simpeg.electromagnetics.time_domain.Simulation3DElectricField.getAsubdiagDeriv#

Simulation3DElectricField.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:

$${\mathbf{B}}_k=-\frac{1}{\mathrm{\Delta }{t}_k}{\mathbf{M}}_{\mathbf{e}\sigma }$$

where $\mathrm{\Delta }{t}_k$ is the step-length and ${\mathbf{M}}_{\mathbf{e}\sigma }$ is the conductivity inner-product matrix on edges.

See the Notes section of the doc strings for Simulation3DElectricField for a full description of the formulation.

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

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

Or the adjoint operation

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

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 for the previous time-step; i.e. ${\mathbf{e}}_{\mathbf{k}\mathbf{-}\mathbf{1}}$.

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.