def RunAcoustic(q):
Nxy = 100
tf = 0.1 # Final time
direction = 0
u0_expr = Expression("100*pow(x[i]-.25,2)*pow(x[i]-0.75,2)*(x[i]<=0.75)*(x[i]>=0.25)", i=direction)
class LeftRight(SubDomain):
def inside(self, x, on_boundary):
return (x[direction] < 1e-16 or x[direction] > 1.0 - 1e-16) and on_boundary
h = 1.0 / Nxy
mesh = UnitSquareMesh(Nxy, Nxy, "crossed")
V = FunctionSpace(mesh, "Lagrange", q)
Vl = FunctionSpace(mesh, "Lagrange", 1)
Dt = h / (q * 10.0)
Wave = AcousticWave({"V": V, "Vl": Vl, "Vr": Vl})
Wave.verbose = True
Wave.lump = True
Wave.set_abc(mesh, LeftRight())
Wave.update(
{
"lambda": 1.0,
"rho": 1.0,
"t0": 0.0,
"tf": tf,
"Dt": Dt,
"u0init": interpolate(u0_expr, V),
"utinit": Function(V),
}
)
sol, tmp = Wave.solve()
def RunAcoustic(q):
Nxy = 100
tf = 0.1 # Final time
direction = 0
u0_expr = Expression(\
'100*pow(x[i]-.25,2)*pow(x[i]-0.75,2)*(x[i]<=0.75)*(x[i]>=0.25)', i=direction)
class LeftRight(SubDomain):
def inside(self, x, on_boundary):
return (x[direction] < 1e-16 or x[direction] > 1.0 - 1e-16) \
and on_boundary
h = 1. / Nxy
mesh = UnitSquareMesh(Nxy, Nxy, "crossed")
V = FunctionSpace(mesh, 'Lagrange', q)
Vl = FunctionSpace(mesh, 'Lagrange', 1)
Dt = h / (q * 10.)
Wave = AcousticWave({'V': V, 'Vl': Vl, 'Vr': Vl})
Wave.verbose = True
Wave.lump = True
Wave.set_abc(mesh, LeftRight())
Wave.update({'lambda':1.0, 'rho':1.0, 't0':0.0, 'tf':tf, 'Dt':Dt,\
'u0init':interpolate(u0_expr, V), 'utinit':Function(V)})
sol, tmp = Wave.solve()
def run_test():
q = 3
Nxy = 100
tf = 0.1
h = 1./Nxy
mesh = UnitSquareMesh(Nxy, Nxy, "crossed")
V = FunctionSpace(mesh, 'Lagrange', q)
Vl = FunctionSpace(mesh, 'Lagrange', 1)
Dt = h/(q*10.)
u0_expr = Expression(\
'100*pow(x[i]-.25,2)*pow(x[i]-0.75,2)*(x[i]<=0.75)*(x[i]>=0.25)', i=0)
Wave = AcousticWave({'V':V, 'Vl':Vl, 'Vr':Vl})
Wave.lump = True
Wave.timestepper = 'backward'
Wave.update({'lambda':1.0, 'rho':1.0, 't0':0.0, 'tf':tf, 'Dt':Dt,\
'u0init':interpolate(u0_expr, V), 'utinit':Function(V)})
K = Wave.K
u = Wave.u0
u.vector()[:] = 1.0
b = Wave.u1
for ii in range(100):
K*u.vector()
(K*u.vector()).array()
b.vector()[:] = (K*u.vector()).array()
b.vector()[:] = 0.0
b.vector().axpy(1.0, K*u.vector())
b.vector()[:] = (K*u.vector()).array() + (K*u.vector()).array()
b.vector()[:] = (K*u.vector() + K*u.vector()).array()
b.vector()[:] = 2.*u.vector().array() + u.vector().array() + \
Dt*u.vector().array()
b.vector()[:] = 0.0
b.vector().axpy(2.0, u.vector())
b.vector().axpy(1.0, u.vector())
b.vector().axpy(Dt, u.vector())
b.assign(u)
b.vector().zero()
def run_test():
q = 3
Nxy = 100
tf = 0.1
h = 1. / Nxy
mesh = UnitSquareMesh(Nxy, Nxy, "crossed")
V = FunctionSpace(mesh, 'Lagrange', q)
Vl = FunctionSpace(mesh, 'Lagrange', 1)
Dt = h / (q * 10.)
u0_expr = Expression(\
'100*pow(x[i]-.25,2)*pow(x[i]-0.75,2)*(x[i]<=0.75)*(x[i]>=0.25)', i=0)
Wave = AcousticWave({'V': V, 'Vl': Vl, 'Vr': Vl})
Wave.lump = True
Wave.timestepper = 'backward'
Wave.update({'lambda':1.0, 'rho':1.0, 't0':0.0, 'tf':tf, 'Dt':Dt,\
'u0init':interpolate(u0_expr, V), 'utinit':Function(V)})
K = Wave.K
u = Wave.u0
u.vector()[:] = 1.0
b = Wave.u1
for ii in range(100):
K * u.vector()
(K * u.vector()).array()
b.vector()[:] = (K * u.vector()).array()
b.vector()[:] = 0.0
b.vector().axpy(1.0, K * u.vector())
b.vector()[:] = (K * u.vector()).array() + (K * u.vector()).array()
b.vector()[:] = (K * u.vector() + K * u.vector()).array()
b.vector()[:] = 2.*u.vector().array() + u.vector().array() + \
Dt*u.vector().array()
b.vector()[:] = 0.0
b.vector().axpy(2.0, u.vector())
b.vector().axpy(1.0, u.vector())
b.vector().axpy(Dt, u.vector())
b.assign(u)
b.vector().zero()
h = 1. / N
mesh = dl.UnitSquareMesh(N, N)
Vl = dl.FunctionSpace(mesh, 'Lagrange', 1)
Vex = dl.FunctionSpace(mesh, 'Lagrange', r)
Pt = PointSources(Vex, [[.5, .5]])
mydelta = Pt[0].array()
def mysrc(tt):
return Ricker(tt) * mydelta
Waveex = AcousticWave({'V': Vex, 'Vm': Vl})
Waveex.timestepper = 'backward'
Waveex.lump = True
Waveex.update({'a':1.0, 'b':1.0, 't0':0.0, 'tf':tf, 'Dt':Dt,\
'u0init':dl.Function(Vex), 'utinit':dl.Function(Vex)})
Waveex.ftime = mysrc
sol, _ = Waveex.solve()
Waveex.exact = dl.Function(Vex)
normex = Waveex.computeabserror()
# plot
myplot.set_varname('u-q' + str(qq))
plotu = dl.Function(Vex)
for index, uu in enumerate(sol):
if index % boolplot == 0:
setfct(plotu, uu[0])
myplot.plot_vtk(plotu, index)
myplot.gather_vtkplots()
print 'Check different spatial sampling'
QQ = [4, 5, 6, 10]
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)})
# parameters
Vm = wavepde.Vm
V = wavepde.V
lenobspts = obsop.PtwiseObs.nbPts
# set up plots:
filename, ext = splitext(sys.argv[0])
if isdir(filename + '/'): rmtree(filename + '/')
myplot = PlotFenics(filename + str(Nxy))
myplot.set_varname('a_target')
myplot.plot_vtk(a_target_fn)
# define objective function:
regul = LaplacianPrior({
myplot.plot_vtk(TargetMed)
# perturbation
# PerturbationMedExpr = dl.Expression(medformula, A=1.0)
# PerturbMed = dl.interpolate(PerturbationMedExpr, Vl)
# myplot.set_varname('perturb_medium')
# myplot.plot_vtk(PerturbMed)
# observation operator:
obspts = [[x / 10.0, 1.0] for x in range(1, 10)]
obsop = TimeObsPtwise({"V": V, "Points": obspts})
# define pde operator:
wavepde = AcousticWave({"V": V, "Vl": Vl, "Vr": Vl})
wavepde.timestepper = "backward"
wavepde.lump = True
wavepde.set_abc(mesh, ABC(), True)
wavepde.update(
{"lambda": TargetMed, "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, 40, fctV)
dd = waveobj.Bp.copy()
waveobj.dd = dd
# Plot observations
# fig = plt.figure()
# for ii in range(len(obspts)):
srcv.zero()
srcv.axpy(Ricker(tt), mydelta)
return srcv
# Computation:
if myrank == 0: print '\n\th = {}, Dt = {}'.format(h, Dt)
Wave = AcousticWave({'V': V, 'Vm': Vl})
#Wave.verbose = True
Wave.timestepper = 'backward'
Wave.lump = True
#Wave.set_abc(mesh, AllFour(), True)
lambda_target = Expression('1.0 + 3.0*(' \
'(x[0]>=0.3)*(x[0]<=0.7)*(x[1]>=0.3)*(x[1]<=0.7))')
lambda_target_fn = interpolate(lambda_target, Vl)
Wave.update({'b':lambda_target_fn, 'a':1.0, \
't0':0.0, 'tf':tf, 'Dt':Dt, 'u0init':Function(V), 'utinit':Function(V)})
Wave.ftime = mysrc
sol, tmp = Wave.solve()
# Observations
myObs = TimeObsPtwise({
'V': V,
'Points': [[.5, .2], [.5, .8], [.2, .5], [.8, .5]]
})
Bp = np.zeros((4, len(sol)))
mytimes = np.zeros(len(sol))
solp = Function(V)
for index, pp in enumerate(sol):
setfct(solp, pp[0])
Bp[:, index] = myObs.obs(solp)
mytimes[index] = pp[1]
Bpf = Bp * mytf.evaluate(mytimes)
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)
"""
for Nxy in NN:
h = 1. / Nxy
mesh = UnitSquareMesh(Nxy, Nxy)
V = FunctionSpace(mesh, 'Lagrange', q)
Vl = FunctionSpace(mesh, 'Lagrange', 1)
Dt = h / (q * alpha * c)
if myrank == 0: print '\n\th = {}, Dt = {}'.format(h, Dt)
Wave = AcousticWave({'V': V, 'Vm': Vl})
#Wave.verbose = True
Wave.timestepper = 'centered'
Wave.lump = True
Wave.set_abc(mesh, LeftRight(), True)
Wave.exact = interpolate(exact_expr, V)
Wave.update({'b':b, 'a':a, 't0':0.0, 'tf':tf, 'Dt':Dt,\
'u0init':interpolate(u0_expr, V), 'utinit':Function(V)})
Wave.ftime = lambda t: 0.0
sol, error = Wave.solve()
ERROR.append(error)
if myrank == 0: print 'relative error = {:.5e}'.format(error)
if not mycomm == None: MPI.barrier(mycomm)
if myrank == 0:
# Order of convergence:
CONVORDER = []
for ii in range(len(ERROR) - 1):
CONVORDER.append(
np.log(ERROR[ii + 1] / ERROR[ii]) / np.log(
(1. / NN[ii + 1]) / (1. / NN[ii])))
print '\n\norder of convergence:', CONVORDER
Vl = FunctionSpace(mesh, 'Lagrange', 1)
c_max = 2
#Dt = h/(q*alpha*c_max)
Dt = 1e-3
checkdt_abc(Dt, h, q, c_max, True, True, 'centered')
if myrank == 0: print '\n\th = {}, Dt = {}'.format(h, Dt)
Wave = AcousticWave({'V':V, 'Vl':Vl, 'Vr':Vl})
#Wave.verbose = True
Wave.timestepper = 'centered'
Wave.lump = True
Wave.set_abc(mesh, LeftRight(), True)
# Medium ppties:
lam_expr = Expression('1.0 + 3.0*(x[i]<=0.25)', i=direction)
lam = interpolate(lam_expr, Vl)
Wave.update({'lambda':lam, 'rho':1.0, 't0':0.0, 'tf':1.5, 'Dt':Dt,\
'u0init':interpolate(u0_expr, V), 'utinit':Function(V)})
Wave.ftime = lambda t: 0.0
sol, tmp = Wave.solve()
if not mycomm == None: MPI.barrier(mycomm)
# Save plots:
try:
boolplot = int(sys.argv[1])
except:
boolplot = 10
if boolplot > 0:
filename, ext = splitext(sys.argv[0])
if myrank == 0:
if isdir(filename + '/'): rmtree(filename + '/')
if not mycomm == None: MPI.barrier(mycomm)
myplot = PlotFenics(filename)
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)
Wave = AcousticWave({'V':V, 'Vl':Vl, 'Vr':Vl})
#Wave.verbose = True
Wave.timestepper = 'backward'
Wave.lump = True
#Wave.set_abc(mesh, AllFour(), True)
lambda_target = Expression('1.0 + 3.0*(' \
'(x[0]>=0.3)*(x[0]<=0.7)*(x[1]>=0.3)*(x[1]<=0.7))')
lambda_target_fn = interpolate(lambda_target, Vl)
Wave.update({'lambda':lambda_target_fn, 'rho':1.0, \
't0':0.0, 'tf':tf, 'Dt':Dt, 'u0init':Function(V), 'utinit':Function(V)})
Wave.ftime = mysrc
sol, tmp = Wave.solve()
if not mycomm == None: MPI.barrier(mycomm)
# Observations
myObs = TimeObsPtwise({'V':V, 'Points':[[.5,.2], [.5,.8], [.2,.5], [.8,.5]]})
Bp = np.zeros((4, len(sol)))
mytimes = np.zeros(len(sol))
solp = Function(V)
for index, pp in enumerate(sol):
setfct(solp, pp[0])
Bp[:,index] = myObs.obs(solp)
mytimes[index] = pp[1]
Bpf = Bp*mytf.evaluate(mytimes)
print 'Compute most accurate solution as reference'
qq = 20
N = int(qq/cmin)
h = 1./N
mesh = dl.UnitSquareMesh(N,N)
Vl = dl.FunctionSpace(mesh, 'Lagrange', 1)
Vex = dl.FunctionSpace(mesh, 'Lagrange', r)
Pt = PointSources(Vex, [[.5,.5]])
mydelta = Pt[0].array()
def mysrc(tt):
return Ricker(tt)*mydelta
Waveex = AcousticWave({'V':Vex, 'Vl':Vl, 'Vr':Vl})
Waveex.timestepper = 'backward'
Waveex.lump = True
Waveex.update({'lambda':1.0, 'rho':1.0, 't0':0.0, 'tf':tf, 'Dt':Dt,\
'u0init':dl.Function(Vex), 'utinit':dl.Function(Vex)})
Waveex.ftime = mysrc
sol,_ = Waveex.solve()
Waveex.exact = dl.Function(Vex)
normex = Waveex.computeabserror()
# plot
myplot.set_varname('u-q'+str(qq))
plotu = dl.Function(Vex)
for index, uu in enumerate(sol):
if index%boolplot == 0:
setfct(plotu, uu[0])
myplot.plot_vtk(plotu, index)
myplot.gather_vtkplots()
print 'Check different spatial sampling'
QQ = [4, 5, 6, 10]
for Nxy in NN:
h = 1. / Nxy
print '\n\th = {}'.format(h)
mesh = UnitSquareMesh(Nxy, Nxy, "crossed")
q = 1 # Polynomial order
V = FunctionSpace(mesh, 'Lagrange', q)
Dt = h / (q * 5. * c)
Wave = AcousticWave({'V': V, 'Vl': V, 'Vr': V})
Wave.timestepper = 'backward'
Wave.lump = True
#Wave.verbose = True
Wave.exact = interpolate(exact_expr, V)
Wave.update({'lambda':lam, 'rho':rho, 't0':0.0, 'tf':tf, 'Dt':Dt,\
'u0init':Function(V), 'utinit':Function(V)})
test = TestFunction(V)
def srcterm(tt):
src_expr = source(tt)
src_vect = assemble(src_expr * test * dx)
return src_vect.array()
Wave.ftime = srcterm
sol, error = Wave.solve()
ERROR.append(error)
print 'relative error = {:.5e}'.format(error)
# Convergence order:
CONVORDER = []
for ii in range(len(ERROR) - 1):
return on_boundary and (x[1] < Y)
Wave = AcousticWave({
'V': V,
'Vm': Vl
}, {
'print': False,
'lumpM': True,
'timestepper': 'backward'
})
Wave.set_abc(mesh, ABCdom(), lumpD=False)
at, bt, ct, _, _ = targetmediumparameters(Vl, X)
a0, b0, _, _, _ = initmediumparameters(Vl, X)
Wave.update({'b':bt, 'a':at, 't0':0.0, 'tf':tf, 'Dt':Dt,\
'u0init':dl.Function(V), 'utinit':dl.Function(V)})
if PRINT:
print 'nb of src={}, nb of timesteps={}'.format(len(Pt.src_loc), Wave.Nt)
sources, timesteps = partition_work(mpicomm_local, mpicomm_global, \
len(Pt.src_loc), Wave.Nt)
mpilocalrank = MPI.rank(mpicomm_local)
mpiglobalrank = MPI.rank(mpicomm_global)
mpiworldsize = MPI.size(dl.mpi_comm_world())
print 'mpiworldrank={}, mpiglobalrank={}, mpilocalrank={}, sources={}, timestep=[{},{}]'.format(\
mpiworldrank, mpiglobalrank, mpilocalrank, sources,\
timesteps[0], timesteps[-1])
# observation operator:
obspts = [[ii * float(X) / float(Nxy), Y] for ii in range(1, Nxy)]
return Expression('6*(sqrt(pow(x[0]-0.5,2)+pow(x[1]-0.5,2))<=t)', t=tt)
for Nxy in NN:
h = 1./Nxy
print '\n\th = {}'.format(h)
mesh = UnitSquareMesh(Nxy, Nxy, "crossed")
q = 1 # Polynomial order
V = FunctionSpace(mesh, 'Lagrange', q)
Dt = h/(q*5.*c)
Wave = AcousticWave({'V':V, 'Vl':V, 'Vr':V})
Wave.timestepper = 'backward'
Wave.lump = True
#Wave.verbose = True
Wave.exact = interpolate(exact_expr, V)
Wave.update({'lambda':lam, 'rho':rho, 't0':0.0, 'tf':tf, 'Dt':Dt,\
'u0init':Function(V), 'utinit':Function(V)})
test = TestFunction(V)
def srcterm(tt):
src_expr = source(tt)
src_vect = assemble(src_expr*test*dx)
return src_vect.array()
Wave.ftime = srcterm
sol, error = Wave.solve()
ERROR.append(error)
print 'relative error = {:.5e}'.format(error)
# Convergence order:
CONVORDER = []
for ii in range(len(ERROR)-1):
CONVORDER.append(np.log(ERROR[ii+1]/ERROR[ii])/np.log((1./NN[ii+1])/(1./NN[ii])))
print '\n\norder of convergence:', CONVORDER
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)