simpeg.electromagnetics.frequency_domain.Simulation3DMagneticFluxDensity.getADeriv#
- Simulation3DMagneticFluxDensity.getADeriv(freq, u, v, adjoint=False)[source]#
Derivative operation for the system matrix times a vector.
The system matrix at each frequency is given by:
\[\mathbf{A} = \mathbf{C M_{e\sigma}^{-1} C^T M_{f\frac{1}{\mu}}} + i\omega \mathbf{I}\]where
\(\mathbf{I}\) is the identity matrix
\(\mathbf{C}\) is the curl operator
\(\mathbf{M_{e\sigma}}\) 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
Simulation3DMagneticFluxDensityfor a full description of the formulation.Where \(\mathbf{m}\) are the set of model parameters defining the electromagnetic properties, \(\mathbf{v}\) is a vector and \(\mathbf{b}\) is the discrete magnetic flux density solution, this method assumes the discrete solution is fixed and returns
\[\frac{\partial (\mathbf{A \, b})}{\partial \mathbf{m}} \, \mathbf{v}\]Or the adjoint operation
\[\frac{\partial (\mathbf{A \, b})}{\partial \mathbf{m}}^T \, \mathbf{v}\]- 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.