simpeg.electromagnetics.time_domain.Simulation3DMagneticField.getAsubdiagDeriv#
- Simulation3DMagneticField.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:
\[\mathbf{B}_k = -\frac{1}{\Delta t_k} \mathbf{M_{e\mu}}\]where \(\Delta t_k\) is the step-length and \(\mathbf{M_{e \mu}}\) is the permeability inner-product matrix on edges.
See the Notes section of the doc strings for
Simulation3DMagneticFieldfor a full description of the formulation.Where \(\mathbf{m}\) are the set of model parameters, \(\mathbf{v}\) is a vector and \(\mathbf{h_{k-1}}\) is the discrete solution for the previous time-step, this method assumes the discrete solution is fixed and returns
\[\frac{\partial (\mathbf{B_k \, h_{k-1}})}{\partial \mathbf{m}} \, \mathbf{v}\]Or the adjoint operation
\[\frac{\partial (\mathbf{B_k \, h_{k-1}})}{\partial \mathbf{m}}^T \, \mathbf{v}\]- Parameters:
- tInd
int 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. \(\mathbf{h_{k-1}}\).
- v
numpy.ndarray The vector. (n_param,) for the standard operation. (n_edges,) for the adjoint operation.
- adjointbool
Whether to perform the adjoint operation.
- tInd
- Returns:
numpy.ndarrayDerivative of system matrix times a vector. (n_edges,) for the standard operation. (n_param,) for the adjoint operation.