Understanding Source Localization of bst_inverse_linear[2018]

Dear Dr.Sylvain
Thank you so much for the reply and the manual that you sent.
I was wondering, is there any specific research paper focusing on why do we need the regularization parameter or the " + lambda |fs| " function? Is it because of the existence of the noise or because the modeling is not completely correct? I understand that adding the regularization parameter makes the solution more sparse and helps with not getting stuck in local minima while performing L2 Norm, but I was wondering, what is the mathematical problem with the Linear MNE that forces us to use regularization parameters, covariance, and lambda?

Thank you
Best