inversion_ideas.Smallness#

class inversion_ideas.Smallness(mesh, *, active_cells=None, cell_weights=None, reference_model=None)#

Smallness regularization.

Regularize a weighted norm of model values.

Parameters:

meshdiscretize.base.BaseMesh: Mesh to use in the regularization.
active_cells(n_cells) array or None, optional: Array full of bools that indicate the active cells in the mesh. It must have the same amount of elements as cells in the mesh.
cell_weights(n_active) array or dict of (n_active) arrays or None, optional: Array with cell weights. For multiple cell weights, pass a dictionary where keys are strings and values are the different weights arrays. If None, no cell weights are going to be used.
reference_model(n_active) array or None, optional: Array with values for the reference model. It must have the same number of elements as active cells in the mesh.

Attributes

`cell_weights`	Regularization weights on cells.
`n_params`	Number of model parameters.
`name`	Name of the objective function.
`weights_matrix`	Diagonal matrix with the square root of regularization weights on cells.

n_active

Methods

`__call__`(model)	Evaluate the regularization on a given model.
`gradient`(model)	Evaluate the gradient of the objective function for a given model.
`hessian`(model)	Evaluate the hessian of the objective function for a given model.
`hessian_diagonal`(model)	Get the main diagonal of the Hessian.
`info`()	Get information about the objective function.
`set_name`(value)	Set name for the objective function.

Notes

Implement a discretized version of the smallness regularization defined as follows:

\[\phi_s(\mathbf{m}) = \int_\Omega w(\mathbf{r}) |m(\mathbf{r}) - m^\text{ref}(\mathbf{r})|^2 \text{d}\mathbf{r},\]

where \(w(\mathbf{r})\) is the value of weights on the point \(\mathbf{r}\), \(m\) and \(m^\text{ref}\) are the model and reference model, respectively.

When discretizing it into the mesh, the smallness regularization can be expressed by:

\[\phi_s(\mathbf{m}) = \sum\limits_{i=0}^M w_i V_i |m_i - m_i^\text{ref}|^2 = \lVert \mathbf{W} \mathbf{V} (\mathbf{m} - \mathbf{m}^\text{ref}) \rVert^2,\]

where \(\mathbf{W} = [\sqrt{w_1}, \dots, \sqrt{w_M}]\) are the square roots of the cell weights, \(\mathbf{V} = [\sqrt{V_1}, \dots, \sqrt{V_M}]\) are the square root of cell volumes, \(\mathbf{m} = [m_1, \dots, m_M]\) and \(\mathbf{m}^\text{ref} = [m_1^\text{ref}, \dots, m_M^\text{ref}]\) are the model and reference model vectors, respectively.

Attributes#

Smallness.cell_weights#: Regularization weights on cells.

Smallness.n_active#

Smallness.n_params#

Smallness.name#: Name of the objective function.

Smallness.weights_matrix#: Diagonal matrix with the square root of regularization weights on cells.

Smallness.active_cells#

Methods#

Methods documentation

Smallness.__call__(model)#

Evaluate the regularization on a given model.

Parameters:

model(n_params) array: Array with model values.

Smallness.gradient(model)#

Evaluate the gradient of the objective function for a given model.

Parameters:

model(n_params) array: Array with model values.

Returns:

(n_params,) array: Gradient vector for the given model.

Smallness.hessian(model)#

Evaluate the hessian of the objective function for a given model.

Parameters:

model(n_params) array: Array with model values.

Returns:

(n_params, n_params) array or LinearOperator: 2D array or LinearOperator that represents the Hessian matrix of the objective funciton, or an approximated version of it.

Smallness.hessian_diagonal(model)#

Get the main diagonal of the Hessian.

Parameters:

model(n_params) array: Array with model values.

Returns:

(n_params,) array: Array containing the diagonal of the Hessian.

Smallness.info()#: Get information about the objective function.

Smallness.set_name(value)#: Set name for the objective function.