Volume 15, Issue 7

Inverse problems of the plane waves scattering of inclined incidence of the SH type by an inhomogeneous half-space (in particular, by a transition layer) or by an inhomogeneous layer with a free boundary are considered. The characteristics of an inhomogeneous elastic medium, i.e., the wave propagation velocity ν( z ) and the density ρ( z ), are functions of depth z and should be determined in the inverse problems. The following are considered: the inverse problems with data for a fixed angle of incidence θ 0 of a plane wave (the angle between the vector of the normal to the plane wave front and z -axis) when the shape φ 0 (ξ, θ 0 ) of an incident wave is known; the inverse problems with data for a family of angles θ 0 both for known and unknown shapes of the incident wave. The functions φ 0 (ξ, θ 0 ) and φ 1 (ξ, θ 0 ) (the shapes of incident and reflected waves), the functions φ 0 (ξ, θ 0 ) and u ( H , ξ, θ 0 ) (the shapes of the incident wave and the free boundary oscillation field), only the function u ( H , ξ, θ 0 ) (in the inverse problem with the unknown function φ 0 (ξ, θ 0 )), or other data are given as data corresponding to any fixed value of θ 0 . Possible application areas of the inverse problems under consideration are specified. This paper is essentially a review. A new result presented here has been obtained for inverse problems with the data {φ 0 (ξ, θ 0 ), φ 1 (ξ, θ 0 } or {φ(ξ, θ 0 ), u ( H , ξ, θ 0 )} for a set of angles θ 0 . The results [Mathematical Problems of Geophysics, V. 5, P. 2. Computer Center of Siberian Branch of USSR Academy of Sciences, Novosibirsk, 1974, 78–107] (the uniqueness theorem, the solution method) are extended to the case when the limiting point θ 0 of this set is zero. Also, a new explicit formula (formula (5.3) from Section 5), which makes it possible to find the functions ν( z ) and ρ( z ) using the data of these inverse problems for θ 0 = 0, has been obtained.

We find explicit representation formulae for the solution to the identification problem (2.1) in a Banach space X , when the linear closed operator , with , generates an analytical semigroup exponentially decreasing at infinity. Such a general result is applied to a specific parabolic boundary value problem.

In this note we present a critical review of the some of the positive features as well as some of the shortcomings of the generalized sensitivity functions (GSF) of Thomaseth–Cobelli in comparison to traditional sensitivity functions (TSF). We do this from a computational perspective of ordinary least squares estimation or inverse problems using two illustrative examples: the Verhulst–Pearl logistic growth model and a recently developed agricultural production network model. Because GSF provide information on the relevance of data measurements for the identification of certain parameters in a typical parameter estimation problems, we argue that they provide the basis for new tools for investigators in design of inverse problem studies.

This paper deals with the problem of recovering a scalar time-dependent function in the source term in an abstract parabolic equation with delay. Both the so-called singular and nonsingular cases are considered. For such problems existence and uniqueness results as well as continuous dependence upon the data are proved. Applications to partial differential equations, with measurement of the flux on the boundary, are given.

This paper summarizes some of the old results obtained for the problems of inverting the two-dimensional attenuated X-ray transform and the attenuated vectorial X-ray transform, both using the fan-beam geometry. These inverse problems are considered on the language of the transport equation and two approaches are described for solving them. The first one, dating back to 1996, gives the inversion formulae on the basis of the theory of so-called A -analytic functions. And the second method (developed by the authors in 2002) yields the inversion formulae for the scalar and vector attenuated X-ray transforms without using the theory of A -analytic functions, but merely by reducing the arising inverse problems to the unattenuated case by the change of variables. Numerical implementation details are also provided.