Constructing trace operator (matrix that restricts function to boundary)

Question

I'm trying to construct a discretization of the trace operator that observes functions along a boundary. That is, suppose $\Gamma$ is some subset of the boundary of $\Omega$ - I want to assemble the wide matrix $B$ such that,
$$\vec{v}^TB\vec{u} = \int_\Gamma v(x) u(x) ds,$$
where $u$ lives in a function space with domain $\Omega$, but $v$ lives in a space with domain $\Gamma$. It is not sufficient to simply compute the trace for a given function - I actually need the matrix.

Is this possible in Fenics?

I tried the following, which is clearly wrong but hopefully illustrates what I'm trying to do:

from dolfin import *
import numpy as np

mesh = UnitSquareMesh(10,10)
V = FunctionSpace(mesh, 'Lagrange', 1)
u = TrialFunction(V)

boundary_mesh = BoundaryMesh(mesh,"exterior")
V_boundary = FunctionSpace(boundary_mesh, 'Lagrange', 1)

v_b = TestFunction(V_boundary)

boundary_observation_operator = u*v_b*ds
B = assemble(boundary_test)

This yielded the error:

An Integral without a Domain is now illegal.
ERROR:UFL:An Integral without a Domain is now illegal.

Comments

This is of interest to me also. But it does not seem to be possible to do this since u and v_b are defined on different meshes. If the space V_boundary can be obtained by restriction of the space V onto the boundary, then you can do some workarounds. For an example see https://github.com/cpraveen/fenics/blob/master/2d/laplace_lagrange_multiplier/demo.py where I use Lagrange multipliers on the boundary to apply DirichletBC. I would be interested to know if there is a better way to do this.