# Nxy = 10
# Dt = 2.5e-3
# t0, tf = 0.0, 7.0
# high frequency:
fpeak = 4.0  # Hz
Nxy = 100
Dt = 2.5e-4
t0, tf = 0.0, 2.0
# parameters
cmin = 2.0
cmax = 3.0
epsmin = -1.0  # medium perturbation
epsmax = 1.0  # medium perturbation
h = 1.0 / Nxy
r = 2
checkdt(Dt, h, r, cmax + epsmax, True)
# mesh
mesh = dl.UnitSquareMesh(Nxy, Nxy)
Vl, V = dl.FunctionSpace(mesh, "Lagrange", 1), dl.FunctionSpace(mesh, "Lagrange", r)
fctV = dl.Function(V)
fctVl = dl.Function(Vl)
# set up plots:
filename, ext = splitext(sys.argv[0])
if isdir(filename + "/"):
    rmtree(filename + "/")
myplot = PlotFenics(filename)
# source:
Ricker = RickerWavelet(fpeak, 1e-10)
Pt = PointSources(V, [[0.5, 1.0]])
mydelta = Pt[0].array()
              for ii in range(1, 10)] + [[ii / 10., 0.0]
                                         for ii in range(1, 10)]
    Pt = PointSources(V, srcloc)
    src = dl.Function(V)
    srcv = src.vector()
    mysrc = [Ricker, Pt, srcv]

    # target medium:
    b_target = dl.Expression(\
    '1.0 + 1.0*(x[0]<=0.7)*(x[0]>=0.3)*(x[1]<=0.7)*(x[1]>=0.3)')
    b_target_fn = dl.interpolate(b_target, Vm)
    a_target = dl.Expression(\
    '1.0 + 0.4*(x[0]<=0.7)*(x[0]>=0.3)*(x[1]<=0.7)*(x[1]>=0.3)')
    a_target_fn = dl.interpolate(a_target, Vm)

    checkdt(Dt, 1. / Nxy, r, np.sqrt(2.0), False)

    # observation operator:
    obspts = [[0.0, ii/10.] for ii in range(1,10)] + \
    [[1.0, ii/10.] for ii in range(1,10)] + \
    [[ii/10., 0.0] for ii in range(1,10)] + \
    [[ii/10., 1.0] for ii in range(1,10)]
    obsop = TimeObsPtwise({'V': V, 'Points': obspts}, [t0, t1, t2, tf])

    # define pde operator:
    wavepde = AcousticWave({'V': V, 'Vm': Vm})
    wavepde.timestepper = 'backward'
    wavepde.lump = True
    wavepde.update({'a':a_target_fn, 'b':b_target_fn, \
    't0':t0, 'tf':tf, 'Dt':Dt, 'u0init':dl.Function(V), 'utinit':dl.Function(V)})
Nxy = 100
Dt = 5e-4  #Dt = h/(r*alpha*c_max)
tf = 2.0
mytf = TimeFilter([0.0, 0.2, tf - 0.2, tf])

#fpeak = .4 # 4Hz => up to 10Hz in input signal
#Nxy = 10
#Dt = 1e-4   #Dt = h/(r*alpha*c_max)
#tf = 7.0
#mytf = TimeFilter([0.,1.,6.,tf])

mesh = UnitSquareMesh(Nxy, Nxy)
h = 1. / Nxy
Vl = FunctionSpace(mesh, 'Lagrange', 1)
r = 2
checkdt(Dt, h, r, 2.0, False)
# Source term:
Ricker = RickerWavelet(fpeak, 1e-10)

# Boundary conditions:
#class AllFour(SubDomain):
#    def inside(self, x, on_boundary):
#        return on_boundary

V = FunctionSpace(mesh, 'Lagrange', r)
Pt = PointSources(V, [[.5, 1.]])
mydelta = Pt[0]
src = Function(V)
srcv = src.vector()

def run_test(fpeak, lambdamin, lambdamax, Nxy, tfilterpts, r, Dt, skip):
    h = 1. / Nxy
    checkdt(Dt, h, r, np.sqrt(lambdamax), True)
    mesh = dl.UnitSquareMesh(Nxy, Nxy)
    Vl = dl.FunctionSpace(mesh, 'Lagrange', 1)
    V = dl.FunctionSpace(mesh, 'Lagrange', r)
    fctV = dl.Function(V)
    # set up plots:
    filename, ext = splitext(sys.argv[0])
    if isdir(filename + '/'): rmtree(filename + '/')
    myplot = PlotFenics(filename)
    # source:
    Ricker = RickerWavelet(fpeak, 1e-10)
    Pt = PointSources(V, [[0.5, 0.5]])
    mydelta = Pt[0].array()

    def mysrc(tt):
        return Ricker(tt) * mydelta

    # target medium:
    lambda_target = dl.Expression('lmin + x[0]*(lmax-lmin)', \
    lmin=lambdamin, lmax=lambdamax)
    lambda_target_fn = dl.interpolate(lambda_target, Vl)
    myplot.set_varname('lambda_target')
    myplot.plot_vtk(lambda_target_fn)
    # initial medium:
    lambda_init = dl.Constant(lambdamin)
    lambda_init_fn = dl.interpolate(lambda_init, Vl)
    myplot.set_varname('lambda_init')
    myplot.plot_vtk(lambda_init_fn)
    # observation operator:
    #obspts = [[0.2, 0.5], [0.5, 0.2], [0.5, 0.8], [0.8, 0.5]]
    obspts = [[0.2, ii/10.] for ii in range(2,9)] + \
    [[0.8, ii/10.] for ii in range(2,9)] + \
    [[ii/10., 0.2] for ii in range(3,8)] + \
    [[ii/10., 0.8] for ii in range(3,8)]
    obsop = TimeObsPtwise({'V': V, 'Points': obspts}, tfilterpts)
    # define pde operator:
    wavepde = AcousticWave({'V': V, 'Vl': Vl, 'Vr': Vl})
    wavepde.timestepper = 'backward'
    wavepde.lump = True
    wavepde.set_abc(mesh, LeftRight(), True)
    wavepde.update({'lambda':lambda_target_fn, 'rho':1.0, \
    't0':t0, 'tf':tf, 'Dt':Dt, 'u0init':dl.Function(V), 'utinit':dl.Function(V)})
    wavepde.ftime = mysrc
    # define objective function:
    waveobj = ObjectiveAcoustic(wavepde)
    waveobj.obsop = obsop
    # data
    print 'generate data'
    waveobj.solvefwd()
    myplot.plot_timeseries(waveobj.solfwd, 'pd', 0, skip, fctV)
    dd = waveobj.Bp.copy()
    # gradient
    print 'generate observations'
    waveobj.dd = dd
    waveobj.update_m(lambda_init_fn)
    waveobj.solvefwd_cost()
    cost1 = waveobj.misfit
    print 'misfit = {}'.format(waveobj.misfit)
    myplot.plot_timeseries(waveobj.solfwd, 'p', 0, skip, fctV)
    # Plot data and observations
    fig = plt.figure()
    if len(obspts) > 9: fig.set_size_inches(20., 15.)
    for ii in range(len(obspts)):
        if len(obspts) == 4: ax = fig.add_subplot(2, 2, ii + 1)
        else: ax = fig.add_subplot(4, 6, ii + 1)
        ax.plot(waveobj.PDE.times, waveobj.dd[ii, :], 'k--')
        ax.plot(waveobj.PDE.times, waveobj.Bp[ii, :], 'b')
        ax.set_title('Plot' + str(ii))
    fig.savefig(filename + '/observations.eps')
    print 'compute gradient'
    waveobj.solveadj_constructgrad()
    myplot.plot_timeseries(waveobj.soladj, 'v', 0, skip, fctV)
    MG = waveobj.MGv.array().copy()
    myplot.set_varname('grad')
    myplot.plot_vtk(waveobj.Grad)
    """
#Nxy = 100
#Dt = 5e-4   #Dt = h/(r*alpha*c_max)
#tf = 1.4
#mytf = TimeFilter([0.,.2,1.2,1.4])

fpeak = .4 # 4Hz => up to 10Hz in input signal
Nxy = 10
Dt = 1e-4   #Dt = h/(r*alpha*c_max)
tf = 7.0
mytf = TimeFilter([0.,1.,6.,tf])

mesh = UnitSquareMesh(Nxy, Nxy)
h = 1./Nxy
Vl = FunctionSpace(mesh, 'Lagrange', 1)
r = 2
checkdt(Dt, h, r, 2.0, True)
# Source term:
Ricker = RickerWavelet(fpeak, 1e-10)

# Boundary conditions:
#class AllFour(SubDomain):
#    def inside(self, x, on_boundary):
#        return on_boundary

V = FunctionSpace(mesh, 'Lagrange', r)
Pt = PointSources(V, [[.5,1.]])
mydelta = Pt[0].array()
def mysrc(tt):
    return Ricker(tt)*mydelta
# Computation:
if myrank == 0: print '\n\th = {}, Dt = {}'.format(h, Dt)
Example #6
0
    #srcloc = [[0.5,1.0]]
    #srcloc = [[ii/10., 1.0] for ii in range(1,10)] + [[ii/10., 0.0] for ii in range(1,10)]
    srcloc = [[ii / 10., 1.0] for ii in range(3, 8, 2)]
    Pt = PointSources(V, srcloc)
    src = dl.Function(V)
    srcv = src.vector()
    mysrc = [Ricker, Pt, srcv]

    # target medium:
    b_target = dl.Expression(\
    '1.0 + 1.0*(x[0]<=0.7)*(x[0]>=0.3)*(x[1]<=0.7)*(x[1]>=0.3)')
    b_target_fn = dl.interpolate(b_target, Vm)
    a_target = dl.Expression('1.0')
    a_target_fn = dl.interpolate(a_target, Vm)

    checkdt(Dt, 1. / Nxy, r, np.sqrt(2.0), True)

    # observation operator:
    obspts = [[0.0, ii/10.] for ii in range(1,10)] + \
    [[1.0, ii/10.] for ii in range(1,10)] + \
    [[ii/10., 0.0] for ii in range(1,10)] + \
    [[ii/10., 1.0] for ii in range(1,10)]
    obsop = TimeObsPtwise({'V': V, 'Points': obspts}, [t0, t1, t2, tf])

    # define pde operator:
    if mpirank == 0: print 'define wave pde'
    wavepde = AcousticWave({'V': V, 'Vm': Vm})
    wavepde.timestepper = 'backward'
    wavepde.lump = True
    wavepde.update({'a':a_target_fn, 'b':b_target_fn, \
    't0':t0, 'tf':tf, 'Dt':Dt, 'u0init':dl.Function(V), 'utinit':dl.Function(V)})
def run_test(fpeak, lambdamin, lambdamax, Nxy, tfilterpts, r, Dt, skip):
    h = 1./Nxy
    checkdt(Dt, h, r, np.sqrt(lambdamax), True)
    mesh = dl.UnitSquareMesh(Nxy, Nxy)
    Vl = dl.FunctionSpace(mesh, 'Lagrange', 1)
    V = dl.FunctionSpace(mesh, 'Lagrange', r)
    fctV = dl.Function(V)
    # set up plots:
    filename, ext = splitext(sys.argv[0])
    if isdir(filename + '/'):   rmtree(filename + '/')
    myplot = PlotFenics(filename)
    # source:
    Ricker = RickerWavelet(fpeak, 1e-10)
    Pt = PointSources(V, [[0.5,0.5]])
    mydelta = Pt[0].array()
    def mysrc(tt):
        return Ricker(tt)*mydelta
    # target medium:
    lambda_target = dl.Expression('lmin + x[0]*(lmax-lmin)', \
    lmin=lambdamin, lmax=lambdamax)
    lambda_target_fn = dl.interpolate(lambda_target, Vl)
    myplot.set_varname('lambda_target')
    myplot.plot_vtk(lambda_target_fn)
    # initial medium:
    lambda_init = dl.Constant(lambdamin)
    lambda_init_fn = dl.interpolate(lambda_init, Vl)
    myplot.set_varname('lambda_init')
    myplot.plot_vtk(lambda_init_fn)
    # observation operator:
    #obspts = [[0.2, 0.5], [0.5, 0.2], [0.5, 0.8], [0.8, 0.5]]
    obspts = [[0.2, ii/10.] for ii in range(2,9)] + \
    [[0.8, ii/10.] for ii in range(2,9)] + \
    [[ii/10., 0.2] for ii in range(3,8)] + \
    [[ii/10., 0.8] for ii in range(3,8)]
    obsop = TimeObsPtwise({'V':V, 'Points':obspts}, tfilterpts)
    # define pde operator:
    wavepde = AcousticWave({'V':V, 'Vl':Vl, 'Vr':Vl})
    wavepde.timestepper = 'centered'
    wavepde.lump = True
    wavepde.set_abc(mesh, LeftRight(), True)
    wavepde.update({'lambda':lambda_target_fn, 'rho':1.0, \
    't0':t0, 'tf':tf, 'Dt':Dt, 'u0init':dl.Function(V), 'utinit':dl.Function(V)})
    wavepde.ftime = mysrc
    # define objective function:
    waveobj = ObjectiveAcoustic(wavepde)
    waveobj.obsop = obsop
    # data
    print 'generate noisy data'
    waveobj.solvefwd()
    myplot.plot_timeseries(waveobj.solfwd, 'pd', 0, skip, fctV)
    dd = waveobj.Bp.copy()
    nbobspt, dimsol = dd.shape
    noiselevel = 0.1   # = 10%
    sigmas = np.sqrt((dd**2).sum(axis=1)/dimsol)*noiselevel
    rndnoise = np.random.randn(nbobspt*dimsol).reshape((nbobspt, dimsol))
    waveobj.dd = dd + sigmas.reshape((len(sigmas),1))*rndnoise
    # gradient
    print 'generate observations'
    waveobj.update_m(lambda_init_fn)
    waveobj.solvefwd_cost()
    cost1 = waveobj.misfit
    print 'misfit = {}'.format(waveobj.misfit)
    myplot.plot_timeseries(waveobj.solfwd, 'p', 0, skip, fctV)
    # Plot data and observations
    fig = plt.figure()
    if len(obspts) > 9: fig.set_size_inches(20., 15.)
    for ii in range(len(obspts)):
        if len(obspts) == 4:    ax = fig.add_subplot(2,2,ii+1)
        else:   ax = fig.add_subplot(4,6,ii+1)
        ax.plot(waveobj.PDE.times, waveobj.dd[ii,:], 'k--')
        ax.plot(waveobj.PDE.times, waveobj.Bp[ii,:], 'b')
        ax.set_title('Plot'+str(ii))
    fig.savefig(filename + '/observations.eps')
    print 'compute gradient'
    waveobj.solveadj_constructgrad()
    myplot.plot_timeseries(waveobj.soladj, 'v', 0, skip, fctV)
    MG = waveobj.MGv.array().copy()
    myplot.set_varname('grad')
    myplot.plot_vtk(waveobj.Grad)
    print 'check gradient with FD'
    Medium = np.zeros((5, Vl.dim()))
    for ii in range(5):
        smoothperturb = dl.Expression('sin(n*pi*x[0])*sin(n*pi*x[1])', n=ii+1)
        smoothperturb_fn = dl.interpolate(smoothperturb, Vl)
        Medium[ii,:] = smoothperturb_fn.vector().array()
    checkgradfd_med(waveobj, Medium, 1e-6, [1e-5, 1e-4])
    print 'check Hessian with FD'
    checkhessfd_med(waveobj, Medium, 1e-6, [1e-1, 1e-2, 1e-3, 1e-4, 1e-5], False)