# SimPEG.regularization.AmplitudeSmoothnessFirstOrder#

Sparse amplitude smoothness (blockiness) regularization.

AmplitudeSmallness is a sparse norm smoothness regularization that acts on the amplitudes of the vectors defining the model. Sparse norm functionality allows the use to recover more blocky regions of higher amplitude vectors. The level of blockiness is controlled by the norm within the regularization function; with more blocky structures being recovered when a smaller norm is used. Optionally, custom cell weights can be included to control the degree of blockiness 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.

orientation{‘x’,’y’,’z’}

The direction along which sparse smoothness is applied.

norm

The norm defining sparseness thoughout the regularization function. Must be within the interval [0,2]. There are several options:

• float: constant sparse norm throughout the domain.

• (n_faces, ) array_like: define the sparse norm independently at each face set by orientation (e.g. x-faces).

• (n_cells, ) array_like: define the sparse norm independently for each cell. Will be averaged to faces specified by orientation (e.g. x-faces).

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. To include the reference model in the regularization, the reference_model_in_smooth property must be set to True.

reference_model_in_smoothbool, optional

Whether to include the reference model in the smoothness terms.

units

Units for the model parameters. Some regularization classes behave differently depending on the units; e.g. ‘radian’.

weights

Custom weights for the least-squares function. Each key points to a numpy.ndarray that is defined on the regularization.RegularizationMesh. A (n_cells, ) numpy.ndarray is used to define weights at cell centers, which are averaged to the appropriate faces internally when weighting is applied. A (n_faces, ) numpy.ndarray is used to define weights directly on the faces specified by the orientation input argument.

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.

Gradient measure used in the IRLS re-weighting. Whether to re-weight using the total gradient or components of the gradient.

Notes

The regularization function (objective function) for sparse amplitude smoothness (blockiness) along the x-direction as:

$\phi (m) = \frac{1}{2} \int_\Omega \, w(r) \, \Bigg | \, \frac{\partial |\vec{m}|}{\partial x} \, \Bigg |^{p(r)} \, dv$

where $$\vec{m}(r)$$ is the 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. Sharper boundaries are recovered in regions where $$p(r)$$ 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 discrete approximation for the regularization function (objective function) is expressed in linear form as:

$\phi (\mathbf{m}) = \frac{1}{2} \sum_i \tilde{w}_i \, \Bigg | \, \frac{\partial |\vec{m}_i|}{\partial x} \, \Bigg |^{p_i}$

where $$\vec{m}_i$$ is the vector defined for mesh cell $$i$$. $$\tilde{w}_i$$ 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:

$\phi \big (\mathbf{m}^{(k)} \big ) = \frac{1}{2} \sum_i \tilde{w}_i \, \left | \, \frac{\partial \big | \vec{m}_i^{(k)} \big | }{\partial x} \right |^{p_i} \approx \frac{1}{2} \sum_i \tilde{w}_i \, r_i^{(k)} \left | \, \frac{\partial \big | \vec{m}_i^{(k)} \big | }{\partial x} \right |^2$

where the IRLS weight $$r_i$$ for iteration $$k$$ is given by:

$r_i^{(k)} = \Bigg [ \Bigg ( \frac{\partial \big | \vec{m}_i^{(k-1)} \big | }{\partial x} \Bigg )^2 + \epsilon^2 \; \Bigg ]^{{p_i}/2 - 1}$

and $$\epsilon$$ is a small constant added for stability (set using irls_threshold).

The global set of model parameters $$\mathbf{m}$$ defined at cell centers is ordered according to its primary ($$p$$), secondary ($$s$$) and tertiary ($$t$$) directions as follows:

$\begin{split}\mathbf{m} = \begin{bmatrix} \mathbf{m}_p \\ \mathbf{m}_s \\ \mathbf{m}_t \end{bmatrix}\end{split}$

We define the amplitudes of the vectors for all cells as:

$\mathbf{\bar{m}} = \Big [ \, \mathbf{m}_p^2 + \mathbf{m}_s^2 + \mathbf{m}_t^2 \Big ]^{1/2}$

The objective function for IRLS iteration $$k$$ is given by:

$\phi \big ( \mathbf{m}^{(k)} \big ) \approx \frac{1}{2} \Big \| \, \mathbf{W}^{(k)} \, \mathbf{G_x} \, \mathbf{\bar{m}}^{(k)} \Big \|^2$

where

• $$\bar{\mathbf{m}}^{(k)}$$ are the discrete vector model amplitudes at iteration $$k$$,

• $$\mathbf{G_x}$$ is the partial cell-gradient operator along x (x-derivative),

• $$\mathbf{W}^{(k)}$$ 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.

Note that since $$\mathbf{G_x}$$ maps from cell centers to x-faces, the weighting matrix acts on variables living on x-faces.

Reference model in smoothness:

Gradients/interfaces within a discrete reference model $$\mathbf{m}^{(ref)}$$ can be preserved by including the reference model the smoothness regularization. In this case, the least-squares problem for IRLS iteration $$k$$ becomes:

$\phi \big ( \mathbf{m}^{(k)} \big ) \approx \frac{1}{2} \Big \| \, \mathbf{W}^{(k)} \, \mathbf{G_x} \, \mathbf{\bar{m}}^{(k)} \Big \|^2$

where

$\mathbf{\bar{m}} = \bigg ( \Big [ \mathbf{m}_p - \mathbf{m}_p^{(ref)} \Big ]^2 + \Big [ \mathbf{m}_s - \mathbf{m}_s^{(ref)} \Big ]^2 + \Big [ \mathbf{m}_t - \mathbf{m}_t^{(ref)} \Big ]^2 \bigg )^{1/2}$

This functionality is used by setting $$\mathbf{m}^{(ref)}$$ with the reference_model property, and by setting the reference_model_in_smooth parameter to True.

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

Let $$\mathbf{w_1, \; w_2, \; w_3, \; ...}$$ each represent an optional set of custom weights defined on the faces specified by the orientation property; i.e. x-faces for smoothness along the x-direction. Each set of weights were either defined directly on the faces or have been averaged from cell centers. And let $$\mathbf{r_x}^{\!\! (k)}$$ represent the IRLS weights for iteration $$k$$. The net weighting applied within the objective function is given by:

$\mathbf{w}^{(k)} = \mathbf{r_x}^{\!\! (k)} \odot \mathbf{v_x} \odot \prod_j \mathbf{w_j}$

where $$\mathbf{v_x}$$ are cell volumes projected to x-faces; i.e. where the x-derivative lives. 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 weights is stored within a dict as an numpy.ndarray. A (n_cells, ) numpy.ndarray is used to define weights at cell centers, which are averaged to the appropriate faces internally when weighting is applied. A (n_faces, ) numpy.ndarray is used to define weights directly on the faces specified by the orientation input argument. The weights can be set all at once during instantiation with the weights keyword argument as follows:

Attributes

 W Weighting matrix.

Methods

 Evaluate the regularization kernel function. Derivative of the regularization kernel function.