Optimal heat problem

from dolfin import *
from dolfin_adjoint import *

mesh = RectangleMesh(-1, -1, 1, 1, 20, 20)
V = FunctionSpace(mesh, "CG", 1)
u = Function(V, name="State")
m = Function(V, name="Control")
v = TestFunction(V)

F = (inner(grad(u), grad(v)) - m*v)*dx
bc = DirichletBC(V, 0.0, "on_boundary")
solve(F == 0, u, bc)

u_desired = Expression("exp(-1/(1-x[0]*x[0])-1/(1-x[1]*x[1]))")

J = Functional((0.5*inner(u-u_desired, u-u_desired))*dx)


reduced_functional = ReducedFunctional(J, SteadyParameter(m))
m_opt = minimize(reduced_functional, method = "L-BFGS-B", bounds =(0.0,0.5),
                        options = {"gtol": 1e-12,"ftol": 1e-12,"disp": True})

proj_error = project(u-u_desired,V)

save_file = File("optim_heat.pvd")

# desire solution
proj_udes = project(u_desired,V)
plot(u_desired, mesh, interactive=True, title="Desired")
# Actual solution
F_opt = replace(F, {m: m_opt})
solve(F_opt == 0, u, bc) 
plot(u, interactive=True, title="Actual")
# L1 error
proj_error = project(u-u_desired,V)
plot(proj_error, interactive=True, title="L1-Error")
# Optimal forcing
plot(m_opt, interactive=True, title="Control")