simpeg.electromagnetics.time_domain.Simulation3DMagneticFluxDensity.getAdiagDeriv#

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

Derivative operation for the diagonal system matrix times a vector.

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

$${\mathbf{A}}_k=\mathbf{C}{\mathbf{M}}_{\mathbf{e}\sigma }^{\mathbf{-}\mathbf{1}}{\mathbf{C}}^{\mathbf{T}}{\mathbf{M}}_{\mathbf{f}\frac{\mathbf{1}}{\mu }}+\frac{1}{\mathrm{\Delta }{t}_k}\mathbf{I}$$

where

• $\mathrm{\Delta }{t}_k$ is the step length

• $\mathbf{I}$ is the identity matrix

• $\mathbf{C}$ is the discrete curl operator

• ${\mathbf{M}}_{\mathbf{e}\sigma }$ is the conductivity inner-product matrix on edges

• ${\mathbf{M}}_{\mathbf{f}\frac{\mathbf{1}}{\mu }}$ is the inverse permeability inner-product matrix on faces

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 ${\mathbf{b}}_{\mathbf{k}}$ is the discrete solution for time-step k, this method assumes the discrete solution is fixed and returns

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

Or the adjoint operation

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

Parameters:

tIndint

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

u(n_faces,) numpy.ndarray

The solution for the fields for the current model; i.e. ${\mathbf{b}}_{\mathbf{k}}$.

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.