Volume 33, Issue 6

The problem of obtaining a 𝐾-sparse binary signal 𝐱 from y = Φ ⁢ x + ν \mathbf{y}=\boldsymbol{\Phi}\mathbf{x}+\boldsymbol{\nu} is often encountered in communications and signal processing. Based on orthogonal least squares (OLS), a greedy algorithm guaranteeing the support recovery of a 𝐾-sparse binary signal is presented in this paper. Using the mutual coherence and restricted isometric properties of Φ \boldsymbol{\Phi} , sufficient conditions ensuring the exact recovery of the support of 𝐱 are given. Finally, compared with binary matching pursuit (BMP) and OLS, the simulation experiments imply the advantage of the proposed algorithm.

Well-posedness questions of reconstruction of the heat transfer coefficients from boundary measurements are considered. The additional measurements are the integrals of a solution with weights over the boundary of a domain. The inverse problem theory is involved. The heat transfer coefficient is sought in the form of a finite segment of the series with unknown Fourier coefficients depending on time. They satisfy a system of integral equations of Volterra type. It is demonstrated that, under certain conditions, a solution to the problem is uniquely determined locally in time and depends continuously on the data of the problem. These conditions include smoothness and compatibility conditions on the data and additional conditions on the kernels of the integral operators used in the additional measurements. The method is constructive and allows to provide a numerical algorithm for solving the problem. The proof rely on a priori estimates and the contraction mapping principle.

In this manuscript, we examine an inverse problem for abstract semilinear differential inclusions. We identify the unknown parameter using the given overdetermined condition. We establish the existence of mild solution to the considered inverse problem. Further, the continuous dependence of the solution to the initial data is discussed.The main tools to obtain the results are Banach fixed point theorem, Grönwall’s inequality, set-valued theory and semigroup theory. Moreover, we provide several examples and some graphical illustrations to validate our findings.

In a parameter estimation problem, the objective is to find the input parameters for some forward model such that the outputs of the model are close to some stated values. These could be measurements from a physical system, or targets in a design problem. A typical approach is to create an optimisation problem, where we search the space of possible parameters and seek to minimise a sum of errors squared. This will often require the forward model to be run many times, which can be computationally expensive. Gaussian process models can be used to model the forward function, which can reduce the computational effort. If a forward function exists, then it may be possible to create an inverse function. This inverse function can then be modelled by a Gaussian process model. We have formulated a method that allows an approximate inverse function to exist for many typical simulation models. We then iteratively create a sequence of Gaussian process models that capture this function. This allows us to solve the parameter estimation problem. This approach uses fewer forward model calculations than typical search algorithms.

In this work, we consider an inverse problem of identify the space-dependent diffusion coefficient in the distributed-order time-fractional diffusion equation from boundary measurement data. A numerical algorithm is developed using the finite difference method to solve the direct problem, numerical results demonstrate the effectiveness and a certain degree of accuracy of this algorithm. To identify the diffusion coefficient, the problem is reformulated into a variational problem using the Tikhonov regularization method. The conjugate gradient method is utilized to tackle the variational problem by leveraging insights from a sensitivity problem and an adjoint problem. Numerical inversions are performed for diffusion coefficients with various functional forms, including scenarios with additional data influenced by random noise. Four numerical instances are presented to exhibit the efficacy and robustness of the method proposed in this paper.

In this paper we consider an inversion problem for the Radon transform in three-dimensional space with a weight function of a special form. Under some natural conditions we prove invertibility for the considering problem and find an explicit formula for the inverse of the problem.

In this study, we introduce an iterated regularization method with a relaxation parameter to compute solutions to ill-posed linear operator equations of the first kind, specifically within the framework of bounded operators. The proposed method is accompanied by a comprehensive regularization theory that establishes its stability and convergence. In addition, we derive convergence results and implement effective stopping criteria based on Morozov’s discrepancy principle. Numerical experiments are performed to validate effectiveness of the iterated regularization method and demonstrate its applicability to deblurring problems.

In this paper we propose a fractional SVIRS epidemic model by directly choosing the survival probability distribution as a power-law decay function in the non-Markovian process, where the model incorporates with the Riemann–Liouville fractional derivative and the coefficients depending upon the order of the derivative. The asymptotic solution of the fractional epidemic model is obtained by using the Laplace Adomian decomposition method (L-ADM), and numerical simulations are performed to reveal the impact of the order of the derivative on the fractional dynamic system. An inverse problem of determining the order of the fractional derivative by one data of the recovered people is transformed to a nonlinear equation with the aid of the asymptotic solution, and a uniqueness of the inverse problem is proved by monotonicity of the nonlinear function on the order.

This paper establishes a stability estimate for the inverse problem of identifying the time-dependent source function in parabolic equations, where coefficients vary with space and time, by employing the Carleman estimate technique.

We study a multidimensional inverse scattering problem under the time-dependent repulsive Hamiltonian of quadratic type. The time-dependent coefficient on the repulsive term decays as the inverse square of time, which is the threshold between the standard free Schrödinger operator and the time-independent repulsive Hamiltonian of quadratic type. Applying the Enss–Weder time-dependent method, we can determine uniquely the short-range potential functions with Coulomb-like singularities from the velocity limit of the scattering operator.

In this paper, we consider the use of an artificial neural network approach to solve the multi-obstacle inverse scattering problems. We propose a two-step algorithm consisting of obstacle perception and shape recognition. In the obstacle perception step, we train a deep neural network to predict the number of obstacles in the system of interest. At the shape recognition step, we train multiple neural networks to invert the shape of each obstacle. A large number of numerical experiments are provided to verify the effectiveness of the proposed method. It is worth pointing out that our algorithm is not only applicable to the full-phase data problem, but also to the phaseless data problem, and even to the finite aperture problem.

The article presents a description of the methodology for three-dimensional modeling of the gaseous pollutants spread in the atmospheric air. The methodology is based on solving a complete hydrodynamic system of equations – the Reynolds-averaged Navier–Stokes equations together with the equations of concentration transfer based on the Fick diffusion law, in conjunction with the equations of the turbulence model. Such a technique will allow for accurate modeling of the transfer of gaseous pollutants, without taking into account empirical patterns. The advantages of such a technique include the ability to take into account the turbulence, terrain, wind rose, underlying surface and geometry of pollution sources, accounting for the compressibility of the medium and the ability to specify an accurate dependence of the stratification of atmospheric air. The article presents the results of numerical modeling of the test case of propane emission from a single source, as well as the problem of dispersion of mixed emissions from a single chimney and determination of the maximum ground concentration of harmful substances. The results of the modeling are evaluated, and compared with experimental data, using the method of calculating the maximum one-time concentrations from emissions of a single point source using the adopted empirical methodology. It is shown that the technique allows estimating all hydrodynamic parameters of polluted air distribution with the required accuracy. The empirical approach is used for the inverse source problem of determining the location of the emission source by known pollutant concentrations. We show a good localization of the source by global optimization method (differential evolution) and that this approximate solution of inverse problem can be considered as good initial guess for iterative methods to solve the inverse source pollutant problem based on the Navier–Stokes equations.