gradient¶
-
emg3d.optimize.gradient(simulation)[source]¶ Compute the discrete gradient using the adjoint-state method.
The discrete adjoint-state gradient for a single source at a single frequency is given by Equation (10) in [PlMu08],
\[\begin{split}\nabla_p \phi(\textbf{p}) = -&\sum_{k,l,m}\mathbf{\bar{\lambda}}_{x; k+\frac{1}{2}, l, m} \frac{\partial S_{k+\frac{1}{2}, l, m}}{\partial \textbf{p}} \textbf{E}_{x; k+\frac{1}{2}, l, m}\\ -&\sum_{k,l,m}\mathbf{\bar{\lambda}}_{y; k, l+\frac{1}{2}, m} \frac{\partial S_{k, l+\frac{1}{2}, m}}{\partial \textbf{p}} \textbf{E}_{y; k, l+\frac{1}{2}, m}\\ -&\sum_{k,l,m}\mathbf{\bar{\lambda}}_{z; k, l, m+\frac{1}{2}} \frac{\partial S_{k, l, m+\frac{1}{2}}}{\partial \textbf{p}} \textbf{E}_{z; k, l, m+\frac{1}{2}}\, ,\end{split}\]where \(\textbf{E}\) is the electric (forward) field and \(\mathbf{\lambda}\) is the back-propagated residual field (from electric and magnetic receivers); \(\bar{~}\) denotes conjugate. The \(\partial S\)-part takes care of the volume-averaged model parameters.
Note
The currently implemented gradient is only for isotropic models without relative electric permittivity nor relative magnetic permeability.
- Parameters
- simulationSimulation
The simulation; a
emg3d.simulations.Simulationinstance.
- Returns
- gradndarray
Adjoint-state gradient (same shape as
simulation.model).