simpeg.electromagnetics.time_domain.Simulation3DMagneticFluxDensity.getAsubdiagDeriv#

Simulation3DMagneticFluxDensity.getAsubdiagDeriv(tInd, u, v, adjoint=False)[source]#

Derivative operation for the sub-diagonal system matrix times a vector.

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

$${\mathbf{B}}_k=-\frac{1}{\mathrm{\Delta }{t}_k}\mathbf{I}$$

where $\mathrm{\Delta }{t}_k$ is the step length and $\mathbf{I}$ is the identity matrix.

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

Where $\mathbf{m}$ are the set of model parameters, $\mathbf{v}$ is a vector and ${\mathbf{b}}_{\mathbf{k}\mathbf{-}\mathbf{1}}$ is the discrete solution for the previous time-step, this method assumes the discrete solution is fixed and returns

$$\frac{\partial ({\mathbf{B}}_{\mathbf{k}}{\mathbf{b}}_{\mathbf{k}\mathbf{-}\mathbf{1}})}{\partial \mathbf{m}}\mathbf{v}=\mathbf{0}$$

Or the adjoint operation

$${\frac{\partial ({\mathbf{B}}_{\mathbf{k}}{\mathbf{b}}_{\mathbf{k}\mathbf{-}\mathbf{1}})}{\partial \mathbf{m}}}^T\mathbf{v}=\mathbf{0}$$

The derivative operation returns a vector of zeros because the sub-diagonal system matrix does not depend on the model!!!

Parameters:

tIndint

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

u(n_faces,) numpy.ndarray

The solution for the fields for the current model for the previous time-step; i.e. ${\mathbf{b}}_{\mathbf{k}\mathbf{-}\mathbf{1}}$.

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.