Аннотация:The gravitational lensing phenomenon can provide us with the information
about luminous and dark matter in our Universe. But robust and
effective tools are needed to extract that valuable information from observations.
In this chapter, two inverse problems arising in gravitational lensing research are
considered. Both problems are ill-posed, so regularization is needed. The first
problem is the one of image reconstruction. Based on the Tikhonov regularization
approach, several modifications of the regularizing algorithm, taking account
of specific properties of gravitational lens systems, are proposed. The numerical
results show that incorporation of all available a priori information allows reconstructing
images of gravitational lens systems quite well. Furthermore, the
algorithm decomposes the images into two parts – point sources and smooth
background, that is needed for further investigations. The second problem concerns
reconstruction of a distant quasar light distribution based on observations
of microlensing events. In this case, a gravitational lens system as a gravitational
telescope and the Tikhonov regularization method as a tool allow getting valuable
information about the distant quasar unresolved by an instrument.