Amplitude regularization

[domfournier]

New implementation for MVI sparse inversion to keep the problem (mostly) linear.

• Implementation of VectorAmplitude regularization

$\phi_s = || \mathbf{W} \mathbf{P} \mathbf{a}(m - m_{ref}) ||_2^2$

• Implementation of Group map as extension of Wire with derivatives of wires returned as a list.

Solution with amplitude regularizer

[codecov]

Codecov Report

Merging #1103 (ff23ad4) into main (e666ab0) will decrease coverage by 5.38%.
The diff coverage is 92.42%.

@@            Coverage Diff             @@
##             main    #1103      +/-   ##
==========================================
- Coverage   81.85%   76.48%   -5.38%     
==========================================
  Files         163      163              
  Lines       24570    24674     +104     
==========================================
- Hits        20111    18871    -1240     
- Misses       4459     5803    +1344     

Impacted Files	Coverage Δ
SimPEG/directives/directives.py	79.65% <72.72%> (-3.80%)	⬇️
SimPEG/regularization/base.py	83.48% <90.00%> (-11.34%)	⬇️
SimPEG/regularization/vector.py	69.71% <96.38%> (-30.29%)	⬇️
SimPEG/maps.py	68.56% <100.00%> (-17.41%)	⬇️
SimPEG/objective_function.py	74.03% <100.00%> (-21.60%)	⬇️
SimPEG/regularization/__init__.py	92.68% <100.00%> (-2.32%)	⬇️
SimPEG/regularization/sparse.py	91.33% <100.00%> (-5.42%)	⬇️

... and 30 files with indirect coverage changes

[domfournier]

@orerocks All your dreams will come true.

[jcapriot]

Hey Dom, I think we're both finding some similar choke points in my cross-reference vector regularization (just opened it at #1214), and this one.

• Had to deal with the UpdateSensitivityWeights directive to handle vector valued models.
  • Mine unconditionally reshapes the output of reg.mapping * jtj_diag to be a (n_cells, -1) array, so then we can always divide by the mesh volume as vol[:, None]
• Needed a way to verify the model weight sizes.
  • Mine allowed for (n_cells,), (n_cells*dim,), and (n_cells, dim) shapes. Then I conditionally handled them in the _W function.

I think we can split a few common things into a BaseVectorRegularizer class, and maybe have them all live inside a SimPEG.regularization.vector.py module?

Some design thoughts, would it be easier if we didn't have to specify a wiremap for the three components in this regularizer? (The simulations already except models like this without mapping them to the different components, so the model should already be in a n_cells * dim array), then it can get reshaped within the calls?

One detail though, why are you passing the model difference (dm) to the mapping.deriv in f_m_deriv? Pretty sure it should just be the m, unless you had a specific reason (this error would only have shown up in a non-linear mapping).

P.S. Can you merge main in? I think there's some duplicate fixes.

[domfournier]

@jcapriot Yea I saw your modifications to the UpdateSensWeights. I like yours better.

Looks like we may be able to share some components indeed. Not sure how you want to proceed?

The wire mapping is kind of nice to keep the operations close to what the linear algebra looks like. And since Wires doesn't have derivatives, I thought we could just implement it in here.

Good point on dm - typo. Won't change anything anyway since the deriv of a wire just returns the projection.

I will merge main in.

[domfournier]

@jcapriot I refactored using your base class.

[domfournier]

@jcapriot Ready for another review.