Mihai,
> In my inverse problem setup with deal.ii I set up an "observation"
> operator that acts on my solution vector to give the FE solution at a
> given set of grid points. I do this using FEFieldFunction for each point
> on this "observation" grid: all observation points are inside my domain
> of course, I identify the active cells that contain the points, and then
> for each point I evaluate the FE solution using FEFieldFunction::value.
>
> Now that that is set up, I need the adjoint of this (linear?) operator.
Of course it's linear -- the point values of (u(x)+v(x)) are the (point
values of u(x)) + (point values of v(x)) ;-)
> How would I go about building the transpose of this (implicit)
> point-solution evaluation operator? Say I have only one observation
> point. I can get the point where my quadrature is evaluated by calling
> FEFieldFunction::compute_point_locations(), right? How do I get the DoFs
> that are associated with the active cell that contains the point, and
> how exactly do they (the DoFs) come into play when the function gets
> evaluated? Does FEFieldFunction use DoFs for evaluation other than those
> associated with the active cell?
If you have only one evaluation point x0, the right hand side of your dual
problem is going to be
F_i = phi_i(x0)
With the function you mention above, you already know which cell this point
lies in and what it's reference coordinates are. You also know that F_i=0
for all DoF indices i of shape functions that are zero on this cell. The
only shape functions that are not zero are those that you get through
cell->get_dof_indices (local_dof_indices)
and the way to implement your right hand side would then be something along
the lines of
dual_rhs = 0;
<cell, reference_coordinates> = output of compute_point_locations;
cell->get_dof_indices (local_dof_indices);
for (unsigned int i=0; i<fe.dofs_per_cell; ++i)
dual_rhs(i) = fe.shape_value (i, reference_coordinates);
W.
-------------------------------------------------------------------------
Wolfgang Bangerth email: [email protected]
www: http://www.math.tamu.edu/~bangerth/
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