Abstract

Geometry plays an important role in inverse problems. For example, reconstruction of second order elliptic selfadjoint differential operator on the manifold through the gauge transformation can be reduced to the reconstruction of Shrodinger operator, corresponding to Beltrami-Laplace operator, i.e. topology of the manifold , riemannian metric on it and potential. The difficulties mostly related to geometric aspects of the problem. If we consider applied invere problems, we also see, that the main problems lies in geometry. For example, in the main problem of geophysics – the so called migration problem, it is necessary to reconstruct high frequency wave fields in the media with complicated geometry with many caustics of different structure. The difficulties of reconstruction of wave fields close to caustics are also of geometric character. To solve the geometric problems it is necessary to have instruments closely related to the geometry of corresponding problem. One of such instruments is Gaussian beams solutions. In the talk the geometric properties of these solutions and their use in direct and inverse problems will be shown. The problems with more complicated Finsler geometry will also be discussed.