Volume 14, Issue 1

In the paper we present new constructive methods of investigation of multidimensional inverse problems for kinetic and other evolution equations. These problems correspond to the author's previous researches and continue them.

A planar graph consisting of strings of variable densities is considered. The spectrum of the Dirichlet problem on the graph and the values of derivatives of the normalized eigenfunctions at the boundary vertices constitute the spectral data. The inverse problem is to recover the structure of the graph and the densities from the spectral data. If the graph doesn't contain cycles (is a tree), it is determined by the spectral data up to a natural isometry on the plane (Belishev, 2004). In the paper this uniqueness result is supplied with an efficient procedure of recovering the tree. The numerical illustration is presented.

We consider an inverse problem of determining a spatially varying source term in a parabolic transmission problem. We establish a Lipschitz stability estimate in determining a source by a single measurement on an arbitrary subdomain over a time interval. Our method is a Carleman estimate for the transmission problem by Doubova, Osses and Puel (2002).

We consider a second-order hyperbolic integro-differential equation governing the third component of the displacement vector in viscoelasticity when the functions accounting for the properties of the viscoelastic medium and the external force are independent of coordinate x 3 . We consider the inverse problem consisting in recovering two functions characterizing both the elastic and the viscoelastic moduli of the medium. As additional information to solve our inverse problem we use three measurements related to the solutions to three direct Cauchy problems with an impulse force supported on suitable planes. Prescribing Cauchy data on the lateral surface of the body assumed to occupy initially a cylindrical region, we obtain a conditional stability estimate for the solution to our inverse problem as well as a uniqueness theorem.

The aim of Doppler tomography is the reconstruction of velocity fields from ultrasonic measurements. The linearized model of this inverse problem is the Doppler transform which integrates projections of the fields along the lines the ultrasonic beams are sent off. Unfortunately the Doppler transform has a non-trivial null space. As a consequence only the solenoidal part of the field can be recovered from the data. This is the reason to develop and investigate methods of defect corrections, which are to improve the reconstruction accuracy. In [ J. Inv. Ill-Posed Prob. , 12(6), 597–626, 2004] and [ Inv. Prob. , 21,75–91, 2005] two approaches for the defect correction are considered which rely on the Helmholtz decomposition of vector fields. They consist of the solution of corresponding boundary value problems. This article is concerned with a comparison of these two approaches. We state error estimates based on orthogonal decompositions of vector fields, which indicate the improvement in accuracy we may hope for.