A Descriptive Regularization Approach for a Class of Ill-Posed Nonlinear Integral Equations

Abstract

There are many problems in material sciences, geophysics, optics and meteorology, where a real–valued vertical profile x(t) with 0 ≤ t ≤ 1 of a physical quantity in a layer of thickness 1 is to be determined. If the profile cannot be measured directly, we have an inverse problem. That means, the unknown profile is to be identified from indirect measurements y(s). The really available observations concern in general a variety of linear or nonlinear functionals y(s) = f _s (x) depending on a real parameter s, for which we in the sequel assume that 0 ≤ s ≤ 1. An especially difficult subproblem occurs if the functionals are nonlinear of the form ${f_s}(x) = \int\limits_0^1 {k(s,x(t))dt}$ with a given kernel function k(s, x) (cf. [3]). This kernel function expresses the transmissibility properties of the layer under consideration with respect to the rays or waves passing through the layer and yielding the measurements y(s) depending on the angle of incidence or wave length s. In our specific problem, which can be written as a nonlinear Urysohn integral equation

$$ \int_0^1 {k(s,x(t))dt = y(s)\quad, \quad \quad 0 \leqslant s \leqslant 1} $$

(1)

we have the situation that the transmissibility conditions depend on the layer level t only via x(t). With respect to the data y(s) this leads to an essential loss of information about the unknown function x(t).