simpeg.electromagnetics.time_domain.Simulation3DMagneticField.getAdiagDeriv

Simulation3DMagneticField.getAdiagDeriv(tInd, u, v, adjoint=False)

Derivative operation for the diagonal system matrix times a vector.

The diagonal system matrix for time-step index k is given by:

\[\mathbf{A}_k = \mathbf{C^T M_{f\rho} C} + \frac{1}{\Delta t_k} \mathbf{M_{e\mu}}\]

where

• \(\Delta t_k\) is the step length

• \(\mathbf{C}\) is the discrete curl operator

• \(\mathbf{M_{f \rho}}\) is the resistivity inner-product matrix on faces

• \(\mathbf{M_{e\mu}}\) is the permeability inner-product matrix on edges

See the Notes section of the doc strings for Simulation3DMagneticField for a full description of the formulation.

Where \(\mathbf{m}\) are the set of model parameters, \(\mathbf{v}\) is a vector and \(\mathbf{h_k}\) is the discrete solution for time-step k, this method assumes the discrete solution is fixed and returns

\[\frac{\partial (\mathbf{A_k \, h_k})}{\partial \mathbf{m}} \, \mathbf{v}\]

Or the adjoint operation

\[\frac{\partial (\mathbf{A_k \, h_k})}{\partial \mathbf{m}}^T \, \mathbf{v}\]

Parameters:

tIndint

The time-step index; between [0, n_steps-1].

u(n_edges,) numpy.ndarray

The solution for the fields for the current model; i.e. \(\mathbf{h_k}\).

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.