Volume 14, Issue 3

In the introduction we show that the inverse problems for transport equations are naturally reduced to the Cauchy problem for the so called A -analytic functions, and hence the solution is given in terms of operator analog of the Cauchy transform. In Section 2 we develop elements of the theory of A -analytic functions and obtain stability estimates for our Cauchy transform. In Section 3 we discuss numerical aspects of this transformation. In Section 4 we apply this algorithm to the 3-dimensional inverse kinematic problem with local data on the Earth surface, using modified Newton method and discuss numerical examples.

We deal with the problem of recovering a memory kernel k ( t, η ), depending on time t and on a spatial variable η , in a parabolic integro-differential equation related to a bounded domain Ω ⊂ 3 . We show that, under suitable assumptions and two pieces of additional information, our identification problem can be uniquely solved locally in time.

We consider regularization of linear ill-posed problems Au = ƒ in Hilbert spaces. Approximations u r to the solution u * can be constructed by the Tikhonov method or by the Lavrentiev method, by iterative or by other methods. We assume that instead of ƒ ∈ R ( A ) noisy data are available with the approximately given noise level δ : in process δ → 0 it holds || − ƒ||/ δ ≤ c with unknown constant c . We propose a new a-posteriori rule for the choice of the regularization parameter r = r ( δ ) guaranteeing u r( δ ) → u * for δ → 0. Note that such convergence is not guaranteed for the parameter choice given by the L-curve rule, by the GCV-rule, by the quasioptimality criterion and also for discrepancy principle || Au r − || = bδ with b < c . The error estimates are given, which in case || − ƒ|| ≤ δ are quasioptimal and order-optimal.

An iterative method for the reconstruction of a stationary three-dimensional temperature field, from Cauchy data given on a part of the boundary, is presented. At each iteration step, a series of mixed well-posed boundary value problems are solved for the heat operator and its adjoint. A convergence proof of this method in a weighted L 2 -space is included.

We consider a boundary data identification problem in quasi-static electromagnetism. Two kinds of boundary conditions are simultaneously imposed on one part of the boundary. The other part of the boundary is unreachable, i.e. the boundary condition is unknown and it has to be determined as a part of the problem. We assume that the permeability outside the domain satisfies μ ≫ μ 0 . Two different numerical approaches to solve this problem are presented, compared and their results are given.

This work deals with the inverse scattering problem for two-dimensional Schrödinger operator. The following problem is studied: To estimate more accurately first nonlinear term from the Born series which corresponds to the scattering data with all energies and all angles in the scattering amplitude. This estimate allows us to conclude that the singularities and the jumps of the unknown potential can be obtained exactly by the Born approximation. Especially, for the potentials from L p -spaces the approximation agrees with the true potential up to the continuous function.

2D motion field is the velocity field which presents the apparent motion from one image to another one in an image sequence. It provides important motion information and is widely used in image processing and computer vision. In this paper, with the objective of accurate estimation of a 2D dense motion field, a hybrid diffusion model is proposed. The present approach differs from those in the literature in the sense that the diffusion model and its associated objective functional are driven by both the flow field and image, through a nonlinear isotropic diffusion term and a linear anisotropic diffusion term, respectively. The diffusion function in the model is required to be non increasing, non negative, differentiable and bounded. Using Schauder's fixed point theorem, we prove the existence, stability and uniqueness of the solution to the proposed hybrid diffusion model. A semi-implicit finite difference scheme is proposed to implement the hybrid diffusion model. We demonstrate its efficiency and accuracy by experiments on both synthetic and real image sequences.