# SimPEG.regularization.LinearCorrespondence#

class SimPEG.regularization.LinearCorrespondence(mesh, wire_map, coefficients=None, **kwargs)[source]#

Linear correspondence regularization for joint inversion with two physical properties.

LinearCorrespondence is used to recover a model where the differences between the model parameter values for two physical property types are minimal. LinearCorrespondence can also be used to minimize the squared L2-norm of a linear combination of model parameters for two physical property types. See the Notes section for a comprehensive description.

Parameters:
mesh

Mesh on which the regularization is discretized. This is not necessarily the same as the mesh on which the simulation is defined.

active_cellsNone, (n_cells, ) numpy.ndarray of bool

Boolean array defining the set of RegularizationMesh cells that are active in the inversion. If None, all cells are active.

wire_mapSimPEG.maps.Wires

Wire map connecting physical properties defined on active cells of the RegularizationMesh to the entire model.

coefficientsNone, (3) numpy.ndarray of float

Coefficients $$\{ \lambda_1, \lambda_2, \lambda_3 \}$$ for the linear relationship between model parameters. If None, the coefficients are set to $$\{ 1, -1, 0 \}$$.

Notes

Let $$\mathbf{m}$$ be a discrete model consisting of two physical property types such that:

$\begin{split}\mathbf{m} = \begin{bmatrix} \mathbf{m_1} \\ \mathbf{m_2} \end{bmatrix}\end{split}$

Where $$\{ \lambda_1 , \lambda_2 , \lambda_3 \}$$ define scalar coefficients for a linear combination of vectors $$\mathbf{m_1}$$ and $$\mathbf{m_2}$$, the regularization function (objective function) is given by:

$\phi (\mathbf{m}) = \frac{1}{2} \big \| \lambda_1 \mathbf{m_1} + \lambda_2 \mathbf{m_2} + \lambda_3 \big \|^2$

Scalar coefficients $$\{ \lambda_1 , \lambda_2 , \lambda_3 \}$$ are set using the coefficients property. For a true linear correspondence constraint, we set $$\{ \lambda_1 , \lambda_2 , \lambda_3 \}$$ to $$\{ 1, -1, 0 \}$$.

Attributes

 coefficients Coefficients for the linear relationship between model parameters.

Methods

 __call__(model) Evaluate the regularization function for the model provided. deriv(model) Gradient of the regularization function evaluated for the model provided. deriv2(model[, v]) Hessian of the regularization function evaluated for the model provided. relation`(model) Computes the relation vector for the model provided.