Pygimli Inversion

[makeabhishek]

Hi Pygimli team,

I have some fundamental questions regarding the inversion framework.
I was reading the article "Rücker, Carsten, Thomas Günther, and Florian M. Wagner. "pyGIMLi: An open-source library for modelling and inversion in geophysics." Computers & Geosciences 109 (2017): 106-123."

It is mentioned that "The default inversion framework is based on the generalized Gauss-Newton method" and further that the application of the Gauss-Newton scheme on minimizing the objective function (\phi) yields the model update in the iteration which is solved using a conjugate-gradient least-squares solver. Aren't Gauss-Newton and conjugate gradient different methods?

The system of above equations has to be solved in every iteration step.
How is the conjugate-gradient method incorporated with the Gauss-Newton method?

Furthermore, how is the smoothness (\phi_m) model defined in Pygimli, used in the objective function? Is it a kind of diagonal matrix?

How the implemented inverse optimisation method is different from conventional Gauss-newton method?

[halbmy]

Gauss-Newton is a minimization method, i.e. a way to iteratively find a model update using the Jacobian matrix (2nd derivative), whereas there are also gradient-based minimization schemes like NLCG (nonlinear conjugate gradients).

So solve the inverse subproblem, we use a conjugate-gradient least-squares solver.

The smoothness matrix is a sparse matrix defining 1st or 2nd (or 0th or mixes of them) order derivatives.

For details on all this please have a look at Günther et al. (2006).

Günther, T., Rücker, C. & Spitzer, K. (2006): Three-dimensional modeling and inversion of dc resistivity data in-
corporating topography – II: Inversion. Geophys. J. Int. 166, 506-517, http://doi.org/10.1111/j.1365-246X.2006.03011.x.

[makeabhishek]

Thanks for the explanation. Yes I was reading the referred article. you mentioned, " Gauss-Newton is a minimization method, i.e. a way to iteratively find a model update using the Jacobian matrix (2nd derivative)". Isn't Jacobian define the 1st derivation and Heasian define 2nd derivative?
However, when I read "Rücker, Carsten, Thomas Günther, and Florian M. Wagner. (2017): pyGIMLi: An open-source library for modelling and inversion in geophysics." Computers & Geosciences, I get confused. Because it mentioned that inversion uses Gauss-Newton method for minimisation and then solved using a conjugate-gradient least-squares solver, as shown in the below image:

[halbmy]

There is no contradiction. The Jacobian J is the derivative of the forward model with respect to the model parameters and the Gauss-Newton method approximates the Hessian (2nd derivative of the objective function) by J.T J. It is written in many text books and papers, but maybe it is a good idea to write it all down in a tutorial, where the individual components are introduced and shown.