simpeg.electromagnetics.frequency_domain.Simulation3DCurrentDensity.getADeriv_rho#
- Simulation3DCurrentDensity.getADeriv_rho(freq, u, v, adjoint=False)[source]#
Resistivity derivative operation for the system matrix times a vector.
The system matrix at each frequency is given by:
\[\mathbf{A} = \mathbf{C M_{e\mu}^{-1} C^T M_{f\rho}} + i\omega \mathbf{I}\]where
\(\mathbf{M_{f\rho}}\) is the inner-product matrix for resistivities projected to faces
\(\mathbf{M_{e\mu}}\) is the inner-product matrix for permeabilities projected to edges
See the Notes section of the doc strings for
Simulation3DCurrentDensityfor a full description of the formulation.Where \(\mathbf{m}_\boldsymbol{\rho}\) are the set of model parameters defining the resistivity, \(\mathbf{v}\) is a vector and \(\mathbf{j}\) is the discrete current density solution, this method assumes the discrete solution is fixed and returns
\[\frac{\partial (\mathbf{A \, j})}{\partial \mathbf{m}_\boldsymbol{\sigma}} \, \mathbf{v}\]Or the adjoint operation
\[\frac{\partial (\mathbf{A \, j})}{\partial \mathbf{m}_\boldsymbol{\sigma}}^T \, \mathbf{v}\]- Parameters:
- freq
float The frequency in Hz.
- u(n_faces,)
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_faces,) for the adjoint operation.
- adjointbool
Whether to perform the adjoint operation.
- freq
- Returns:
numpy.ndarrayDerivative of system matrix times a vector. (n_faces,) for the standard operation. (n_param,) for the adjoint operation.