myplot.set_varname('m_target')
myplot.plot_vtk(mtrue)
myplot.set_varname('m_targetVm')
myplot.plot_vtk(mtrueVm)
else:
myplot = None
if mpirank == 0: print 'Compute noisy data'
OPPS = 15
NB = OPPS + 1.
Points = np.array([[[ii/NB,jj/NB] for ii in range(1,OPPS+1)]\
for jj in range(1,OPPS+1)])
Points.resize((OPPS * OPPS, 2))
ObsOp = ObsPointwise({'V': V, 'Points': Points}, mpicomm)
ObsOp.noise = False
goal = ObjFctalElliptic(V, Vme, bc, bc, [f], ObsOp, [], [], [], False, mpicomm)
goal.update_m(mtrue)
goal.solvefwd()
np.random.seed(11)
noisepercent = 0.02 # e.g., 0.02 = 2% noise level
UD = goal.U[0]
rndnb = np.random.randn(UD.size)
rndnb = rndnb / np.linalg.norm(rndnb)
noiseres = noisepercent * np.linalg.norm(UD)
UDnoise = UD + rndnb * noiseres
if mpirank == 0: print 'noiseres={}'.format(noiseres)
if PLOT:
myplot.set_varname('u_target')
myplot.plot_vtk(goal.u)
# Define regularization:
def run_problem():
# Domain, f-e spaces and boundary conditions:
mesh = UnitSquareMesh(20,20)
V = FunctionSpace(mesh, 'Lagrange', 2) # space for state and adjoint variables
Vm = FunctionSpace(mesh, 'Lagrange', 1) # space for medium parameter
Vme = FunctionSpace(mesh, 'Lagrange', 5) # sp for target med param
# Define zero Boundary conditions:
def u0_boundary(x, on_boundary):
return on_boundary
u0 = Constant("0.0")
bc = DirichletBC(V, u0, u0_boundary)
# Define target medium and rhs:
mtrue_exp = Expression('1 + 7*(pow(pow(x[0] - 0.5,2) +' + \
' pow(x[1] - 0.5,2),0.5) > 0.2)')
mtrue = interpolate(mtrue_exp, Vme)
f = Expression("1.0")
# Compute target data:
noisepercent = 0.05 # e.g., 0.02 = 2% noise level
ObsOp = ObsEntireDomain({'V': V,'noise':noisepercent})
goal = ObjFctalElliptic(V, Vme, bc, bc, [f], ObsOp)
goal.update_m(mtrue)
goal.solvefwd()
UDnoise = goal.U
# Solve reconstruction problem:
Regul = LaplacianPrior({'Vm':Vm,'gamma':1e-3,'beta':1e-14})
ObsOp.noise = False
InvPb = ObjFctalElliptic(V, Vm, bc, bc, [f], ObsOp, UDnoise, Regul, [], False)
InvPb.update_m(1.0) # Set initial medium
InvPb.solvefwd_cost()
# Choose between steepest descent and Newton's method:
METHODS = ['sd','Newt']
meth = METHODS[1]
if meth == 'sd': alpha_init = 1e3
elif meth == 'Newt': alpha_init = 1.0
nbcheck = 0 # Grad and Hessian checks
nbLS = 20 # Max nb of line searches
# Prepare results outputs:
PP = PostProcessor(meth, Vm, mtrue)
PP.getResults0(InvPb) # Get results for index 0 (before first iteration)
PP.printResults()
# Start iteration:
maxiter = 100
for it in range(1, maxiter+1):
InvPb.solveadj_constructgrad()
# Check gradient and Hessian:
if nbcheck and (it == 1 or it % 10 == 0):
checkgradfd(InvPb, nbcheck)
checkhessfd(InvPb, nbcheck)
# Compute search direction:
if it == 1: gradnorm_init = InvPb.getGradnorm()
if meth == 'Newt':
if it == 1: maxtolcg = .5
else: maxtolcg = CGresults[3]
else: maxtolcg = None
CGresults = compute_searchdirection(InvPb, meth, gradnorm_init, maxtolcg)
# Compute line search:
LSresults = bcktrcklinesearch(InvPb, nbLS, alpha_init)
InvPb.plotm(it) # Plot current medium reconstruction
# Print results:
PP.getResults(InvPb, LSresults, CGresults)
PP.printResults()
if PP.Stop(): break # Stopping criterion
alpha_init = PP.alpha_init() # Initialize next alpha when using sd
InvPb.gatherm() # Create one plot for all intermediate reconstructions
if it == maxiter: print "Max nb of iterations reached."
return on_boundary
u0 = Constant("0.0")
bc = DirichletBC(V, u0, u0_boundary)
# Define target medium and rhs:
mtrue_exp = Expression("1 + 7*(pow(pow(x[0] - 0.5,2) +" + " pow(x[1] - 0.5,2),0.5) > 0.2)")
mtrue = interpolate(mtrue_exp, Vme)
f = Expression("1.0")
if myrank == 0:
print "Compute target data".format(myrank)
obspts = [[ii / 5.0, jj / 5.0] for ii in range(1, 5) for jj in range(1, 5)]
noisepercent = 0.10 # e.g., 0.02 = 2% noise level
ObsOp = ObsPointwise({"V": V, "Points": obspts, "noise": noisepercent}, mycomm)
goal = ObjFctalElliptic(V, Vme, bc, bc, [f], ObsOp, [], [], [], False, mycomm)
goal.update_m(mtrue)
goal.solvefwd()
UDnoise = goal.U
if myrank == 0:
print "Define prior and sample from prior"
m0 = Function(Vm)
m0.vector()[:] = 1.0
myprior = BilaplacianPrior({"Vm": Vm, "gamma": 10.0, "beta": 10.0, "m0": m0})
filename, ext = splitext(sys.argv[0])
if isdir(filename + "/"):
rmtree(filename + "/")
plotprior = PlotFenics(filename + "/Prior/")
plotprior.set_varname("m_prior")
for ii in range(10):
# Define zero Boundary conditions:
def u0_boundary(x, on_boundary):
return on_boundary
u0 = Constant("0.0")
bc = DirichletBC(V, u0, u0_boundary)
# Define target medium and rhs:
mtrue_exp = Expression('1 + 7*(pow(pow(x[0] - 0.5,2) +' + \
' pow(x[1] - 0.5,2),0.5) > 0.2)')
mtrue = interpolate(mtrue_exp, Vme)
f = Expression("1.0")
print 'p{}: Compute target data'.format(myrank)
obspts = [[ii/5.,jj/5.] for ii in range(1,5) for jj in range(1,5)]
noisepercent = 0.05 # e.g., 0.02 = 2% noise level
ObsOp = ObsPointwise({'V': V, 'Points':obspts,'noise':noisepercent}, mycomm)
goal = ObjFctalElliptic(V, Vme, bc, bc, [f], ObsOp, [], [], [], False, mycomm)
goal.update_m(mtrue)
goal.solvefwd()
UDnoise = goal.U
print 'p{}: Solve reconstruction problem'.format(myrank)
Regul = LaplacianPrior({'Vm':Vm,'gamma':2e-8,'beta':1e-14})
plot_option = False
ObsOp.noise = False
InvPb = ObjFctalElliptic(V, Vm, bc, bc, [f], ObsOp, UDnoise, Regul, \
[], plot_option, mycomm)
InvPb.update_m(1.0) # Set initial medium
InvPb.solvefwd_cost()
# Choose between steepest descent and Newton's method:
METHODS = ['sd','Newt']
meth = METHODS[1]
if mpirank == 0 and isdir(filename + '/'):
rmtree(filename + '/')
MPI.barrier(mpicomm)
myplot = PlotFenics(filename)
MPI.barrier(mpicomm)
myplot.set_varname('m_target')
myplot.plot_vtk(mtrue)
myplot.set_varname('m_targetVm')
myplot.plot_vtk(mtrueVm)
else:
myplot = None
if mpirank == 0: print 'Compute noisy data'
ObsOp = ObsEntireDomain({'V': V}, mpicomm)
ObsOp.noise = False
goal = ObjFctalElliptic(V, Vme, bc, bc, [f], ObsOp, [], [], [], False, mpicomm)
goal.update_m(mtrue)
goal.solvefwd()
# noise
np.random.seed(11)
noisepercent = 0.1 # e.g., 0.02 = 2% noise level
UD = goal.U[0]
rndnb = np.random.randn(UD.size)
rndnb = rndnb / np.linalg.norm(rndnb)
noiseres = noisepercent * np.linalg.norm(UD)
UDnoise = UD + rndnb * noiseres
if mpirank == 0: print 'noiseres={}, rndnb*noiseres={}'.format(\
MPI.sum(mpicomm, noiseres), MPI.sum(mpicomm, np.linalg.norm(rndnb*noiseres)))
if PLOT:
myplot.set_varname('u_target')
myplot.plot_vtk(goal.u)
Vme = FunctionSpace(mesh, 'Lagrange', 5) # sp for target med param
# Define zero Boundary conditions:
def u0_boundary(x, on_boundary):
return on_boundary
u0 = Constant("0.0")
bc = DirichletBC(V, u0, u0_boundary)
# Define target medium and rhs:
mtrue_exp = Expression('1 + 7*(pow(pow(x[0] - 0.5,2) +' + \
' pow(x[1] - 0.5,2),0.5) > 0.2)')
mtrue = interpolate(mtrue_exp, Vme)
f = Expression("1.0")
print 'p{}: Compute target data'.format(myrank)
noisepercent = 0.00 # e.g., 0.02 = 2% noise level
ObsOp = ObsEntireDomain({'V': V,'noise':noisepercent}, mycomm)
goal = ObjFctalElliptic(V, Vme, bc, bc, [f], ObsOp, [], [], [], False, mycomm)
goal.update_m(mtrue)
goal.solvefwd()
print 'p{}'.format(myrank)
UDnoise = goal.U
print 'p{}: Solve reconstruction problem'.format(myrank)
Regul = LaplacianPrior({'Vm':Vm,'gamma':1e-5,'beta':1e-14})
ObsOp.noise = False
InvPb = ObjFctalElliptic(V, Vm, bc, bc, [f], ObsOp, UDnoise, Regul, [], False, mycomm)
InvPb.update_m(1.0) # Set initial medium
InvPb.solvefwd_cost()
# Choose between steepest descent and Newton's method:
METHODS = ['sd','Newt']
meth = METHODS[1]
if meth == 'sd': alpha_init = 1e3