# fig = plt.figure()
# fig.set_size_inches(20., 15.)
# for ii in range(lenobspts):
# ax = fig.add_subplot(6,6,ii+1)
# ax.plot(waveobj.PDE.times, waveobj.dd[ii,:], 'k--')
# ax.plot(waveobj.PDE.times, waveobj.Bp[ii,:], 'b')
# ax.set_title('obs'+str(ii))
# fig.savefig(filename + '/observations' + str(iter) + '.eps')
# plt.close(fig)
# stopping criterion (gradient)
if gradnorm < gradnorm0 * tolgrad or gradnorm < 1e-12:
print '\nGradient sufficiently reduced -- optimization stopped'
break
# search direction
tolcg = min(0.5, np.sqrt(gradnorm / gradnorm0))
cgiter, cgres, cgid, tolcg = compute_searchdirection(
waveobj, 'Newt', tolcg)
myplot.set_varname('srchdir' + str(iter))
myplot.plot_vtk(waveobj.srchdir)
# line search
cost_old = waveobj.cost
statusLS, LScount, alpha = bcktrcklinesearch(waveobj, 12)
cost = waveobj.cost
print '{:11.3f} {:12.2e} {:10d}'.\
format(alpha, tolcg, cgiter)
# stopping criterion (cost)
if np.abs(cost - cost_old) / np.abs(cost_old) < tolcost:
print 'Cost function stagnates -- optimization stopped'
break
# Compute eigenspectrum
waveobj.alpha_reg = 0.0 # no regularization
def inversion(self, initial_medium, target_medium, mpicomm, \
parameters_in=[], myplot=None):
""" solve inverse problem with that objective function """
parameters = {'tolgrad':1e-10, 'tolcost':1e-14, 'maxnbNewtiter':50, \
'maxtolcg':0.5}
parameters.update(parameters_in)
maxnbNewtiter = parameters['maxnbNewtiter']
tolgrad = parameters['tolgrad']
tolcost = parameters['tolcost']
tolcg = parameters['maxtolcg']
mpirank = MPI.rank(mpicomm)
self.update_m(initial_medium)
self._plotm(myplot, 'init')
if mpirank == 0:
print '\t{:12s} {:10s} {:12s} {:12s} {:12s} {:10s} \t{:10s} {:12s} {:12s}'.format(\
'iter', 'cost', 'misfit', 'reg', '|G|', 'medmisf', 'a_ls', 'tol_cg', 'n_cg')
dtruenorm = np.sqrt(target_medium.vector().\
inner(self.MM*target_medium.vector()))
self.solvefwd_cost()
for it in xrange(maxnbNewtiter):
self.solveadj_constructgrad() # compute gradient
if it == 0: gradnorm0 = self.Gradnorm
diff = self.m.vector() - target_medium.vector()
medmisfit = np.sqrt(diff.inner(self.MM * diff))
if mpirank == 0:
print '{:12d} {:12.4e} {:12.2e} {:12.2e} {:11.4e} {:10.2e} ({:4.2f})'.\
format(it, self.cost, self.misfit, self.regul, \
self.Gradnorm, medmisfit, medmisfit/dtruenorm),
self._plotm(myplot, str(it))
self._plotgrad(myplot, str(it))
if self.Gradnorm < gradnorm0 * tolgrad or self.Gradnorm < 1e-12:
if mpirank == 0:
print '\nGradient sufficiently reduced -- optimization stopped'
break
# Compute search direction:
tolcg = min(tolcg, np.sqrt(self.Gradnorm / gradnorm0))
self.assemble_hessian() # for regularization
cgiter, cgres, cgid, tolcg = compute_searchdirection(
self, 'Newt', tolcg)
self._plotsrchdir(myplot, str(it))
# Line search:
cost_old = self.cost
statusLS, LScount, alpha = bcktrcklinesearch(self, 12)
if mpirank == 0:
print '{:11.3f} {:12.2e} {:10d}'.format(alpha, tolcg, cgiter)
if self.PD: self.Regul.update_w(self.srchdir.vector(), alpha)
if np.abs(self.cost - cost_old) / np.abs(cost_old) < tolcost:
if mpirank == 0:
if tolcg < 1e-14:
print 'Cost function stagnates -- optimization aborted'
break
tolcg = 0.001 * tolcg
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."
nbLS = 20 # Max nb of line searches
# Prepare results outputs:
PP = PostProcessor(meth, Vm, mtrue, mycomm)
PP.getResults0(InvPb) # Get results for index 0 (before first iteration)
PP.printResults(myrank)
# 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(myrank)
if PP.Stop(myrank): 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."