simpeg.electromagnetics.frequency_domain.Simulation3DMagneticFluxDensity.getADeriv_sigma#

Simulation3DMagneticFluxDensity.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:

$$\mathbf{A}=\mathbf{C}{\mathbf{M}}_{\mathbf{e}\sigma }^{\mathbf{-}\mathbf{1}}{\mathbf{C}}^{\mathbf{T}}{\mathbf{M}}_{\mathbf{f}\frac{\mathbf{1}}{\mu }}+i\omega \mathbf{I}$$

where

• $\mathbf{I}$ is the identity matrix

• $\mathbf{C}$ is the curl operator

• ${\mathbf{M}}_{\mathbf{e}\sigma }$ is the inner-product matrix for conductivities projected to edges

• ${\mathbf{M}}_{\mathbf{f}\frac{\mathbf{1}}{\mu }}$ is the inner-product matrix for inverse permeabilities projected to faces

See the Notes section of the doc strings for Simulation3DMagneticFluxDensity for a full description of the formulation.

Where ${\mathbf{m}}_{\mathit{\sigma }}$ are the set of model parameters defining the conductivity, $\mathbf{v}$ is a vector and $\mathbf{b}$ is the discrete magnetic flux density solution, this method assumes the discrete solution is fixed and returns

$$\frac{\partial (\mathbf{A}\mathbf{b})}{\partial {\mathbf{m}}_{\mathit{\sigma }}}\mathbf{v}$$

Or the adjoint operation

$${\frac{\partial (\mathbf{A}\mathbf{b})}{\partial {\mathbf{m}}_{\mathit{\sigma }}}}^T\mathbf{v}$$

Parameters:

freqfloat

The frequency in Hz.

u(n_faces,) numpy.ndarray

The solution for the fields for the current model at the specified frequency.

vnumpy.ndarray

The vector. (n_param,) for the standard operation. (n_faces,) for the adjoint operation.

adjointbool

Whether to perform the adjoint operation.

Returns:

numpy.ndarray

Derivative of system matrix times a vector. (n_faces,) for the standard operation. (n_param,) for the adjoint operation.