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."
meth = METHODS[1]
if meth == 'sd': alpha_init = 1e3
elif meth == 'Newt': alpha_init = 1.0
nbcheck = 2 # Grad and Hessian checks
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