simpeg.electromagnetics.frequency_domain.Simulation3DElectricField.getADeriv_mui#

Simulation3DElectricField.getADeriv_mui(freq, u, v, adjoint=False)[source]#

Inverse permeability derivative operation for the system matrix times a vector.

The system matrix at each frequency is given by:

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

where

• ${\mathbf{M}}_{\mathbf{e}\sigma }$ 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 Simulation3DElectricField for a full description of the formulation.

Where ${\mathbf{m}}_{\mathit{\mu }}$ are the set of model parameters defining the permeability, $\mathbf{v}$ is a vector and $\mathbf{e}$ is the discrete electric field solution, this method assumes the discrete solution is fixed and returns

$$\frac{\partial (\mathbf{A}\mathbf{e})}{\partial {\mathbf{m}}_{\mathit{\mu }}}\mathbf{v}$$

Or the adjoint operation

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

Parameters:

freqfloat

The frequency in Hz.

u(n_edges,) 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_edges,) for the adjoint operation.

adjointbool

Whether to perform the adjoint operation.

Returns:

numpy.ndarray

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