inversion_ideas.SparseSmallness

class inversion_ideas.SparseSmallness(mesh, *, norm, active_cells=None, cell_weights=None, reference_model=None, threshold=1e-08, cooling_factor=1.25, model_previous=None, irls=False)

Smallness regularization using lp norm.

Parameters:

meshdiscretize.base.BaseMesh

Mesh to use in the regularization.

normfloat

Norm used in the regularization (p).

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.

thresholdfloat, optional

IRLS threshold. Symbolized with \(\epsilon\) in Fournier and Oldenburg (2019).

cooling_factorfloat, optional

Factor used to cool down the threshold when updating the IRLS.

model_previous(n_params) array or None, optional

Array with previous model in the iterations. This model is used to build the R matrix. If None, an array full of zeros will be assigned.

irlsbool, optional

Flag to activate or deactivate IRLS. If False, the class would work as an L2 smallness term. If True, the R matrix will be built using the model_previous.

Attributes

R	R matrix to approximate lp norm using Lawson's algorithm.
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 objective function for a given model.
activate_irls(model_previous)	Activate IRLS.
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.
update_irls(model)	Update IRLS parameters.

Attributes

SparseSmallness.R

R matrix to approximate lp norm using Lawson’s algorithm.

SparseSmallness.cell_weights

Regularization weights on cells.

SparseSmallness.n_active

SparseSmallness.n_params

SparseSmallness.name

Name of the objective function.

SparseSmallness.weights_matrix

Diagonal matrix with the square root of regularization weights on cells.

SparseSmallness.active_cells

Methods

Methods documentation

SparseSmallness.__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.

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SparseSmallness.activate_irls(model_previous)

Activate IRLS.

Parameters:

model_previous(n_params) array

Inverted model obtained after the first stage (l2 inversion).

Notes

Activate IRLS on the regularization, assign model_previous with the model_previous obtained after the first stage (the l2 inversion, before IRLS gets activated), and estimate the initial threshold as:

\[\epsilon = \lVert \mathbf{m}_\text{prev} \rVert_\infty = \text{max}(|\mathbf{m}_\text{prev}|)\]

where \(\mathbf{m}_\text{prev}\) is the model_previous argument.

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SparseSmallness.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.

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SparseSmallness.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.

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SparseSmallness.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.

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SparseSmallness.info()

Get information about the objective function.

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SparseSmallness.set_name(value)

Set name for the objective function.

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SparseSmallness.update_irls(model)

Update IRLS parameters.

Cool down the threshold and assign the model as the new model_previous attribute.

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