inversion_ideas.SimpleSmallness#

class inversion_ideas.SimpleSmallness(n_params, weights=None, reference_model=None)#

Simple smallness regularization.

Implement a simple smallness regularization that evaluates the norm of the model vector.

Hint

Use this regularization in non mesh-based inversions, in which we don’t need to include mesh details such as cell volumes.

Parameters:

n_paramsint: Number of elements in the model array.
weights(n_params) array or dict of (n_params) arrays or None, optional: Array with regularization weights. For multiple weights, pass a dictionary where keys are strings and values are the different weights arrays. If None, no weights are going to be used.
reference_model(n_params) array or None, optional: Array with values for the reference model.

Attributes

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

Methods

`__call__`(model)	Evaluate the objective function for 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 simple smallness regularization that evaluates the weighted L2 model norm:

\[\phi(\mathbf{m}) = \sum\limits_{i=1}^M w_i |m_i - m_i^\text{ref}|^2 = \lVert \mathbf{W} (\mathbf{m} - \mathbf{m}^\text{ref}) \rVert^2\]

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

Attributes#

SimpleSmallness.n_params#: Number of model parameters.

SimpleSmallness.name#: Name of the objective function.

SimpleSmallness.weights#: Regularization weights.

SimpleSmallness.weights_matrix#: Diagonal matrix with the square root of the regularization weights.

Methods#

Methods documentation

SimpleSmallness.__call__(model)#

Evaluate the objective function for a given model.

Parameters:

model(n_params) array: Array with model values.

Returns:

float: Value of the objective function for the given model.

SimpleSmallness.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.

SimpleSmallness.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.

SimpleSmallness.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.

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

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