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We consider an inverse problem for the Helmholtz equation of reconstructing a solution from measurements taken on a segment inside a semi-infinite strip. Homogeneous Neumann conditions are prescribed on both side boundaries of the strip and an unknown Dirichlet condition on the remaining part of the boundary. Additional complexity is that the radiation condition at infinity is unknown. Our aim is to find the unknown function in the Dirichlet boundary condition and the radiation condition. Such problems appear in acoustics to determine acoustical sources and surface vibrations from acoustic field measurements. The problem is split into two sub-problems, a well-posed and an ill-posed problem. We analyse the theoretical properties of both problems; in particular, we show that the radiation condition is described by a stable non-linear problem. The second problem is ill-posed, and we use the Landweber iteration method together with the discrepancy principle to regularize it. Numerical tests show that the approach works well. [ABSTRACT FROM AUTHOR]

This article addresses the deconvolution problem of estimating a square-integrable probability density from observations contaminated with additive measurement errors having a known density. The estimator begins with a density estimate of the contaminated observations and minimizes a reconstruction error penalized by an integrated squared mth derivative. Theory for deconvolution has mainly focused on kernel- or wavelet-based techniques, but other methods including spline-based techniques and this smoothness-penalized estimator have been found to outperform kernel methods in simulation studies. This article fills in some of these gaps by establishing asymptotic guarantees for the smoothness-penalized approach. Consistency is established in mean integrated squared error, and rates of convergence are derived for Gaussian, Cauchy, and Laplace error densities, attaining some lower bounds already in the literature. The assumptions are weak for most results; the estimator can be used with a broader class of error densities than the deconvoluting kernel. Our application example estimates the density of the mean cytotoxicity of certain bacterial isolates under random sampling; this mean cytotoxicity can only be measured experimentally with additive error, leading to the deconvolution problem. We also describe a method for approximating the solution by a cubic spline, which reduces to a quadratic program. for this article are available online. [ABSTRACT FROM AUTHOR]

A quadratic minimization problem is considered in Hilbert spaces under constraints given by a linear operator equation and a convex quadratic inequality. The main feature of the problem statement is that the practically available approximations to the exact linear operators specifying the criterion and the constraints converge to them only strongly pointwise rather than in the uniform operator norm, which makes it impossible to justify the use of the classical regularization methods. We propose a regularization method that is applicable in the presence of error estimates for approximate operators in pairs of other operator norms, which are weaker than the original ones. For each of the operators, the pair of corresponding weakened operator norms is obtained by strengthening the norm in the domain of the operator and weakening the norm in its range. The weakening of operator norms usually makes it possible to estimate errors in operators where this was fundamentally impossible in the original norms, for example, in the finite-dimensional approximation of a noncompact operator. From the original optimization formulation, a transition is made to the problem of finding a saddle point of the Lagrange function. The proposed numerical method for finding a saddle point is an iterative regularized extragradient two-stage procedure. At the first stage of each iteration, an approximation to the optimal value of the criterion is refined; at the second stage, the approximate solution with respect to the main variable is refined. Compared to the methods previously developed by the authors and working under similar information conditions, this method is preferable for practical implementation, since it does not require the gradient step size to converge to zero. The main result of the work is the proof of the strong convergence of the approximations generated by the method to one of the exact solutions to the original problem in the norm of the original space. [ABSTRACT FROM AUTHOR]

Identification of physical properties of materials is very important because they are in general unknown. Furthermore, their direct experimental measurement could be costly and inaccurate. In such a situation, a cheap and efficient alternative is to mathematically formulate an inverse, but difficult, problem that can be solved, in general, numerically; the challenge being that the problem is, in general, nonlinear and ill-posed. In this paper, the reconstruction of a lower-order unknown time-dependent coefficient in a Cahn–Hilliard-type fourth-order equation from an additional integral observation, which has application to characterizing the nonlinear saturation of the collisional trapped-ion mode in a tokamak, is investigated. The local existence and uniqueness of the solution to such inverse problem is established by utilizing the Rothe method. Moreover, the continuous dependence of the unknown coefficient upon the measured data is derived. Next, the Tikhonov regularization method is applied to recover the unknown coefficient from noisy measurements. The stability estimate of the minimizer is derived by investigating an auxiliary linear fourth-order inverse source problem. Henceforth, the variational source condition can be verified. Then the convergence rate is obtained under such source condition. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the problem of reconstructing the past distribution for diffusion equations in multidimensional space with the presence of a poly-fractional operator which takes the following form$ {\mathbb{P}_\alpha }\left({{{\left({ - \Delta } \right)}^\gamma }} \right)u: = \sum\limits_{i = 1}^m {{\alpha _i}} {\left({ - \Delta } \right)^{{\gamma _i}}}u, $for $ m\in {\mathbb N} $, $ 0 < {\gamma _1} \le \cdots \le {\gamma _m} \le 1 $ and $ {\alpha _i} > 0 $ for $ i = \overline{1, m} $. The problem essentially arises from the superposition of multiple stochastic processes with different scales, including classical random walks and Lévy flights. We prove that the investigated problem is extremely ill-posed in the sense of Hadamard. Using an appropriate filtering method, we construct a regularized solution which converges to the original solution in $ L^2 $-sense. If we further impose a Gevrey-type a-priori condition, a Hölder convergence rate can be obtained. Three numerical examples using Matlab are also performed to demonstrate the ill-posedness of the problem and effectiveness of the regularization method. [ABSTRACT FROM AUTHOR]

In this paper, we first prove that the heat equation backward in time with Neumann and integral boundary conditions is an ill-posed problem. Then, we establish a stability estimate of Hölder type for this ill-posed problem.

Affected by the insufficient information of single baseline observation data, the three-stage method assumes the Ground-to-Volume Ratio (GVR) to be zero so as to invert the vegetation height. However, this assumption introduces much biases into the parameter estimates which greatly limits the accuracy of the vegetation height inversion. Multi-baseline observation can provide redundant information and is helpful for the inversion of GVR. Nevertheless, the similar model parameter values in a multi-baseline model often lead to ill-posed problems and reduce the inversion accuracy of conventional algorithm. To this end, we propose a new step-by-step inversion method applied to the multi-baseline observations. Firstly, an adjustment inversion model is constructed by using multi-baseline volume scattering dominant polarization data, and the regularized estimates of model parameters are obtained by regularization method. Then, the reliable estimates of GVR are determined by the MSE (mean square error) analysis of each regularized parameter estimation. Secondly, the estimated GVR is used to extracts the pure volume coherence, and then the vegetation height parameter is inverted from the pure volume coherence by least squares estimation. The experimental results show that the new method can improve the vegetation height inversion result effectively. The inversion accuracy is improved by 26% with respect to the three-stage method and the conventional solution of multi-baseline. All of these have demonstrated the feasibility and effectiveness of the new method.

We consider solving a severely ill-posed problem for determining the surface temperature and heat flux distribution of the nonhomogeneous sideways heat equation by a quasi-reversibility regularization method. An analytical solution is deduced based on the Fourier transform, and then, the logarithmic-type error estimate for the regularized solution is achieved. Finally, three numerical examples, including smooth and non-smooth functions, validate the feasibility and effectiveness of the proposed approach.

In a lot of engineering applications, one has to recover the temperature of a heat body having a surface that cannot be measured directly. This raises an inverse problem of determining the temperature on the surface of a body using interior measurements, which is called the sideways problem. Many papers investigated the problem (with or without fractional derivatives) by using the Cauchy data u(x0,t),ux(x0,t)$$ u\left({x}_0,t\right),{u}_x\left({x}_0,t\right) $$ measured at one interior point x=x0∈(0,L)$$ x&#x0003D;{x}_0\in \left(0,L\right) $$ or using an interior data u(x0,t)$$ u\left({x}_0,t\right) $$ and the assumption limx→∞u(x,t)=0$$ {\lim}_{x\to \infty }u\left(x,t\right)&#x0003D;0 $$. However, the flux ux(x0,t)$$ {u}_x\left({x}_0,t\right) $$ is not easy to measure, and the temperature assumption at infinity is inappropriate for a bounded body. Hence, in the present paper, we consider a fractional sideways problem, in which the interior measurements at two interior point, namely, x=1$$ x&#x0003D;1 $$ and x=2$$ x&#x0003D;2 $$, are given by continuous data with deterministic noises and by discrete data contaminated with random noises. We show that the problem is severely ill‐posed and further constructs an a posteriori optimal truncation regularization method for the deterministic data, and we also construct a nonparametric regularization for the discrete random data. Numerical examples show that the proposed method works well. [ABSTRACT FROM AUTHOR]

In this paper, an inverse problem of determining a source in a time-fractional diffusion equation is investigated. A Fourier extension scheme is used to approximate the solution to avoid the impact on smoothness caused by directly using singular system eigenfunctions for approximation. A modified implicit iteration method is proposed as a regularization technique to stabilize the solution process. The convergence rates are derived when a discrepancy principle serves as the principle for choosing the regularization parameters. Numerical tests are provided to further verify the efficacy of the proposed method. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Geodesy & Geodynamics (1671-5942) is the property of Editorial Board Journal of Geodesy & Geodynamics and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this article, we consider an inverse problem of determining an unknown heat source term from the final temperature in a radial domain. The iterative fractional Tikhonov–Landweber regularized prior and posterior method is conducted to solve the problem and obtain a regularized approximation with the optimal order error estimate. Numerical examples confirm the effectiveness of the iterative fractional Tikhonov–Landweber regularization approach. [ABSTRACT FROM AUTHOR]

This paper studies simultaneous inversion of initial value and source term of a time‐fractional Black‐Scholes equation. This problem is ill‐posed, and we use Tikhonov regularization method to solve it. Under the selection rules of a priori and a posteriori regularization parameters, a priori and a posteriori Hölder type error estimates are derived. Finally, numerical experiments demonstrate the stability and effectiveness of the proposed regularization method. [ABSTRACT FROM AUTHOR]

The determination of an unknown time-dependent source term is investigated in a Kuramoto–Sivashinsky equation from given additional integral-type measurement. Based on Schauder's fixed point theorem, the existence and uniqueness of such inverse problem are obtained under certain assumptions on the input data. In order to calculate the unknown source term, a time-discrete system is established, and its solution shall be applied to approximate the unknown quantity. The existence, uniqueness and some estimates to the time-discrete system are derived, and the convergence rates are deduced rigorously for both exact and noisy observation, respectively. Finally, the theoretical convergence rate results are verified, and accurate and stable solutions to the inverse problem are computed numerically by two numerical experiments. [ABSTRACT FROM AUTHOR]

Background: Calibrating material models to experimental measurements is crucial for realistic computational analysis of components. For complex material models, however, optimization-based identification procedures can become time-consuming, particularly if the optimization problem is ill-posed. Objective: The objective of this paper is to assess the feasibility of using machine learning to identify the parameters of a Chaboche-type material model that describes copper alloys. Specifically, we apply and analyze this identification approach using short-term uniaxial relaxation tests on a C19010 copper alloy. Methods: A genetic algorithm forms the basis for identifying the parameters of the Chaboche-type material model. The approach is accelerated by replacing the numerical simulation of the experimental setup by a neural network surrogate. The neural networks-based approach is compared against a classic approach using both, synthetic and experimental data. Results: The results show that on the one hand, a sufficiently accurate identification of the material model parameters can be achieved by a classic but time-consuming genetic algorithm. On the other hand, it is shown that machine learning enables a much more time-efficient identification procedure, however, suffering from the ill-posedness of the identification problem. Conclusion: Compared to classic parameter identification approaches, machine learning techniques can significantly accelerate the identification procedure for parameters of Chaboche-type material models with acceptable loss of accuracy. [ABSTRACT FROM AUTHOR]

To solve the Laplacian problems, we adopt a meshless method with the multiquadric radial basis function (MQRBF) as a basis whose center is distributed inside a circle with a fictitious radius. A maximal projection technique is developed to identify the optimal shape factor and fictitious radius by minimizing a merit function. A sample function is interpolated by theMQ-RBF to provide a trial coefficient vector to compute the merit function. We can quickly determine the optimal values of the parameters within a preferred rage using the golden section search algorithm. The novel method provides the optimal values of parameters and, hence, an optimal MQ-RBF; the performance of the method is validated in numerical examples. Moreover, nonharmonic problems are transformed to the Poisson equation endowed with a homogeneous boundary condition; this can overcome the problem of these problems being ill-posed. The optimal MQ-RBF is extremely accurate. We further propose a novel optimal polynomial method to solve the nonharmonic problems, which achieves high precision up to an order of 10-11. [ABSTRACT FROM AUTHOR]

This paper is concerned with the backward problem of recovering initial value for a homogeneous time-space fractional pseudo parabolic differential equation. Since the problem is ill-posed, a version of modified quasi boundary value method is used as the method of regularization for obtaining stable approximations. Error analysis and parameter choice strategies are done for both a priori and a posteriori cases. It is shown that, the order of the convergence rate can exceed 2 3. Hence, the obtained rate improves upon the previously known rates of order 2 3 for the a priori case, and 1 2 for the a posteriori case for the considered problem. [ABSTRACT FROM AUTHOR]

In the current paper, an inverse boundary value problem so-called the sideways problem for the multi-term time-fractional diffusion equation is investigated. The problem of interest includes the recovering of the diffusion distribution from the boundary data. We prove that the problem is ill-posed as the solution does not continuously depend on the boundary data. We further propose a fractional filter method to regularize the problem. The stability and convergence of the proposed method are gingerly analyzed. Two numerical examples, with the support from the fast Fourier transform (FFT), are implemented to illustrate the theoretical results. The numerical results are consistent with the theoretical analysis. [ABSTRACT FROM AUTHOR]

Standard beams are mainly used for the calibration of strain sensors using their load reconstruction models. However, as an ill-posed inverse problem, the solution to these models often fails to converge, especially when dealing with dynamic loads of different frequencies. To overcome this problem, a piecewise Tikhonov regularization method (PTR) is proposed to reconstruct dynamic loads. The transfer function matrix is built both using the denoised excitations and the corresponding responses. After singular value decomposition (SVD), the singular values are divided into submatrices of different sizes by utilizing a piecewise function. The regularization parameters are solved by optimizing the piecewise submatrices. The experimental result shows that the MREs of the PTR method are 6.20% at 70 Hz and 5.86% at 80 Hz. The traditional Tikhonov regularization method based on GCV exhibits MREs of 28.44% and 29.61% at frequencies of 70 Hz and 80 Hz, respectively, whereas the L-curve-based approach demonstrates MREs of 29.98% and 18.42% at the same frequencies. Furthermore, the PREs of the PTR method are 3.54% at 70 Hz and 3.73% at 80 Hz. The traditional Tikhonov regularization method based on GCV exhibits PREs of 27.01% and 26.88% at frequencies of 70 Hz and 80 Hz, respectively, whereas the L-curve-based approach demonstrates PREs of 29.50% and 15.56% at the same frequencies. All in all, the method proposed in this paper can be extensively applied to load reconstruction across different frequencies. [ABSTRACT FROM AUTHOR]

The influence of different error functions (standard deviation, normalized standard deviation, and chi-square) as objective functions in the calculation of the relaxation time spectrum is studied, as well as the influence of different minimization procedures available in MATLAB (fmincon, fminimax, and patternsearch). Two simulated spectra were extracted from the literature to validate the proposed methodology, each with its corresponding dynamic modulus. These dynamic moduli were then used to calculate the spectrum, which was subsequently compared with the initial simulated spectra. This calculus implies a well-known ill-posed problem solved by simultaneously minimizing the regularized error function and a constraint function. The proposed regularization is based on Tikhonov regularization, with a second spectrum differential as the operator and an optimal regularization parameter generated using a modified L Curve. A new way of comparing the different procedures used is proposed, which involves the calculation of the mean between the resulting spectra and its subsequent use as an initial point in the different minimizations studied until convergence is achieved. Finally, the results show that the best objective function studied is the normalized standard deviation and that the best minimization procedure is fmincon.

Image deblurring (ID) plays a vital role in various applications, including photography, medical imaging, and surveillance. Traditional ID methods often face challenges in preserving fine details and handling complex blurring scenarios due to expensive calculations. In this paper, a novel approach utilizing conformable fractional derivative (CFD) to address these challenges and improve the effectiveness of ID is presented. CFD offer a flexible framework for capturing and exploiting the non-local and non-linear properties inherent in images. Additionally, we propose a new circulant preconditioned matrix that ensures a fast convergence rate. The proven analytical property of the new preconditioner ensures fast convergence rates. The efficiency and efficacy of our algorithm is demonstrated by numerical experiments.

An ill-posed problem in the form of a linear operator equation is considered. It is assumed that the solution to the equation in the one-dimensional case can be represented in the form of a sum of three components: the first component contains discontinuities, the second contains discontinuities in the derivative, and the third is continuous. To construct a stable approximate solution, the three-component Tikhonov method is used. In this case, the stabilizer is the sum of three functionals: BVp-norm of the first component, BVp-norm of the derivative for the second component and the norm of the Sobolev space for the third component, and each functional depends on only one component. The convergence of the sum of regularized components to the solution of the original equation is proved. In addition, piecewise uniform convergence of approximate solutions is established. The results of numerical experiments on reconstructing a three-component model solution for the Fredholm equation of the first kind are presented. [ABSTRACT FROM AUTHOR]

In this paper, we discuss properties of the singular value decomposition (SVD-decomposition) within the framework of the analysis of the numerical rank in ill-posed problems for determining frequency properties of discrete sequences consisting of trigonometric binomials. The properties of the numerical rank of the SVD-decomposition are indicated. We propose an algorithm for determining the frequencies of trigonometric binomials involved in the original function that forms a discrete sequence; this algorithm is based on estimates of the numerical rank. We obtain a stable criterion for estimating the numerical rank based on the arithmetic-mean estimators of the pseudo-null-space of the trajectory matrix. Also, we present the results of numerical experiments that demonstrate the consistency of the arithmetic-mean estimator of the pseudo-zero-space for the analysis of the spectrum of noisy sequences. [ABSTRACT FROM AUTHOR]

In this article, we investigate a spherically symmetric backward heat conduction problem, starting from the final temperature. This problem is severely ill posed: the solution (if it exists) does not depend continuously on the final data. A conditional stability result of its solution is given. Further, we propose a quasi-boundary value regularization method to solve this ill-posed problem. Two Hölder type error estimates between the approximate solution and its exact solution are obtained under an a priori and an a posteriori regularization parameter choice rule, respectively.

In this paper, we delve into the challenge of identifying an unknown source in a space-time fractional diffusion-wave equation. Through an analysis of the exact solution, it becomes evident that the problem is ill-posed. To address this, we employ both the Tikhonov regularization method and the Quasi-boundary regularization method, aiming to restore the stability of the solution. By adhering to both a priori and a posteriori regularization parameter choice rules, we derive error estimates that quantify the discrepancies between the regularization solutions and the exact solution. Finally, we present numerical examples to illustrate the effectiveness and stability of the proposed methods. [ABSTRACT FROM AUTHOR]

The paper presents an overview of methods for solving an ill-posed problem of constrained convex quadratic minimization based on the Fejér-type iterative methods, which widely use the ideas and approaches developed in the works of I. I. Eremin, the founder of the Ural research school of mathematical programming. Along with a problem statement of general form, we consider variants of the original problem with constraints in the form of systems of equalities and inequalities, which have numerous applications. In addition, particular formulations of the problem are investigated, including the problem of finding a metric projection and solving a linear program, which are of independent interest. A distinctive feature of these methods is that not only convergence but also stability with respect to errors in the input data are established for them; i.e., the methods generate regularizing algorithms in contrast to the direct methods, which do not have this property. [ABSTRACT FROM AUTHOR]

This paper aims to retrieve the initial value for a non-local fractional Sobolev–Galpern problem. The given data are subject to noise by the discrete random model. We show that the solution to the problem is ill-posed in the sense of Hadamard. In this paper, we applied the Fourier truncation method to construct the regularized solution. We estimate the convergence between the solution and the regularized solution. In addition, the numerical example is also proposed to assess the efficiency of the theory. [ABSTRACT FROM AUTHOR]

The paper proposes a stable method for constructing a normal to a surface given approximately. The normal is calculated as the gradient of the function in the surface equation. As is known, the problem of calculating the derivative is ill-posed. In the paper, an approach is adopted to solving this problem as to the problem of calculating the values of an unbounded operator. To construct its stable solution, the principle of minimum of the smoothing functional in Morozov’s formulation is used. The normal is obtained in the form of a Fourier series in the expansion in terms of eigenfunctions of the Laplace operator in a rectangle with boundary conditions of the second kind. The functional stabilizer uses the Laplacian, which makes it possible to obtain a normal in the form of a Fourier series that converges uniformly to the exact normal vector as the error in the surface definition tends to zero. The resulting approximate normal vector can be used to solve various problems of mathematical physics using surface integrals, normal derivatives, simple and double layer potentials.

In this paper, an inverse problem of determining a source in a time-fractional diffusion equation is investigated. A Fourier extension scheme is used to approximate the solution to avoid the impact on smoothness caused by directly using singular system eigenfunctions for approximation. A modified implicit iteration method is proposed as a regularization technique to stabilize the solution process. The convergence rates are derived when a discrepancy principle serves as the principle for choosing the regularization parameters. Numerical tests are provided to further verify the efficacy of the proposed method.

In this article, the inverse time problem is investigated. Regarding the ill-posed linear problem, utilize the quasi-reversibility method first. This problem has been regularized and after that provides an iterative regularizing strategy for noisy input data that are based on homotopy. For the regularizing solution, the error analysis is proved when we employ noisy measurement data as our initial guess. Finally, numerical implementations are presented.

We investigate the ill-posedness of the inverse biosensor problem when the biosensor signals are corrupted by noise. To solve the problem, we employ feed-forward and convolutional neural networks. Computational experiments were performed with different levels of additive and multiplicative noises for the batch and flow injection analysis modes of the biosensor. Obtained results show that the largest errors of recovered concentrations are located on the edges of the training domain. We have found that the inverse problem is less ill-posed in the flow injection analysis mode and concentrations can be reliably recovered for higher levels of noise compared to the batch mode. This finding is confirmed by the application of the DIRECT global optimization method to the considered inverse biosensor problem.

In this paper, an explicit boundary-type numerical procedure, including a constraint-type fictitious time integration method (FTIM) and a two-point boundary solution of the Lie-group shooting method (LGSM), is constructed to tackle nonlinear nonhomogeneous backward heat conduction problems (BHCPs). Conventional methods cannot effectively overcome numerical instability to solve inverse problems that lack initial conditions and take a long time to calculate, even using different variable transformations and regularization techniques. Therefore, an explicit-type numerical procedure is developed from the FTIM and the LGSM to avoid numerical instability and numerical iterations. First, a two-point boundary solution of the LGSM is introduced into the numerical algorithm. Then, the maximum and minimum values of the initial guess value can be determined linearly from the boundary conditions at the initial and final times. Finally, an explicit-type boundary-type numerical procedure, including a boundary value solution and constraint-type FTIM, can directly avoid the numerical iterative problems of BHCPs. Several nonlinear examples are tested. Based on the numerical results shown, this boundary-type numerical procedure using a two-point solution can directly obtain an approximated solution and can achieve stable convergence to boundary conditions, even if numerical iterations occur. Furthermore, the numerical efficiency and accuracy are better than in the previous literature, even with an increased computational time span without the regularization technique. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Harbin University of Science & Technology is the property of Journal of Harbin University of Science & Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This paper deals with convolution-type Urysohn equations of the first kind. Finding a solution for such equations is an ill-posed problem. For it to be solved, regularization algorithms and the continuous wavelet transform are used. Similar to the Fourier transform, the continuous wavelet transform is applied to convolution-type equations (based on the Fourier and wavelet transforms) and to Urysohn equations with unknown shift. The wavelet transform is preferable for the cases with approximated right-hand sides and for type 1 equations. We demonstrated that the application of the wavelet transform to Urysohn-type equations with unknown shift translates into a solution of a nonlinear equation with an oscillating kernel. Depending on the availability of a priori information, a combination of regularization and iterative algorithms with the use of close equations are effective for solving convolution-type equations based on the continuous wavelet transform and Urysohn equation. [ABSTRACT FROM AUTHOR]

The paper studies the problem of determining the error introduced by inaccuracy in determining the thickness of a protective heat-resistant coating of composite materials. The mathematical problem is the heat equation on an inhomogeneous half-line. The temperature on the outer side of the half-line () is considered unknown over an infinite time interval. To find it, the temperature is measured at the interface of the media at the point . An analytical study of the direct problem is carried out and enables a rigorous statement of the inverse problem and determining the functional spaces in which it is natural to solve the inverse problem. The main difficulty that the present paper aims at solving is obtaining an estimate for the error of the approximate solution. To estimate the conditional correctness modulus, the projection regularization method is used; this allows obtaining order-accurate estimates. [ABSTRACT FROM AUTHOR]

The article solves the problem of determining the temperature on the inner wall of a hollow cylinder. Using a time Fourier transform, the problem is reduced to an ordinary differential equation, with the help of which the Fourier transform of an exact solution of the inverse boundary value problem is found. A projection regularization method is considered, which makes it possible to obtain a stable solution to the problem and an accurate in order of magnitude estimate of the error of the approximate solution. Since high accuracy requirements are imposed on the solutions of such problems in numerical calculations, an algorithm is developed to improve the accuracy and reliability of processing the results of thermal test data. To check the performance of the algorithm, test calculations are carried out. [ABSTRACT FROM AUTHOR]

An ill-posed problem of localization (determining the position) of discontinuity lines of a function of two variables is considered: outside the discontinuity lines, the function is smooth, and, at each point on the line, it has a discontinuity of the first kind. Under the Lipschitz conditions on the discontinuity line, averaging procedures are constructed and global discrete regularizing algorithms of localization are studied. A parametric family of fractal lines is constructed for which all conditions can be checked analytically. A fractal having a large fractal dimension is indicated for which the efficiency of the constructed methods can be guaranteed. [ABSTRACT FROM AUTHOR]

The article describes a method for determining the heat flux density on the surface of a thin foil that is inaccessible for thermal imaging measurements based on thermal imaging data on the other side of the foil, available for measurements. Mathematically, the problem is reduced to solving the Cauchy problem for an elliptic equation, which is ill-posed. The problem is solved using noise smoothing in the initial temperature field and a correct restriction on the number of expansion terms in the Fourier series. According to the measurement results, the heat flux density reaches its maximum in the area of the contact line and amounts to . [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate an inverse problem of recovering a space-dependent source term governed by distributed order time-fractional diffusion equations in Hilbert scales. Such a problem is ill-posed and has important practical applications. For this problem, we propose a general regularization method based on the idea of the filter method. With a suitable source condition, we prove that the method is of optimal order under various choices of regularization parameter. One is based on the a priori regularization parameter choice rule and another one is the discrepancy principle. Finally, the capabilities of our method are illustrated by both the Tikhonov and the Landweber method. [ABSTRACT FROM AUTHOR]