Gradient values with respect to a particular quantity of interest can be efficiently calculated by solving the adjoint equation.
Methods based on solution of adjoint equations are used in wing shape optimization, fluid flow control and uncertainty quantification.
Consider the following linear, scalar advection-diffusion equation for the primal solution
with Dirichlet boundary conditions: Let the output of interest be the following linear functional: Derive the weak form by multiplying the primal equation with a weighting function
and performing integration by parts: where, Then, consider an infinitesimal perturbation to
which produces an infinitesimal change in
Using the weak form above and the definition of the adjoint
Next, use integration by parts to transfer derivatives of
: The adjoint PDE and its boundary conditions can be deduced from the last equation above.
, in order for the volume term to vanish.
is generally non-zero at the boundary, we require
to be zero there in order for the first boundary term to vanish.
The second boundary term vanishes trivially since the primal boundary condition requires
Therefore, the adjoint problem is given by: Note that the advection term reverses the sign of the convective velocity
in the adjoint equation, whereas the diffusion term remains self-adjoint.
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