Stable Identification of Sources Located on Interface of Nonhomogeneous Media

This paper presents a stable method for the identification of sources located on the separation interface of two homogeneous media (where one of them is contained by the other one), from measurement yielded by those sources on the exterior boundary of the media. This is an ill-posed problem because numerical instability is presented, i.e., minimal errors in the measurement can result in significant changes in the solution. To obtain the proposed stable method the identification problem is categorized into three subproblems, two of which present numerical instability and regularization methods must be applied to obtain their solution in a stable form. To manage the numerical instability due to the ill-posedness of these subproblems, the Tikhonov regularization and sequential smoothing methods are used. We illustrate this methodology in a circular and irregular region to demonstrate the feasibility of the proposed method, which yields convergent and stable solutions for input data with and without noise.

On the property of ”statistical identity” of solutions to some classical problems of the radiative transfer theory

A probabilistic interpretation of the classical solution of the diffuse reflection problem (DRP) of radiation from a semi-infinite homogeneous scattering-absorbing medium on the language of random events in the simple case of monochromatic and isotropic scattering is constructed. A certain property of the so-called ”statistical identity” is specially defined. By using these two circumstances, it is possible to construct a simple symbolic scheme for the direct transformation of the solution mentioned above in the particular case of DRP into solutions to more general cases of DRP, which taking into account the anisotropy and incoherence of scattering, as well as the temporal dependence of the task on the act of absorption. Moreover, some generalization of the primary scheme makes it possible to directly obtain solutions of the DRP also for nonhomogeneous media and for general case of time dependence (on absorption acts and free flights between them) for the quanta diffusion process. At the same time, both the well-known results of the DRP decisions and some new ones were obtained.

The method of finite spheres in acoustic wave propagation through nonhomogeneous media: Inf-sup stability conditions

When the method of finite spheres is used for the solution of time-harmonic acoustic wave propagation problems in nonhomogeneous media, a mixed (or saddle-point) formulation is obtained in which the unknowns are the pressure fields and the Lagrange multiplier fields defined at the interfaces between the regions with distinct material properties. Then certain inf-sup conditions must be satisfied by the discretized spaces in order for the finite-dimensional problems to be well-posed. We discuss in this paper the analysis and use of these conditions. Since the conditions involve norms of functionals in fractional Sobolev spaces, we derive ‘stronger’ conditions that are simpler in form. These new conditions pave the way for the inf-sup testing, a tool for assessing the stability of the discretized problems.