Special Issue for the Fourth Young Scholars Symposium on Inverse Problems

Quantum systems described by the fractional powers of the negative Laplacian and the interaction potentials are considered. If the potential function slowly decays and the Dollard-type modified wave operators exist and are asymptotically complete, we prove that the factional Laplacian does not possess the standard wave operators. This result suggests the borderline between the short- and long-range behaviour.

Two weak formulations for the Lamé system with the boundary conditions of third and fourth types are proposed. It is shown that the regularity of the solutions and properties of the boundary surface guarantee the equivalence of variational and standard formulations of the problem. Moreover, if the boundary of $Ω$ is a Lipschitz polyhedron or if $\mathcal{S}$ ($x$) = 0 on $∂Ω$, the decoupling results of [8] are derived from the weak formulations.

We study normwise backward error estimates for the equality constrained indefinite least squares problem and introduce an explicit sub-optimal linearisation estimate. Numerical examples show the reliability and efficiency of the estimate presented.

An inverse problem to recover small inclusions inside a two-layer structure is considered. Integral representations for the solution of two-layer inhomogeneous conductivity problem are derived and asymptotic expansions of a perturbed electrical field are obtained. Moreover, the uniqueness of the recovery of the locations and conductivities of small inclusions is proved.

Maxwell exterior transmission eigenvalue problems arising in scattering problem for a penetrable cavity are considered. Properties of such eigenvalues for a weak formulation of the transmission problem in unbounded domains are studied and the Calderon operator for Maxwell's system is described. In particular, the absence of pure imaginary eigenvalues and the discreteness of the set of exterior transmission eigenvalues are established. Applications to exterior invisibility cloaking are also considered. Using the Maxwell-Herglotz approximation, we can generate nearly non-scattering waves corresponding to the exterior eigenvalues of an electromagnetic medium.

An inverse time-dependent source problem for a multi-dimensional fractional diffusion wave equation is considered. The regularity of the weak solution for the direct problem under strong conditions is studied and the unique solvability of the inverse problem is proved. The regularised variational problem is solved by the conjugate gradient method combined with Morozov's discrepancy principle. Numerical examples show the stability and efficiency of the method.

Finite-time dynamic output feedback control for a class of discrete-time nonlinear quadratic systems with stochastic parametric uncertainty, exogenous disturbance and missing measurements, modeled by a Bernoulli distributed stochastic variable, is considered and sufficient conditions for FTSB under a dynamic output feedback controller are provided. As a consequence, a sufficient condition for FTSS is derived. A numerical example demonstrates the effectiveness of the method.

An inverse source problem for elastic waves in isotropic homogeneous media is considered and a Fourier transform based multi-frequency approach to determine the elastic source is proposed. This is an easily implementable fast non-iterative inversion scheme. The stability estimate is derived. Numerical examples demonstrate the effectiveness of the method.

The ill-posed problems of the identification of unknown source terms in a time-fractional radial heat conduction problem are studied. To overcome the difficulties caused by the ill-posedness, a fractional Tikhonov regularisation method is proposed. Employing Mittag-Leffler function, we obtain error estimates under a priori and a posteriori regularisation parameter choices. In the last situation, the Morozov discrepancy principle is also used. Numerical examples show that the method is a stable and effective tool in the reconstruction of smooth and non-smooth source terms and, generally, outperforms the classical Tikhonov regularisation.

An inverse source problem for time-harmonic Maxwell equation is studied and a direct sampling method to recover the location and the moments of electric current dipoles from near- and far-field data at a fixed frequency is developed. The method is based on a new indicator function and requires the evaluation of simple integrals only. It is easily implementable and efficient. Numerical examples demonstrate the robustness of the method with respect to noisy measurements and its capability in identifying the source location and the moments.

The probability-guaranteed $H$_∞-filtering problem for 2$D$-discrete time systems with random nonlinearities and missing measurements is studied. To characterise the statistical uncertain parameters, the uniform distribution is used. The filter parameters are derived by LMI technique and a special parameter-box is used to guarantee $H$_∞-performance with pre-specified probability constraints. An example shows the effectiveness of the filtering scheme developed.