Volume 23, Issue 5

We consider the identification of nonlinear heat conduction laws in stationary and instationary heat transfer problems. Only a single additional measurement of the temperature on a curve on the boundary is required to determine the unknown parameter function on the range of observed temperatures. We first present a new proof of Cannon's uniqueness result for the stationary case, which allows us to derive a corresponding stability estimate, and then extend our argument to instationary problems which are close to steady state.

The coefficient inverse problem for the two-dimensional wave equation is solved. We apply the Gelfand–Levitan approach to transform the nonlinear inverse problem to a family of linear integral equations. We consider the Monte Carlo method for solving the Gelfand–Levitan equation. We obtain the estimation of the solution of the Gelfand–Levitan equation in one specific point, due to the properties of the method. That allows the Monte Carlo method to be more effective in terms of span cost, compared with regular methods of solving linear system. Results of numerical simulations are presented.

In this paper we study advantages and limitations of the Generalized Cross Validation (GCV) approach for selecting a regularization parameter in the case of a partially stochastic linear irregular operator equation. The research has been motivated by an inverse problem in epidemiology, where the goal was to reconstruct a time dependent treatment recovery rate for plasmodium falciparum , the most dangerous form of malaria. Initial numerical simulations gave rise to a theoretical analysis of the expected value of the GCV function and the efficiency of the GCV method for different noise levels. It was shown that, as opposed to L-curve, the GCV does not necessarily generate a systematic error in the value of the regularization parameter for Tikhonov's stabilizing algorithm.

In this article, we consider an inverse problem for the integral equation of the convolution type in a multidimensional case. This problem is severely ill-posed. To deal with this problem, using a priori information (sourcewise representation) based on optimal recovery theory we propose a new method. The regularization and optimization properties of this method are proved. An optimal minimal a priori error of the problem is found. Moreover, a so-called optimal regularized approximate solution and its corresponding error estimation are considered. Efficiency and applicability of this method are demonstrated in a numerical example of the image deblurring problem with noisy data.

In this paper, multilevel Jacobi and Gauss–Seidel type iteration methods with compression technique are developed for solving ill-posed integral equations by making use of the multiscale structure of the matrix representation of the integral operator. The methods are based on the combination of Tikhonov regularization and multiscale Galerkin methods, and lead to fast solutions of discrete regularization methods for the equations. Choice for an a posteriori regularization parameter is proposed. An optimal convergence order for the method with the choices of parameters is established. Numerical experiments are given to illustrate the efficiency of the method.

We investigate the feasibility of quantifying properties of a composite dielectric material through the reflectance, where the permittivity is described by the Lorentz model in which an unknown probability measure is placed on the model parameters. We summarize the computational and theoretical framework (the Prohorov metric framework) developed by our group in the past two decades for nonparametric estimation of probability measures using a least-squares method, and point out the limitation of the existing computational algorithms for this particular application. We then improve the algorithms, and demonstrate the feasibility of our proposed methods by numerical results obtained for both simulated data and experimental data for inorganic glass when considering the resonance wavenumber as a distributed parameter. Finally, in the case where the distributed parameter is taken as the relaxation time, we show using simulated data how the addition of derivative measurements improves the accuracy of the method.

In this work the process of gas-lift in the oil production is considered. The process is described by partial differential equations of hyperbolic type. A small parameter is introduced, which is the inverse of the well depth. Gas-lift process is investigated behind the front of sound wave. The initial system of hyperbolic equations is reduced to the nonlinear ordinary differential equation (NODE) of the first order relatively to the gas volume and gas liquid (GLM), which depends on the coordinates of wells and hydraulic resistance coefficient (HRC). An asymptotic solution of NODE is obtained and this solution is calculated at the point. It is shown that for the determination of HRC statistical data of well is required (volume of injected gas at the wellhead of the annular space and GLM at the end of lift (debit)). Then on the basis of these results, by constituting the corresponding functional, which is the quadratic deviation of the statistics and calculated asymptotic solutions, the functional gradient is derived that allows one to calculate HRC in first approximation relative to small parameter. An example for the specific case from the practice shows that HRC in first approximation differs from the value on the order of 10 -2 continuous case on the order of 10 -1 in the discrete case.

Four simple mathematical models of pharmacokinetic, competition between immune and tumor cells, infectious disease and tuberculosis epidemic are considered. An optimization approach for identification those models based on gradient type methods is introduced. Inverse problems are formulated in the form of an operator equation and then reduced to the minimization of the corresponding misfit functionals. The adjoint problems are used for the calculation of gradients. A mathematical model of competition between immune and tumor cells is considered numerically. The results of a numerical experiment are demonstrated.

Electrical impedance tomography (EIT) is a well-known technique to estimate the conductivity distribution γ of a body Ω with unknown electromagnetic properties. EIT is a severely ill-posed inverse problem. In this paper, we formulate the EIT problem in the Bayesian framework using mixed total variation (TV) and non-convex ℓ p regularization prior. We use the Markov Chain Monte Carlo (MCMC) method for sampling the posterior distribution to solve the ill-posed inverse problem in EIT. We present simulations to estimate the distribution for each pixel for the image reconstruction of the conductivity in EIT.

In this paper, a Volterra integral equation of the first kind in convolutions on the semiaxis when the integral operator kernel and the right-hand side of the equation have a bounded support is considered. An inverse problem of reconstructing the solution to the equation and the integral operator kernel from values of the right-hand side is formulated. Necessary and sufficient conditions for the inverse problem solvability are obtained. A uniqueness and stability theorem is proved. Explicit formulas for reconstruction of the solution and kernel are obtained.