simpeg.regularization.LinearCorrespondence

class simpeg.regularization.LinearCorrespondence(mesh, wire_map, coefficients=None, **kwargs)

Bases: BaseSimilarityMeasure

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:

meshsimpeg.regularization.RegularizationMesh, discretize.base.BaseMesh

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 \(\{ 1, -1, 0 \}\).

Attributes

W	Weighting matrix.
active_cells	Active cells defined on the regularization mesh.
coefficients	Coefficients for the linear relationship between model parameters.
mapping	Mapping from the inversion model parameters to the regularization mesh.
model	The model parameters.
nP	Number of model parameters.
parent	The parent objective function
reference_model	Reference model.
regularization_mesh	Regularization mesh.
units	Units for the model parameters.
weights_keys	Return the keys for the existing cell weights
wire_map	Mapping from model to physical properties defined on the regularization mesh.

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.
f_m(m)	Not implemented for BaseRegularization class.
f_m_deriv(m)	Not implemented for BaseRegularization class.
get_weights(key)	Cell weights for a given key.
map_class	alias of IdentityMap
relation(model)	Computes the relation vector for the model provided.
remove_weights(key)	Removes the weights for the key provided.
set_weights(**weights)	Adds (or updates) the specified weights to the regularization.
test([x, num, random_seed])	Run a convergence test on both the first and second derivatives.

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:

\[\phi (\mathbf{m}) = \big \| \lambda_1 \mathbf{m_1} + \lambda_2 \mathbf{m_2} + \lambda_3 \big \|^2\]

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 \(\{ 1, -1, 0 \}\).