A family of generalized Gauss-Newton methods for 2D inverse
gravimetry problem

Alexandra Smirnova
Dept. of Mathematics and Statistics
Georgia State University, 30 Pryor St., Atlanta GA 30303
smirn@mathstat.gsu.edu

We consider a generalized Gauss-Newton's scheme

$\displaystyle x_{n+1}=\xi-\theta(F^{\prime*}(x_n)F'(x_n),\alpha_n) F^{\prime*}(x_n)\{F(x_n)-f-F'(x_n)(x_n-\xi)\} $

for solving nonlinear unstable operator equation $ \,F(x)=f\,$ in a Hilbert space. In case of noisy data we propose a novel a posteriori stopping rule

$\displaystyle \vert\vert F(x_N)-f_\delta\vert\vert^2\le \tau \delta < \vert\vert F(x_n)-f_\delta\vert\vert^2,\quad 0\le n< N,\quad \tau >1, $

and prove a convergence theorem under a source type condition on the solution. As a consequence of this theorem we obtain convergence rates for five different generating functions, $ \,\theta=\theta(\lambda,\alpha),\,$ of a spectral parameter $ \lambda$ and $ \alpha>0$.

The new algorithms are tested on the 2D inverse gravimetry problem reduced to a nonlinear integral equation of the first kind:

$\displaystyle F(x):=g \,\triangle \sigma \int^b_a\int^d_c\left\{ \frac{1}{[\,(\... ...- \frac{1}{[\,(\xi-t)^2+(\nu-s)^2+h^2\,]^{1/2}} \right\}\,d\xi\,d\nu = f(t,s), $

where $ g$ is the gravitational constant, $ \triangle \sigma$ is the density jump on the interface, and $ f(t,s)$ is the gravitational strength anomaly. The results of numerical simulations are presented and some practical recommendations on the choice of parameters are given.