Volume 12, Issue 6

Multidimensional systems of kinetic equations are considered in this paper. New representations for solutions and coefficients are given and operator formulas are obtained for solving a certain inverse problem.

An algorithm for the simultaneous determination of two coefficients in an inverse problem for equations of the parabolic type is studied. The model under investigation has been proposed by H. de Vriend and M. Capobianco to describe the long-time coastal profile evolution. The iterative inversion procedure is based on the minimization of a suitable cost functional. Results of numerical tests are shown to illustrate the performance of the algorithm. The real measured data of the coastal profile evolution have been numerically processed to validate the aforementioned diffusion type model. The paper is targetly oriented to attract attention of professionals, who study Inverse Problems, to such new area of applications as Coastal Engineering.

We consider one- and multi-dimensional inverse problems for acoustic equation u tt = c 2 (Δ u – ∇ ln ρ · ∇ u ) in a medium filling interior of a cylinder infinite with respect to the variable z . For the one-dimensional problem of reconstruction of acoustic medium parameters (the velocity c ( r ) and the density ρ ( r )) we present necessary and sufficient conditions of unique solvability. In multi-dimensional case we investigate the inverse problem of reconstruction of the velocity c ( r, φ, z ) assuming the density ρ ( r ) is known. Estimates of conditional stability are obtained in the both cases.

The problem of vector field tomography consists of reconstructing a vector field from integrals of certain projections of the field over geodesic lines. In case of an homogeneous medium the underlying Riemannian metric is the Euclidean one and the geodesics are straight lines. Modeling inhomogeneous media we take also non-Euclidean metrics into account. It is well known that the corresponding integral transform has a nontrivial kernel consisting of potential fields with potentials being equal to a constant on the boundary of the domain. Thus, the reconstruction from these data may contain a potential part of this type. Since we consider only solenoidal fields, e.g.velocity fields of incompressible fluids, the accuracy of the solution can be improved detecting the undesirable potential part of the reconstruction and subtracting it. Depending on the boundary condition the sought potential is the solution of either a Dirichlet or a Neumann boundary value problem. In this article two approaches to recover the potential are suggested: One deals with the solution of the Dirichlet problem with a finite difference scheme, the other solves the Neumann problem with the help of boundary element methods.

In this paper we show that an additional a priori information about a sourcewise representation of the solution of nonlinear ill-posed problem can be used for constructing regularizing algorithms with a posteriori error estimates. We describe these algorithms. Algorithms for the case when operators are linear can be obtained from these general algorithms.

We present a new procedure in the finite dimensional setting for the recently developed method of mollifiers for obtaining stable approximate solutions for ill-posed equations. A particular case of the derived error estimate improves upon similar results obtained recently by Rieder and Schuster (2000).

A class of nonlinear inverse problems that is widely studied in many applications are parameter identification problems. In this paper, we consider a modified Landweber iterative method for solving a class of parameter identification problems. We carry out the analysis with a weak source condition and also illustrate the theoretical aspects numerically.