Volume 9, Issue 6

- As is known, boundary spectral data of the compact Riemannian manifold Ω (spectrum of Laplacian with zero Dirichlet boundary condition plus traces of normal derivatives of eigenfunctions at ∂Ω) determine its boundary dynamical data (dynamical Dirichlet-to-Neumann map) R 2T for all T > 0. In the paper the procedures recovering spectral data of the submanifold Ω T = {x ∈ Ω | dist(x, ∂Ω) < T} via given R2T with any prescribed T > 0 and continuing R 2T from ∂Ω × (0, 2T) onto ∂Ω×(0,∞) are proposed. The procedures do not invoke solving the inverse problems; main fragment is the constructing (via R2T ) and the use of a model of dynamical system associated with Ω T .

- A formula is given for recovering the boundary values of the coefficient γ of an elliptic operator, divγ∇, from the Dirichlet to Neumann map. The main point is that one may recover γ without any a priori smoothness assumptions. The formula allows one to recover the value of γ pointwise.

- The regularization problem of degenerated nonlinear system of equations/ inequalities in Banach spaces is considered. The proposed approach is based on the explicit parametrization of input data and on the utilization of the multi-valued mapping techniques. Two regularization methods are suggested: the extended minimization and the regularized partial penalty function one. According to the first method an ill-posed problem should be replaced by a search of the minimal norm element in the join of solution sets of the family of problems that are equivalent to the initial one with respect to input data accuracy. The second one is based on separation from the initial system some regular part. Suggested approach is illustrated on some examples (analytical and numerical), including a singular system of ordinary differential and finite equations.

- We are concerned with the problem of recovering the radial kernel k , depending also on time, in the parabolic integro-differential equation A being a uniformly elliptic second-order linear operator in divergence form. We single out a special class of operators A and two pieces of suitable additional information for which the problem of identifying k can be uniquely solved locally in time when the domain under consideration is a ball or a disk.

- In this paper we numerically compare several recently investigated iterative regularization schemes for the solution of a parameter estimation problem in a hyperbolic partial differential equation. Moreover, we introduce a framework of monotone iterative regularization techniques for general ill-posed problems. This concept relies on monotone fixed point iteration schemes.

- We recover, in addition to the solution and the free boundary, the unknown time-dependent kernel of an integro-differential one-phase Stefan problem in the framework of H¨older spaces.

- Three-dimensional models of radiation transport in the P n -approximation of the spherical harmonic method are considered. Algorithms for atmosphere correction of satellite images of the Earth’s surface are developed.