# 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)
#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)