_rsp_dtpr,
                           return_dictofvelstrs=True)

            dictofvelstrs = snu.solve_nse(**soldict)

            return cou.extract_output(strdict=dictofvelstrs, tmesh=trange,
                                      c_mat=cmat, load_data=dou.load_npa)

    # differences in the initial vector
    # print np.dot(c_mat_reg, v_ss_nse)
    # print np.dot(np.dot(c_mat_reg, tr),
    #              np.dot(tl.T, stokesmatsc['M']*v_ss_nse))

        jsonstr = fdstr + stp_rsp_dtpr + '_Nred{0}_t0tENts{1}{2}{3}.json'.\
            format(tl.shape[1], t0, tE, Nts)
        btu.compare_stepresp(tmesh=np.linspace(t0, tE, Nts),
                             a_mat=f_mat, c_mat=c_mat_reg, b_mat=b_mat,
                             m_mat=stokesmatsc['M'],
                             tr=tr, tl=tl, iniv=v_ss_nse,
                             # ss_rhs=ssv_rhs,
                             fullresp=fullstepresp_lnse, fsr_soldict=soldict,
                             plot=True, jsonstr=jsonstr)

# compute the regulated system
    trange = np.linspace(t0, tE, Nts)

    if closed_loop is False:
        return

    elif closed_loop == 'full_state_fb':
        zwc = dou.load_npa(fdstr + '__zwc')
        zwo = dou.load_npa(fdstr + '__zwo')

        mtxtb = pru.get_mTzzTtb(stokesmatsc['M'].T, zwc, b_mat)

        def fv_tmdp_fullstatefb(time=None, curvel=None,
                                linv=None, tb_mat=None, tbxm_mat=None, **kw):
            """realizes a full state static feedback as a function

            that can be passed to a solution routine for the
            unsteady Navier-Stokes equations

            Parameters
            ----------
            time : real
                current time
            curvel : (N,1) nparray
                current velocity
            linv : (N,1) nparray
                linearization point for the linear model
            tb_mat : (N,K) nparray
                input matrix containing the input weighting
            tbxm_mat : (N,K) nparray
                `tb_mat * gain * mass`

            Returns
            -------
            actua : (N,1) nparray
                current contribution to the right-hand side
            , : dictionary
                dummy `{}` for consistency
            """

            actua = -lau.comp_uvz_spdns(tb_mat, tbxm_mat, curvel-linv)
            # actua = 0*curvel
            print '\nnorm of deviation', np.linalg.norm(curvel-linv)
            # print 'norm of actuation {0}'.format(np.linalg.norm(actua))
            return actua, {}

        tmdp_fsfb_dict = dict(linv=v_ss_nse, tb_mat=b_mat, tbxm_mat=mtxtb.T)

        fv_tmdp = fv_tmdp_fullstatefb
        fv_tmdp_params = tmdp_fsfb_dict
        fv_tmdp_memory = None

    elif closed_loop == 'red_output_fb':
        try:
            xok = dou.load_npa(fdstr+truncstr+'__xok')
            xck = dou.load_npa(fdstr+truncstr+'__xck')
            ak_mat = dou.load_npa(fdstr+truncstr+'__ak_mat')
            ck_mat = dou.load_npa(fdstr+truncstr+'__ck_mat')
            bk_mat = dou.load_npa(fdstr+truncstr+'__bk_mat')
        except IOError:
            print 'couldn"t load the red system - compute it'
            zwc = dou.load_npa(fdstr + '__zwc')
            zwo = dou.load_npa(fdstr + '__zwo')

            ak_mat = np.dot(tl.T, f_mat*tr)
            ck_mat = lau.mm_dnssps(c_mat_reg, tr)
            bk_mat = lau.mm_dnssps(tl.T, b_mat)

            tltm, trtm = tl.T*stokesmatsc['M'], tr.T*stokesmatsc['M']
            xok = np.dot(np.dot(tltm, zwo), np.dot(zwo.T, tltm.T))
            xck = np.dot(np.dot(trtm, zwc), np.dot(zwc.T, trtm.T))

            dou.save_npa(xok, fdstr+truncstr+'__xok')
            dou.save_npa(xck, fdstr+truncstr+'__xck')
            dou.save_npa(ak_mat, fdstr+truncstr+'__ak_mat')
            dou.save_npa(ck_mat, fdstr+truncstr+'__ck_mat')
            dou.save_npa(bk_mat, fdstr+truncstr+'__bk_mat')

        obs_bk = np.dot(xok, ck_mat.T)
        DT = (tE - t0)/(Nts-1)

        sysmatk_inv = np.linalg.inv(np.eye(ak_mat.shape[1]) - DT*(ak_mat -
                                    np.dot(np.dot(xok, ck_mat.T), ck_mat) -
                                    np.dot(bk_mat, np.dot(bk_mat.T, xck))))

        def fv_tmdp_redoutpfb(time=None, curvel=None, memory=None,
                              linvel=None,
                              ipsysk_mat_inv=None,
                              obs_bk=None, cts=None,
                              b_mat=None, c_mat=None,
                              xck=None, bk_mat=None,
                              **kw):
            """realizes a reduced static output feedback as a function

            that can be passed to a solution routine for the
            unsteady Navier-Stokes equations

            For convinience the
            Parameters
            ----------
            time : real
                current time
            curvel : (N,1) nparray
                current velocity. For consistency, the full state is taken
                as input. However, internally, we only use the observation
                `y = c_mat*curvel`
            memory : dictionary
                contains values from previous call, in particular the
                previous state estimate
            linvel : (N,1) nparray
                linearization point for the linear model
            ipsysk_mat_inv : (K,K) nparray
                inverse of the system matrix that defines the update
                of the state estimate
            obs_bk : (K,NU) nparray
                input matrix in the observer
            cts : real
                time step length
            b_mat : (N,NU) sparse matrix
                input matrix of the full system
                c_mat=None,
            c_mat : (NY,N) sparse matrix
                output matrix of the full system
            xck : (K,K) nparray
                reduced solution of the CARE
            bk_mat : (K,NU) nparray
                reduced input matrix

            Returns
            -------
            actua : (N,1) nparray
                the current actuation
            memory : dictionary
                to be passed back in the next timestep

            """
            xk_old = memory['xk_old']
            buk = cts*np.dot(obs_bk,
                             lau.mm_dnssps(c_mat, (curvel-linvel)))
            xk_old = np.dot(ipsysk_mat_inv, xk_old + buk)
            #         cts*np.dot(obs_bk,
            #                 lau.mm_dnssps(c_mat, (curvel-linvel))))
            memory['xk_old'] = xk_old
            actua = -lau.mm_dnssps(b_mat,
                                   np.dot(bk_mat.T, np.dot(xck, xk_old)))
            print '\nnorm of deviation', np.linalg.norm(curvel-linvel)
            print 'norm of actuation {0}'.format(np.linalg.norm(actua))
            return actua, memory

        fv_rofb_dict = dict(cts=DT, linvel=v_ss_nse, b_mat=b_mat,
                            c_mat=c_mat_reg, obs_bk=obs_bk, bk_mat=bk_mat,
                            ipsysk_mat_inv=sysmatk_inv, xck=xck)

        fv_tmdp = fv_tmdp_redoutpfb
        fv_tmdp_params = fv_rofb_dict
        fv_tmdp_memory = dict(xk_old=np.zeros((tl.shape[1], 1)))

    else:
        fv_tmdp = None
        fv_tmdp_params = {}
        fv_tmdp_memory = {}

    perturbini = perturbpara*np.ones((NV, 1))
    reg_pertubini = lau.app_prj_via_sadpnt(amat=stokesmatsc['M'],
                                           jmat=stokesmatsc['J'],
                                           rhsv=perturbini)

    soldict.update(fv_stbc=rhsd_stbc['fv'],
                   trange=trange,
                   iniv=v_ss_nse + reg_pertubini,
                   lin_vel_point=None,
                   clearprvdata=True, data_prfx=fdstr + truncstr,
                   fv_tmdp=fv_tmdp,
                   comp_nonl_semexp=True,
                   fv_tmdp_params=fv_tmdp_params,
                   fv_tmdp_memory=fv_tmdp_memory,
                   return_dictofvelstrs=True)

    outstr = truncstr + '{0}'.format(closed_loop) \
        + 't0{0}tE{1}Nts{2}N{3}Re{4}'.format(t0, tE, Nts, N, Re)
    if paraoutput:
        soldict.update(paraviewoutput=True,
                       vfileprfx='results/vel_'+outstr,
                       pfileprfx='results/p_'+outstr)

    dictofvelstrs = snu.solve_nse(**soldict)

    yscomplist = cou.extract_output(strdict=dictofvelstrs, tmesh=trange,
                                    c_mat=c_mat, load_data=dou.load_npa)

    dou.save_output_json(dict(tmesh=trange.tolist(), outsig=yscomplist),
                         fstring=fdstr + truncstr + '{0}'.format(closed_loop) +
                         't0{0}tE{1}Nts{2}'.format(t0, tE, Nts) +
                         'inipert{0}'.format(perturbpara))

    dou.plot_outp_sig(tmesh=trange, outsig=yscomplist)
示例#4
0
def optcon_nse(problemname='drivencavity',
               N=10, Nts=10, nu=1e-2, clearprvveldata=False,
               ini_vel_stokes=False, stst_control=False,
               closed_loop=True,
               outernwtnstps=1,
               t0=None, tE=None,
               use_ric_ini_nu=None,
               alphau=1e-9, gamma=1e-3,
               spec_tip_dict=None,
               nwtn_adi_dict=None,
               linearized_nse=False,
               stokes_flow=False,
               ystar=None):

    tip = time_int_params(Nts, t0=t0, tE=tE)
    if spec_tip_dict is not None:
        tip.update(spec_tip_dict)
    if nwtn_adi_dict is not None:
        tip['nwtn_adi_dict'] = nwtn_adi_dict

    problemdict = dict(drivencavity=dnsps.drivcav_fems,
                       cylinderwake=dnsps.cyl_fems)

    problemfem = problemdict[problemname]
    femp = problemfem(N)

    # output
    ddir = 'data/'
    try:
        os.chdir(ddir)
    except OSError:
        raise Warning('need "' + ddir + '" subdir for storing the data')
    os.chdir('..')

    if linearized_nse and not outernwtnstps == 1:
        raise Warning('Linearized problem can have only one Newton step')

    if closed_loop:
        if stst_control:
            data_prfx = ddir + 'stst_' + problemname + '__'
        else:
            data_prfx = ddir + 'tdst_' + problemname + '__'

    else:
        data_prfx = ddir + problemname + '__'

    if stokes_flow:
        data_prfx = data_prfx + 'stokes__'

    # specify in what spatial direction Bu changes. The remaining is constant
    if problemname == 'drivencavity':
        uspacedep = 0
    elif problemname == 'cylinderwake':
        uspacedep = 1

    stokesmats = dts.get_stokessysmats(femp['V'], femp['Q'], nu)
    rhsd_vf = dts.setget_rhs(femp['V'], femp['Q'],
                             femp['fv'], femp['fp'], t=0)

    # remove the freedom in the pressure
    stokesmats['J'] = stokesmats['J'][:-1, :][:, :]
    stokesmats['JT'] = stokesmats['JT'][:, :-1][:, :]
    rhsd_vf['fp'] = rhsd_vf['fp'][:-1, :]

    # reduce the matrices by resolving the BCs
    (stokesmatsc, rhsd_stbc,
     invinds, bcinds, bcvals) = dts.condense_sysmatsbybcs(stokesmats,
                                                          femp['diribcs'])

    print 'Dimension of the div matrix: ', stokesmatsc['J'].shape
    # pressure freedom and dirichlet reduced rhs
    rhsd_vfrc = dict(fpr=rhsd_vf['fp'], fvc=rhsd_vf['fv'][invinds, ])

    # add the info on boundary and inner nodes
    bcdata = {'bcinds': bcinds, 'bcvals': bcvals, 'invinds': invinds}
    femp.update(bcdata)

    # casting some parameters
    NV = len(femp['invinds'])

    soldict = stokesmatsc  # containing A, J, JT
    soldict.update(femp)  # adding V, Q, invinds, diribcs
    # soldict.update(rhsd_vfrc)  # adding fvc, fpr
    soldict.update(fv=rhsd_stbc['fv']+rhsd_vfrc['fvc'],
                   fp=rhsd_stbc['fp']+rhsd_vfrc['fpr'],
                   N=N, nu=nu,
                   trange=tip['tmesh'],
                   get_datastring=get_datastr,
                   data_prfx=data_prfx,
                   clearprvdata=clearprvveldata,
                   paraviewoutput=tip['ParaviewOutput'],
                   vfileprfx=tip['proutdir']+'vel_',
                   pfileprfx=tip['proutdir']+'p_')

#
# Prepare for control
#

    contp = ContParams(femp['odcoo'], ystar=ystar, alphau=alphau, gamma=gamma)
    # casting some parameters
    NY, NU = contp.NY, contp.NU

    contsetupstr = problemname + '__NV{0}NU{1}NY{2}'.format(NV, NU, NY)

    # get the control and observation operators
    try:
        b_mat = dou.load_spa(ddir + contsetupstr + '__b_mat')
        u_masmat = dou.load_spa(ddir + contsetupstr + '__u_masmat')
        print 'loaded `b_mat`'
    except IOError:
        print 'computing `b_mat`...'
        b_mat, u_masmat = cou.get_inp_opa(cdcoo=femp['cdcoo'], V=femp['V'],
                                          NU=NU, xcomp=uspacedep)
        dou.save_spa(b_mat, ddir + contsetupstr + '__b_mat')
        dou.save_spa(u_masmat, ddir + contsetupstr + '__u_masmat')
    try:
        mc_mat = dou.load_spa(ddir + contsetupstr + '__mc_mat')
        y_masmat = dou.load_spa(ddir + contsetupstr + '__y_masmat')
        print 'loaded `c_mat`'
    except IOError:
        print 'computing `c_mat`...'
        mc_mat, y_masmat = cou.get_mout_opa(odcoo=femp['odcoo'],
                                            V=femp['V'], NY=NY)
        dou.save_spa(mc_mat, ddir + contsetupstr + '__mc_mat')
        dou.save_spa(y_masmat, ddir + contsetupstr + '__y_masmat')

    # restrict the operators to the inner nodes
    mc_mat = mc_mat[:, invinds][:, :]
    b_mat = b_mat[invinds, :][:, :]

    # for further use:
    c_mat = lau.apply_massinv(y_masmat, mc_mat, output='sparse')

    if contp.ystarx is None:
        c_mat = c_mat[NY:, :][:, :]  # TODO: Do this right
        mc_mat = mc_mat[NY:, :][:, :]  # TODO: Do this right
        y_masmat = y_masmat[:NY, :][:, :NY]  # TODO: Do this right

    mct_mat_reg = lau.app_prj_via_sadpnt(amat=stokesmatsc['M'],
                                         jmat=stokesmatsc['J'],
                                         rhsv=mc_mat.T,
                                         transposedprj=True)

    # set the weighing matrices
    contp.R = contp.alphau * u_masmat

#
# solve the differential-alg. Riccati eqn for the feedback gain X
# via computing factors Z, such that X = -Z*Z.T
#
# at the same time we solve for the affine-linear correction w
#

    # tilde B = BR^{-1/2}
    tb_mat = lau.apply_invsqrt_fromright(contp.R, b_mat, output='sparse')
    # tb_dense = np.array(tb_mat.todense())

    trct_mat = lau.apply_invsqrt_fromright(y_masmat,
                                           mct_mat_reg, output='dense')

    if closed_loop:
        cntpstr = 'NV{3}NY{0}NU{1}alphau{2}gamma{4}'.\
            format(contp.NU, contp.NY, contp.alphau, NV, contp.gamma)
    else:
        cntpstr = ''

    # we gonna use this quite often
    M, A = stokesmatsc['M'], stokesmatsc['A']
    datastrdict = dict(time=None, meshp=N, nu=nu, Nts=Nts,
                       data_prfx=data_prfx)

    # compute the uncontrolled steady state (Navier-)Stokes solution
    # as initial value
    if ini_vel_stokes:
        # compute the uncontrolled steady state Stokes solution
        ini_vel, newtonnorms = snu.solve_steadystate_nse(vel_nwtn_stps=0,
                                                         vel_pcrd_stps=0,
                                                         **soldict)
        soldict.update(dict(iniv=ini_vel))
    else:
        ini_vel, newtonnorms = snu.solve_steadystate_nse(**soldict)
        soldict.update(dict(iniv=ini_vel))

    if closed_loop:
        if stst_control:
            if stokes_flow:
                convc_mat = sps.csr_matrix((NV, NV))
                rhs_con, rhsv_conbc = np.zeros((NV, 1)), np.zeros((NV, 1))
                lin_point = None
            else:
                lin_point, newtonnorms = snu.solve_steadystate_nse(**soldict)
                (convc_mat, rhs_con,
                 rhsv_conbc) = snu.get_v_conv_conts(prev_v=lin_point,
                                                    invinds=invinds,
                                                    V=femp['V'],
                                                    diribcs=femp['diribcs'])
            # infinite control horizon, steady target state
            cdatstr = get_datastr(time=None, meshp=N, nu=nu,
                                  Nts=None, data_prfx=data_prfx)

            try:
                Z = dou.load_npa(cdatstr + cntpstr + '__Z')
                print 'loaded ' + cdatstr + cntpstr + '__Z'
            except IOError:
                if use_ric_ini_nu is not None:
                    cdatstr = get_datastr(nwtn=None, time=None, meshp=N,
                                          nu=use_ric_ini_nu, Nts=None,
                                          data_prfx=data_prfx)
                    try:
                        zini = dou.load_npa(ddir + cdatstr
                                            + cntpstr + '__Z')
                        print 'Initialize Newton ADI by Z from ' + cdatstr
                    except IOError:
                        raise Warning('No data for initialization of '
                                      ' Newton ADI -- need ' + cdatstr
                                      + '__Z')
                    cdatstr = get_datastr(meshp=N, nu=nu,
                                          data_prfx=data_prfx)
                else:
                    zini = None

                parnadi = pru.proj_alg_ric_newtonadi
                Z = parnadi(mmat=M, amat=-A-convc_mat,
                            jmat=stokesmatsc['J'],
                            bmat=tb_mat, wmat=trct_mat, z0=zini,
                            nwtn_adi_dict=tip['nwtn_adi_dict'])['zfac']

                dou.save_npa(Z, fstring=cdatstr + cntpstr + '__Z')
                print 'saved ' + cdatstr + cntpstr + '__Z'

                if tip['compress_z']:
                    Zc = pru.compress_Zsvd(Z, thresh=tip['comprz_thresh'],
                                           k=tip['comprz_maxc'])
                    Z = Zc

            fvnstst = rhs_con + rhsv_conbc + rhsd_stbc['fv'] + rhsd_vfrc['fvc']

            # X = -ZZ.T
            mtxtb_stst = -pru.get_mTzzTtb(M.T, Z, tb_mat)
            mtxfv_stst = -pru.get_mTzzTtb(M.T, Z, fvnstst)

            fl = mc_mat.T * contp.ystarvec(0)

            wft = lau.solve_sadpnt_smw(amat=A.T+convc_mat.T,
                                       jmat=stokesmatsc['J'],
                                       rhsv=fl+mtxfv_stst,
                                       umat=mtxtb_stst,
                                       vmat=tb_mat.T)[:NV]

            auxstrg = cdatstr + cntpstr
            dou.save_npa(wft, fstring=cdatstr + cntpstr + '__w')
            dou.save_npa(mtxtb_stst, fstring=cdatstr + cntpstr + '__mtxtb')
            feedbackthroughdict = {None:
                                   dict(w=auxstrg + '__w',
                                        mtxtb=auxstrg + '__mtxtb')}

            cns = 0
            soldict.update(data_prfx=data_prfx+'_cns{0}'.format(cns))
            if linearized_nse:
                soldict.update(vel_pcrd_stps=0,
                               vel_nwtn_stps=1,
                               lin_vel_point={None: lin_point})
            dictofvels = snu.\
                solve_nse(return_dictofvelstrs=True,
                          closed_loop=True,
                          static_feedback=True,
                          tb_mat=tb_mat,
                          stokes_flow=stokes_flow,
                          clearprvveldata=True,
                          feedbackthroughdict=feedbackthroughdict, **soldict)

        else:  # time dep closed loop

            cns_data_prfx = 'data/cnsvars'
            invd = init_nwtnstps_value_dict
            curnwtnsdict = invd(tmesh=tip['tmesh'],
                                data_prfx=cns_data_prfx)
            # initialization: compute the forward solution
            if stokes_flow:
                dictofvels = None
            else:
                dictofvels = snu.solve_nse(return_dictofvelstrs=True,
                                           stokes_flow=stokes_flow,
                                           **soldict)

            # dbs.plot_vel_norms(tip['tmesh'], dictofvels)

            # function for the time depending parts
            # -- to be passed to the solver
            def get_tdpart(time=None, dictofvalues=None, feedback=False,
                           V=None, invinds=None, diribcs=None, **kw):

                if stokes_flow:
                    convc_mat = sps.csr_matrix((NV, NV))
                    rhs_con, rhsv_conbc = np.zeros((NV, 1)), np.zeros((NV, 1))
                else:
                    curvel = dou.load_npa(dictofvalues[time])
                    convc_mat, rhs_con, rhsv_conbc = \
                        snu.get_v_conv_conts(prev_v=curvel, invinds=invinds,
                                             V=V, diribcs=diribcs)

                return convc_mat, rhsv_conbc+rhs_con

            gttdprtargs = dict(dictofvalues=dictofvels,
                               V=femp['V'],
                               diribcs=femp['diribcs'],
                               invinds=invinds)

            # old version rhs
            # ftilde = rhs_con + rhsv_conbc + rhsd_stbc['fv']
            for cns in range(outernwtnstps):

                datastrdict.update(data_prfx=data_prfx+cntpstr+'_cns{0}'.
                                   format(cns))
                soldict.update(data_prfx=data_prfx+cntpstr+'_cns{0}'.
                               format(cns))

                sfd = sdr.solve_flow_daeric
                feedbackthroughdict = \
                    sfd(mmat=M, amat=A, jmat=stokesmatsc['J'],
                        bmat=b_mat,
                        # cmat=ct_mat_reg.T,
                        mcmat=mct_mat_reg.T,
                        v_is_my=True, rmat=contp.alphau*u_masmat,
                        vmat=y_masmat, rhsv=rhsd_stbc['fv'],
                        gamma=contp.gamma,
                        rhsp=None,
                        tmesh=tip['tmesh'], ystarvec=contp.ystarvec,
                        nwtn_adi_dict=tip['nwtn_adi_dict'],
                        comprz_thresh=tip['comprz_thresh'],
                        comprz_maxc=tip['comprz_maxc'], save_full_z=False,
                        get_tdpart=get_tdpart, gttdprtargs=gttdprtargs,
                        curnwtnsdict=curnwtnsdict,
                        get_datastr=get_datastr, gtdtstrargs=datastrdict)

                # for t in tip['tmesh']:  # feedbackthroughdict.keys():
                #     curw = dou.load_npa(feedbackthroughdict[t]['mtxtb'])
                #     print cns, t, np.linalg.norm(curw)

                cdatstr = get_datastr(time='all', meshp=N, nu=nu,
                                      Nts=None, data_prfx=data_prfx)

                if linearized_nse:
                    dictofvels = snu.\
                        solve_nse(return_dictofvelstrs=True,
                                  closed_loop=True, tb_mat=tb_mat,
                                  lin_vel_point=dictofvels,
                                  feedbackthroughdict=feedbackthroughdict,
                                  vel_nwtn_stps=1,
                                  vel_pcrd_stps=0,
                                  **soldict)
                else:
                    dictofvels = snu.\
                        solve_nse(return_dictofvelstrs=True,
                                  closed_loop=True, tb_mat=tb_mat,
                                  stokes_flow=stokes_flow,
                                  feedbackthroughdict=feedbackthroughdict,
                                  vel_pcrd_stps=1,
                                  vel_nwtn_stps=2,
                                  **soldict)

                # for t in dictofvels.keys():
                #     curw = dou.load_npa(dictofvels[t])
                #     print cns, t, np.linalg.norm(curw)

                gttdprtargs.update(dictofvalues=dictofvels)
    else:
        # no control
        feedbackthroughdict = None
        tb_mat = None
        cdatstr = get_datastr(meshp=N, nu=nu, time='all',
                              Nts=Nts, data_prfx=data_prfx)

        soldict.update(clearprvdata=True)
        dictofvels = snu.solve_nse(feedbackthroughdict=feedbackthroughdict,
                                   tb_mat=tb_mat, closed_loop=closed_loop,
                                   stokes_flow=stokes_flow,
                                   return_dictofvelstrs=True,
                                   static_feedback=stst_control,
                                   **soldict)

    (yscomplist,
     ystarlist) = dou.extract_output(dictofpaths=dictofvels,
                                     tmesh=tip['tmesh'],
                                     c_mat=c_mat, ystarvec=contp.ystarvec)

    save_output_json(yscomplist, tip['tmesh'].tolist(), ystar=ystarlist,
                     fstring=cdatstr + cntpstr + '__sigout')

    costfunval = eval_costfunc(W=y_masmat, V=contp.gamma*y_masmat,
                               R=None, tbmat=tb_mat, cmat=c_mat,
                               ystar=contp.ystarvec,
                               tmesh=tip['tmesh'], veldict=dictofvels,
                               fbftdict=feedbackthroughdict)

    print 'Value of cost functional: ', costfunval

    costfunval = eval_costfunc(W=y_masmat, V=contp.gamma*y_masmat,
                               R=None, tbmat=tb_mat, cmat=c_mat,
                               ystar=contp.ystarvec, penau=False,
                               tmesh=tip['tmesh'], veldict=dictofvels,
                               fbftdict=feedbackthroughdict)

    print 'Value of cost functional not considering `u`: ', costfunval

    print 'dim of v :', femp['V'].dim()
    charlene = .15 if problemname == 'cylinderwake' else 1.0
    print 'Re = charL / nu = {0}'.format(charlene/nu)