Nonlinear inversion and Gauss–Newton

At xₖ, approximate F(xₖ+h) by F(xₖ)+Jₖh, where Jₖ is the derivative. The Gauss–Newton step minimizes the squared norm of Jₖh−rₖ with rₖ=yδ−F(xₖ). In finite dimensions with full column rank it has the usual least-squares solution. If Jₖ is nearly rank deficient or is a compact operator with nonclosed range, that subproblem can itself be unstable. A good forward derivative calculation therefore does not eliminate the need for inverse regularization.

Adding αₖ||h||²/2 to the local squared residual gives the Levenberg–Marquardt-type equation (Jₖ*Jₖ+αₖI)h=Jₖ*rₖ. This penalizes the step, not the absolute object. An iteratively regularized Gauss–Newton method that penalizes xₖ+h relative to a fixed reference has a different right-hand side and different convergence analysis. The distinction matters: a small damped step may indicate strong damping or a weak derivative, rather than an accurate reconstructed object.

A smooth map can be locally invertible near a point yet have several preimages globally. The scalar map x↦x² has nonzero derivative at every nonzero x, but positive data have two opposite roots on the unrestricted real domain. Restricting the admissible class to positive x removes that branch ambiguity away from zero. Near zero, however, the derivative degenerates and the square-root inverse loses Lipschitz stability. The domain restriction, derivative bound, and neighborhood must accompany a local identifiability statement.

In a finite-dimensional zero-residual problem, a sufficiently smooth map with a full-column-rank derivative at the solution supports familiar local Gauss–Newton convergence results from a sufficiently close starting point. Those hypotheses do not establish convergence from arbitrary initialization or for a noisy inconsistent residual. In infinite-dimensional inverse problems, bounded invertibility can fail even at the truth. Local regularization theory then uses quantitative conditions on the nonlinear remainder, source behavior, and parameter or stopping rules rather than borrowing a finite-matrix theorem unchanged.

A trust-region or line-search test compares the actual residual reduction with that predicted by the linear model. Poor agreement can require a shorter step or stronger damping. Such a safeguard controls local model use but does not reveal an unobserved branch or prove that a local minimum is the desired physical solution. Constraints and multiple initializations may help explore possibilities, but their empirical success should not be presented as a general global recovery theorem.

With noisy data, continuing to reduce the nonlinear residual can fit noise just as in linear inversion. Discrepancy stopping needs a justified data norm and uncertainty level, while parameter updates need to respect the nonlinear method's assumptions. An inaccurately implemented adjoint or derivative can also create an apparently stationary step. Checking the derivative against finite differences and verifying a local normal-equation residual are computational checks; establishing identifiability or a physical error bound requires separate mathematical and modeling evidence.