simpeg.electromagnetics.frequency_domain.Simulation3DElectricField.getADeriv_sigma#
- Simulation3DElectricField.getADeriv_sigma(freq, u, v, adjoint=False)[source]#
Conductivity derivative operation for the system matrix times a vector.
The system matrix at each frequency is given by:
\[\mathbf{A} = \mathbf{C^T M_{f\frac{1}{\mu}} C} + i\omega \mathbf{M_{e\sigma}}\]where
\(\mathbf{M_{e\sigma}}\) is the inner-product matrix for conductivities projected to edges
\(\mathbf{M_{f\frac{1}{\mu}}}\) is the inner-product matrix for inverse permeabilities projected to faces
See the Notes section of the doc strings for
Simulation3DElectricFieldfor a full description of the formulation.Where \(\mathbf{m}_\boldsymbol{\sigma}\) are the set of model parameters defining the conductivity, \(\mathbf{v}\) is a vector and \(\mathbf{e}\) is the discrete electric field solution, this method assumes the discrete solution is fixed and returns
\[\frac{\partial (\mathbf{A \, e})}{\partial \mathbf{m}_\boldsymbol{\sigma}} \, \mathbf{v}\]Or the adjoint operation
\[\frac{\partial (\mathbf{A \, e})}{\partial \mathbf{m}_\boldsymbol{\sigma}}^T \, \mathbf{v}\]- Parameters:
- freq
float The frequency in Hz.
- u(n_edges,)
numpy.ndarray The solution for the fields for the current model at the specified frequency.
- v
numpy.ndarray The vector. (n_param,) for the standard operation. (n_edges,) for the adjoint operation.
- adjointbool
Whether to perform the adjoint operation.
- freq
- Returns:
numpy.ndarrayDerivative of system matrix times a vector. (n_edges,) for the standard operation. (n_param,) for the adjoint operation.