SimPEG.regularization.SparseSmallness#

class SimPEG.regularization.SparseSmallness(mesh, norm=2.0, irls_scaled=True, irls_threshold=1e-08, **kwargs)[source]#

Bases: BaseSparse, Smallness

Sparse smallness (compactness) regularization.

SparseSmallness is used to recover models comprised of compact structures. The level of compactness is controlled by the norm within the regularization function; with more compact structures being recovered when a smaller norm is used. Optionally, custom cell weights can be included to control the degree of compactness being enforced throughout different regions the model.

See the Notes section below for a comprehensive description.

Parameters:

meshregularization.RegularizationMesh: Mesh on which the regularization is discretized. Not the mesh used to define the simulation.
normfloat, (n_cells, ) array_like: The norm defining sparseness in the regularization function. Use a float to define the same norm for all mesh cells, or define an independent norm for each cell. All norm values must be within the interval [0, 2].
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.
mappingNone, SimPEG.maps.BaseMap: The mapping from the model parameters to the active cells in the inversion. If None, the mapping is the identity map.
reference_modelNone, (n_param, ) numpy.ndarray: Reference model. If None, the reference model in the inversion is set to the starting model.
unitsNone, str: Units for the model parameters. Some regularization classes behave differently depending on the units; e.g. ‘radian’.
weightsNone, dict: Weight multipliers to customize the least-squares function. Each key points to a (n_cells, ) numpy.ndarray that is defined on the regularization.RegularizationMesh.
irls_scaledbool: If True, scale the IRLS weights to preserve magnitude of the regularization function. If False, do not scale.
irls_thresholdfloat: Constant added to IRLS weights to ensures stability in the algorithm.

Notes

We define the regularization function (objective function) for sparse smallness (compactness) as:

\[\phi (m) = \frac{1}{2} \int_\Omega \, w(r) \, \Big | \, m(r) - m^{(ref)}(r) \, \Big |^{p(r)} \, dv\]

where \(m(r)\) is the model, \(m^{(ref)}(r)\) is the reference model, \(w(r)\) is a user-defined weighting function and \(p(r) \in [0,2]\) is a parameter which imposes sparseness throughout the recovered model. More compact structures are recovered in regions where \(p\) is small. If the same level of sparseness is being imposed everywhere, the exponent becomes a constant.

For implementation within SimPEG, the regularization function and its variables must be discretized onto a mesh. The discretized approximation for the regularization function (objective function) is expressed in linear form as:

\[\phi (\mathbf{m}) = \frac{1}{2} \sum_i \tilde{w}_i \, \Big | m_i - m_i^{(ref)} \Big |^{p_i}\]

where \(m_i \in \mathbf{m}\) are the discrete model parameters defined on the mesh. \(\tilde{w}_i \in \mathbf{\tilde{w}}\) are amalgamated weighting constants that 1) account for cell dimensions in the discretization and 2) apply user-defined weighting. \(p_i \in \mathbf{p}\) define the norm for each cell (set using norm).

It is impractical to work with the general form directly, as its derivatives with respect to the model are non-linear and discontinuous. Instead, the iteratively re-weighted least-squares (IRLS) approach is used to approximate the sparse norm by iteratively solving a set of convex least-squares problems. For IRLS iteration \(k\), we define:

\[\phi \big (\mathbf{m}^{(k)} \big ) = \frac{1}{2} \sum_i \tilde{w}_i \, \Big | m_i^{(k)} - m_i^{(ref)} \Big |^{p_i} \approx \frac{1}{2} \sum_i \tilde{w}_i \, r_i^{(k)} \Big | m_i^{(k)} - m_i^{(ref)} \Big |^2\]

where the IRLS weight \(r_i\) for iteration \(k\) is given by:

\[r_i^{(k)} = \bigg [ \Big ( m_i^{(k-1)} - m_i^{(ref)} \Big )^2 + \epsilon^2 \; \bigg ]^{{p_i}/2 - 1}\]

and \(\epsilon\) is a small constant added for stability (set using irls_threshold). For the set of model parameters \(\mathbf{m}\) defined at cell centers, the objective function for IRLS iteration \(k\) can be expressed as follows:

\[\phi \big ( \mathbf{m}^{(k)} \big ) \approx \frac{1}{2} \Big \| \, \mathbf{W}^{\! (k)} \big [ \mathbf{m}^{(k)} - \mathbf{m}^{(ref)} \big ] \Big \|^2\]

where

\(\mathbf{m}^{(k)}\) are the discrete model parameters at iteration \(k\),

\(\mathbf{m}^{(ref)}\) is a reference model (optional, set with reference_model),

\(\mathbf{W}^{(k)}\) is the weighting matrix for iteration \(k\). It applies the IRLS weights, user-defined weighting, and accounts for cell dimensions when the regularization function is discretized.

IRLS weights, user-defined weighting and the weighting matrix:

Let \(\mathbf{w_1, \; w_2, \; w_3, \; ...}\) each represent an optional set of custom cell weights. And let \(\mathbf{r_s}^{\!\! (k)}\) represent the IRLS weights for iteration \(k\). The net weighting applied within the objective function is given by:

\[\mathbf{w}^{(k)} = \mathbf{r_s}^{\!\! (k)} \odot \mathbf{v} \odot \prod_j \mathbf{w_j}\]

where \(\mathbf{v}\) are the cell volumes. For a description of how IRLS weights are updated at every iteration, see the documentation for update_weights().

The weighting matrix used to apply the weights is given by:

\[\mathbf{W}^{(k)} = \textrm{diag} \Big ( \sqrt{\mathbf{w}^{(k)} \, } \Big )\]

Each set of custom cell weights is stored within a dict as an (n_cells, ) numpy.ndarray. The weights can be set all at once during instantiation with the weights keyword argument as follows:

>>> reg = SparseSmallness(mesh, weights={'weights_1': array_1, 'weights_2': array_2})

or set after instantiation using the set_weights method:

>>> reg.set_weights(weights_1=array_1, weights_2=array_2})

Methods

update_weights(m)

Update the IRLS weights for sparse smallness regularization.