Regularization¶
If there is one model that has a misfit that equals the desired tolerance, then there are infinitely many other models which can fit to the same degree. The challenge is to find that model which has the desired characteristics and is compatible with a priori information. A single model can be selected from an infinite ensemble by measuring the length, or norm, of each model. Then a smallest, or sometimes largest, member can be isolated. Our goal is to design a norm that embodies our prior knowledge and, when minimized, yields a realistic candidate for the solution of our problem. The norm can penalize variation from a reference model, spatial derivatives of the model, or some combination of these.
Tikhonov Regularization¶
Here we will define regularization of a model, m, in general however, this should be thought of as (mm_ref) but otherwise it is exactly the same:
Our discrete gradient operator works on cell centers and gives the derivative on the cell faces, which is not where we want to be evaluating this integral. We need to average the values back to the cellcenters before we integrate. To avoid null spaces, we square first and then average. In 2D with ij notation it looks like this:
If we let D_1 be the derivative matrix in the x direction
Where d_1 is the one dimensional derivative:
Recall that this is really a just point wise multiplication, or a diagonal matrix times a vector. When we multiply by something in a diagonal we can interchange and it gives the same results (i.e. it is point wise)
and the transpose also is true (but the sizes have to make sense…):
So R(m) can simplify to:
We will define W_x as:
And then W as a tall matrix of all of the different regularization terms:
Then we can write
The API¶

class
SimPEG.Regularization.
RegularizationMesh
(mesh, **kwargs)¶ Regularization Mesh
This contains the operators used in the regularization. Note that these are not necessarily true differential operators, but are constructed from a SimPEG Mesh.
 param discretize.base.BaseMesh mesh
problem mesh
 param numpy.ndarray indActive
bool array, size nC, that is True where we have active cells. Used to reduce the operators so we regularize only on active cells
Required Properties:
indActive (
Array
): active indices in mesh, a list or numpy array of <class ‘bool’>, <class ‘int’> with shape (*)

regularization_type
= None¶

property
indActive
¶ indActive (
Array
): active indices in mesh, a list or numpy array of <class ‘bool’>, <class ‘int’> with shape (*)

property
vol
¶ reduced volume vector
 Return type
 Returns
reduced cell volume

property
Pac
¶ projection matrix that takes from the reduced space of active cells to full modelling space (ie. nC x nindActive)
 Return type
 Returns
active cell projection matrix

property
Pafx
¶ projection matrix that takes from the reduced space of active xfaces to full modelling space (ie. nFx x nindActive_Fx )
 Return type
 Returns
active facex projection matrix

property
Pafy
¶ projection matrix that takes from the reduced space of active yfaces to full modelling space (ie. nFy x nindActive_Fy )
 Return type
 Returns
active facey projection matrix

property
Pafz
¶ projection matrix that takes from the reduced space of active zfaces to full modelling space (ie. nFz x nindActive_Fz )
 Return type
 Returns
active facez projection matrix

property
aveFx2CC
¶ averaging from active cell centers to active xfaces
 Return type
 Returns
averaging from active cell centers to active xfaces

property
aveCC2Fx
¶ averaging from active xfaces to active cell centers
 Return type
 Returns
averaging matrix from active xfaces to active cell centers

property
aveFy2CC
¶ averaging from active cell centers to active yfaces
 Return type
 Returns
averaging from active cell centers to active yfaces

property
aveCC2Fy
¶ averaging from active yfaces to active cell centers
 Return type
 Returns
averaging matrix from active yfaces to active cell centers

property
aveFz2CC
¶ averaging from active cell centers to active zfaces
 Return type
 Returns
averaging from active cell centers to active zfaces

property
aveCC2Fz
¶ averaging from active zfaces to active cell centers
 Return type
 Returns
averaging matrix from active zfaces to active cell centers

property
cellDiffx
¶ cell centered difference in the xdirection
 Return type
 Returns
differencing matrix for active cells in the xdirection

property
cellDiffy
¶ cell centered difference in the ydirection
 Return type
 Returns
differencing matrix for active cells in the ydirection

property
cellDiffz
¶ cell centered difference in the zdirection
 Return type
 Returns
differencing matrix for active cells in the zdirection

property
faceDiffx
¶ xface differences
 Return type
 Returns
differencing matrix for active faces in the xdirection

property
faceDiffy
¶ yface differences
 Return type
 Returns
differencing matrix for active faces in the ydirection

property
faceDiffz
¶ zface differences
 Return type
 Returns
differencing matrix for active faces in the zdirection

property
cellDiffxStencil
¶ cell centered difference stencil (no cell lengths include) in the xdirection
 Return type
 Returns
differencing matrix for active cells in the xdirection

property
cellDiffyStencil
¶ cell centered difference stencil (no cell lengths include) in the ydirection
 Return type
 Returns
differencing matrix for active cells in the ydirection

property
cellDiffzStencil
¶ cell centered difference stencil (no cell lengths include) in the ydirection
 Return type
 Returns
differencing matrix for active cells in the ydirection