Volume 33, Issue 1

We study an inverse boundary value problem for a polyharmonic operator in two dimensions. We show that the Cauchy data uniquely determine all the anisotropic perturbations of orders at most m - 1 {m-1} and several perturbations of orders m to 2 ⁢ m - 2 {2m-2} under some restriction. The uniqueness proof relies on the ∂ ¯ {\bar{\partial}} -techniques and the method of stationary phase.

This paper considers homogeneous equation of convolution type of the first kind on a finite interval. An analogue of the well-known Titchmarsh theorem on supports in convolution is obtained. The results of the work were obtained under the condition that the kernel function in the integral operator is equal to zero in the neighborhood of zero.

The question of finding the optimal location and number of sensors is important when solving applied inverse problems of remote sensing. An incorrect answer to this question can significantly affect (1) the accuracy of reconstructing the unknown parameters of the object under study, and/or (2) the computational complexity of the inverse problem being solved. The work proposes an economical algorithm that allows among all equivalent (in the sense of the computational complexity of obtaining a solution to the inverse problem) sensor configurations, to select the one that will give a more accurate result when processing experimental data. This algorithm also allows to make a conclusion about whether it is really necessary to process all the experimental data obtained in the experiment, or whether it is possible to limit ourselves to only part of them without a significant loss of accuracy in the reconstructed solution. The algorithm is based on (1) the application of the mosaic-skeleton approximation method to the matrix of the system of equations to which the inverse problem being solved is reduced, and (2) the calculation of the compression rate of the approximating matrix relative to the original dense matrix for a given permissible relative approximation error. Thus, the algorithm is a priori, that is, it does not require a preliminary search for a solution to the inverse problem for the sensor configuration under study. Moreover, if the formulas that define the elements of the system matrix are known, then the algorithm requires the calculation of not all of these elements, but only part of them.

The inverse problem of reconstructing two space-varying coefficients in a system of one-dimensional (1-d) time-dependent advection-diffusion-reaction (ADR) equations is considered. The ADR system can be used as a water quality model which describes the evolution of the biochemical oxygen demand (BOD) and dissolved oxygen (DO) in a river or stream. The coefficients to be reconstructed represents the effect of the deoxygenation and superficial reaeration processes on the DO and BOD concentration in water. Hölder stability estimates for the coefficients of interest are established using the Carleman estimate technique.

Inverse spectral problems are considered for the discontinuous Dirac operator with complex-value weight and the spectral parameter boundary conditions. We investigate some properties of spectral characteristics and show that the potential can be uniquely determined by the Weyl-type function or by two spectra on the whole interval.

In this paper, we present a theoretical analysis of the ℓ p / ℓ 1 {\ell_{p}/\ell_{1}} minimization method with 0 < p < 1 {0<p<1} for sparse signal recovery. We provide a verifiable sufficient condition for the exact noiseless sparse recovery and establish reconstruction error bounds using q -ratio constrained minimal singular values (CMSV) and restricted isometry property (RIP) tools. Additionally, we adopt an efficient algorithm to solve the optimization problem and conduct numerical experiments to demonstrate its superior performance.

The inverse problem of the reconstruction of the Sturm–Liouville problem on a half-line from its Weyl function is considered. Given the Weyl function, we obtain the potential in the Schrödinger equation and the boundary condition at the origin. If the boundary condition is known, this problem is equivalent to the inverse scattering problem of the recovery of the potential, which is zero on a half-line, from a given reflection coefficient. We develop a simple and direct method for solving the inverse problem. The method consists of two steps. First, the Jost solution is computed from the Weyl function, by solving a homogeneous Riemann boundary value problem. Second, a system of linear algebraic equations is constructed for the coefficients of series representations of three solutions of the Schrödinger equation. The potential and the boundary condition are then recovered from the first component of the solution vector of the system. A numerical illustration of the functionality of the method is presented.

The paper considers an inverse problem and a direct problem for the Burgers equation in a domain with movable boundaries. With the help of an additional condition, a formula is obtained for determining the desired function from a direct problem for the loaded Burgers equation for the solvability of which we require a condition on the functions according to which the boundaries of the domain change. The solvability of direct problems is proved using a priori estimates and the methods of Faedo–Galerkin and functional analysis.