Search

In this paper, we study the Kuramoto–Sivashinsky (KS) equation backward in time. First, we prove a stability estimate of Hölder type. Then the ill-posed problem is regularized by the Tikhonov regularization method, and an error estimate of Hölder type is obtained. Finally, we apply a physics-informed neural network method to solve the problem numerically. [ABSTRACT FROM AUTHOR]

The inverse Helmholtz problem is crucial in many fields like non-destructive testing and heat conduction analysis, emphasizing the need for efficient numerical solutions. This paper investigates the parameter identification problems of the Helmholtz equation, based on the stabilized Lagrange interpolation collocation method (SLICM) associated with least-squares solution. This method circumvents the limitations of traditional meshfree methods that cannot perform accurate integrations. It offers advantages of high accuracy, good stability, and high computational efficiency, rendering it suitable for solving inverse problems. Additionally, considering potential errors in measurement data, this study employs the least squares method to directly utilize all available information from the measurement data, minimizing errors and avoiding the iterative calculations based on measurement data in the Galerkin methods. To balance the numerical errors among measurement locations, boundaries, and within the domain, this paper studies the optimal weights for the overdetermined system based on the least squares functional obtained through SLICM, achieving a global error balance. Moreover, to further mitigate the noise in measurement data, this paper introduces the Tikhonov regularization technique and selects suitable regularization parameters to process noisy data through the L-curve. Numerical results in 1D, 2D and even 3D complicated domains indicate that SLICM can attain accurate and convergent results in parameter identification, even when the noise level is as high as 10%. [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]

In this paper, we consider the inverse problem for identifying the source term and the initial value of time-fractional diffusion equation with Caputo-like counterpart hyper-Bessel operator. Firstly, we prove that the problem is ill-posed and give the conditional stability result. Then, we choose the Tikhonov regularization method to solve this ill-posed problem, and give the error estimates under a priori and a posteriori regularization parameter selection rules. Finally, we give numerical examples to illustrate the effectiveness of this method. [ABSTRACT FROM AUTHOR]

The mesoscale interaction between sea surface temperature (SST) and wind is a crucial factor influencing oceanic and atmospheric conditions. To investigate the mesoscale coupling characteristics of the Yellow Sea and East China Sea, we applied a locally weighted regression filtering method to extract mesoscale signals from Quik-SCAT wind field data and AMSR-E SST data and found that the mesoscale coupling intensity is stronger in the Yellow Sea during the spring and winter seasons. We calculated the mesoscale coupling coefficient to be approximately 0.009 N·m−2/°C. Subsequently, the Tikhonov regularization method was used to establish a mesoscale empirical coupling model, and the feedback effect of mesoscale coupling on the ocean was studied. The results show that the mesoscale SST–wind field coupling can lead to the enhancement of upwelling in the offshore area of the East China Sea, a decrease in the upper ocean temperature, and an increase in the eddy kinetic energy in the Yellow Sea. Diagnostic analyses suggested that mesoscale coupling-induced variations in horizontal advection and surface heat flux contribute most to the variation in SST. Moreover, the increase in the wind energy input to the eddy is the main factor explaining the increase in the eddy kinetic energy. [ABSTRACT FROM AUTHOR]

This study aims to find the time-dependent potential terms in the two inverse problems of the third-order pseudo-parabolic with initial and various boundary conditions supplemented by the overdetermination data. The nonlinear inverse problems have significant applications in physics and engineering fields. We proved the existence and uniqueness of the solution of the two problems are being proved, but they still need to be proposed (since tiny perturbations in input data cause considerable errors in the output potential term). Consequently, the regularized methods should be employed. A finite difference schema is used for solving direct problems. In contrast, the inverse problems were reformulated as nonlinear least-square minimization and solved efficiently by optimizing MATLAB routine lsqnonlin. Tikhonov's regularization method was applied to get stable results. The numerical results were explained by presenting a test example for each problem. In addition, the stability was discussed by utilizing the Von Neumann stability analysis. The results showed that the time-dependent potential terms were reconstructed successfully and were stable and accurate.

In the present paper, we propose a new approach to solving a class of generalized split problems. This approach will open some new directions for research to solve the other split problems, for instance, the split common zero point problem and the split common fixed point problem. More precisely, we study the split common solution problem for monotone operator equations in real Hilbert spaces. To find a solution to this problem, we propose and establish the strong convergence of the two new iterative methods by using the Tikhonov regularization method. Meantime, we also study the stability of the iterative methods. Finally, two numerical examples are also given to illustrate the effectiveness of the proposed methods. [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 work, the pseudoparabolic problem of the fourth order is investigated to identify the time -dependent potential term under periodic conditions, namely, the integral condition and overdetermination condition. The existence and uniqueness of the solution to the inverse problem are provided. The proposed method involves discretizing the pseudoparabolic equation by using a finite difference scheme, and an iterative optimization algorithm to resolve the inverse problem which views as a nonlinear least-square minimization. The optimization algorithm aims to minimize the difference between the numerical computing solution and the measured data. Tikhonov’s regularization method is also applied to gain stable results. Two examples are introduced to explain the reliability of the proposed scheme. Finally, the results showed that the time dependent potential terms are successfully reconstructed, stable and accurate, even in inclusion of noise. [ABSTRACT FROM AUTHOR]

The modeling of many problems of practical interest leads to nonlinear ill-posed equations (for example, the parameter identification problem (see the Numerical section)). In this article, we introduce a new source condition (SC) and a new parameter choice strategy (PCS) for the Tikhonov regularization (TR) method for nonlinear ill-posed problems. The new PCS is introduced using a new SC to compute the regularization parameter (RP) before computing the regularized solution. The theoretical results are verified using a numerical example.

The modeling of many problems of practical interest leads to nonlinear ill-posed equations (for example, the parameter identification problem (see the Numerical section)). In this article, we introduce a new source condition (SC) and a new parameter choice strategy (PCS) for the Tikhonov regularization (TR) method for nonlinear ill-posed problems. The new PCS is introduced using a new SC to compute the regularization parameter (RP) before computing the regularized solution. The theoretical results are verified using a numerical example. [ABSTRACT FROM AUTHOR]

In this paper, we employ Tikhonov regularization method with the projection methods using Jacobi polynomial bases to the first kind of Fredholm integral equations to find the approximate solution. We discuss the convergence analysis and obtain the convergence rates in L w α , β 2 norm under a priori parameter choice strategy. We also consider the Engl-type discrepancy principle as a posteriori parameter strategy for finding the regularization parameter and also evaluate the convergence rate which is of optimal order. Finally, we provide the numerical experiments to justify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we apply the discrete Legendre Galerkin and multi-Galerkin methods to find the approximate solution of the Tikhonov regularized equation of the Fredholm integral equations of the first kind. We evaluate the error bounds for the approximate solutions with the exact solution in the infinity norm. We provide an a priori parameter choice strategy to find the convergence rates under the infinity norm. Since smoothness of the solution is not known in applied problems, we discuss an adaptive parameter choice rule to choose the regularization parameter, and then using this regularization parameter, we obtain the order of convergence in infinity norm. We give test examples to justify the theoretical estimates. [ABSTRACT FROM AUTHOR]

The mesoscale interaction between sea surface temperature (SST) and wind is a crucial factor influencing oceanic and atmospheric conditions. To investigate the mesoscale coupling characteristics of the Yellow Sea and East China Sea, we applied a locally weighted regression filtering method to extract mesoscale signals from Quik-SCAT wind field data and AMSR-E SST data and found that the mesoscale coupling intensity is stronger in the Yellow Sea during the spring and winter seasons. We calculated the mesoscale coupling coefficient to be approximately 0.009 N·m−2/°C. Subsequently, the Tikhonov regularization method was used to establish a mesoscale empirical coupling model, and the feedback effect of mesoscale coupling on the ocean was studied. The results show that the mesoscale SST–wind field coupling can lead to the enhancement of upwelling in the offshore area of the East China Sea, a decrease in the upper ocean temperature, and an increase in the eddy kinetic energy in the Yellow Sea. Diagnostic analyses suggested that mesoscale coupling-induced variations in horizontal advection and surface heat flux contribute most to the variation in SST. Moreover, the increase in the wind energy input to the eddy is the main factor explaining the increase in the eddy kinetic energy.

This paper solves the inverse source problem of heat conduction in which the source term only varies with time. The application of the discrete regularization method, a kind of effective radial symmetry and axisymmetric heat conduction problem source identification that does not involve the grid integral numerical method, is put forward. Taking the fundamental solution as the fundamental function, the classical Tikhonov regularization method combined with the L-curve criterion is used to select the appropriate regularization parameters, so the problem is transformed into a class of ill-conditioned linear algebraic equations to solve with an optimal solution. Several numerical examples of inverse source problems are given. Simultaneously, a few numerical examples of inverse source problems are given, and the effectiveness and superiority of the method is shown by the results. [ABSTRACT FROM AUTHOR]

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 Rayleigh–Stokes equation with a fractional derivative is widely used in many fields. In this paper, we consider the inverse initial value problem of the Rayleigh–Stokes equation. Since the problem is ill-posed, we adopt the Tikhonov regularization method to solve this problem. In addition, this paper not only analyzes the ill-posedness of the problem but also gives the conditional stability estimate. Finally, the convergence estimates are proved under two regularization parameter selection rules. [ABSTRACT FROM AUTHOR]

Copyright of Journal Of Sichuan University (Natural Sciences Division) / Sichuan Daxue Xuebao-Ziran Kexueban is the property of Editorial Department of Journal of Sichuan University Natural Science Edition 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 examines the finding of spacewise dependent heat source function in pseudoparabolic equation with initial and homogeneous Dirichlet boundary conditions, as well as the final time value / integral specification as additional conditions that ensure the uniqueness solvability of the inverse problem. However, the problem remains ill-posed because tiny perturbations in input data cause huge errors in outputs. Thus, we employ Tikhonov’s regularization method to restore this instability. In order to choose the best regularization parameter, we employ L-curve method. On the other hand, the direct (forward) problem is solved by a finite difference scheme while the inverse one is reformulated as an optimization problem. The later problem is accomplished by employing lsqnolin subroutine from MATLAB. Two test examples are presented to show the efficiency and accuracy of the employed method by including many noises level and various regularization parameters. [ABSTRACT FROM AUTHOR]

It is known that the standard Tikhonov regularization methods oversmoothen the solution x ^ of the ill-posed equation T (x) = y , so the computed approximate solution lacks many inherent details that are expected in the desired solution. To rectify this problem, Fractional Tikhonov Regularization (FTR) method have been introduced. Kanagaraj et al. (J Appl Math Comput 63(1):87–105, 2020), studied FTR method for solving ill-posed problems. Techniques are developed to study the Finite Dimensional FTR (FDFTR) method. We also study Raus and Gfrerer type discrepancy principle for FDFTR method and compare the numerical results with other discrepancy principles of the same type. [ABSTRACT FROM AUTHOR]

We propose a multilevel iteration method for the numerical solution of nonlinear ill-posed problems in the Hilbert space by using the Tikhonov regularization method. This leads to fast solutions of the discrete regularization methods for the nonlinear ill-posed equations. An adaptive choice of an a posteriori rule is suggested to choose the stopping index of iteration, and the rates of convergence are also derived. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, a generalized version of the auto-convolution Volterra integral equation of the first kind as an ill-posed problem is studied. We apply the piecewise polynomial collocation method to reduce the numerical solution of this equation to a system of algebraic equations. According to the proposed numerical method, for $n=0$ and $n=1,\ldots, N-1$, we obtain a nonlinear and linear system, respectively. We have to distinguish between two cases, nonlinear and linear systems of algebraic equations. A double iteration process based on the modified Tikhonov regularization method is considered to solve the nonlinear algebraic equations. In this process, the outer iteration controls the evolution path of the unknown vector $U_0^{\delta}$ in the selected direction $\tilde{u}_0$, which is determined from the inner iteration process. For the linear case, we apply the Lavrentiev $\tilde{m}$ times iterated regularization method to deal with the ill-posed linear system. The validity and efficiency of the proposed method are demonstrated by several numerical experiments.

The unknown boundary condition identification problems of the 2-D Laplace equation are considered in this paper. Based on the good characteristics of the B-spline wavelet and Tikhonov regularization method (TRM), a new regularization B-spline wavelet method (RBPWM) is proposed. The novel algorithm could be regarded as one kind of wavelet mesh-free, non-iterative numerical scheme that converts the boundary condition identification problem into a large-scale algebraic equation system that can be solved in a single step. However, the coefficient matrix of the algebraic equation system is ill-conditioned, which will lead to an unstable solution for the case of higher-level noise. The Tikhonov regularization method (TRM) is used to achieve a steady numerical solution to this problem. The current work of this paper has studied four examples with different simulated noise levels for different boundary conditions. The efficiency and accuracy of the presented algorithm are verified with the numerical simulation. [ABSTRACT FROM AUTHOR]

In this paper, a generalized version of the auto-convolution Volterra integral equation of the first kind as an ill-posed problem is studied. We apply the piecewise polynomial collocation method to reduce the numerical solution of this equation to a system of algebraic equations. According to the proposed numerical method, for n = 0 and n = 1,. .. ,N - 1, we obtain a nonlinear and linear system, respectively. We have to distinguish between two cases, nonlinear and linear systems of algebraic equations. A double iteration process based on the modified Tikhonov regularization method is considered to solve the nonlinear algebraic equations. In this process, the outer iteration controls the evolution path of the unknown vector Uδ 0 in the selected direction ˜u0, which is determined from the inner iteration process. For the linear case, we apply the Lavrentiev ˜m times iterated regularization method to deal with the ill-posed linear system. The validity and efficiency of the proposed method are demonstrated by several numerical experiments. [ABSTRACT FROM AUTHOR]

The paper considers mathematical methods of correction of thermographic images (thermograms) in the form of temperature distribution on the surface of the object under study, obtained using a thermal imager. The thermogram reproduces the image of the heat-generating structures located inside the object under study. This image is transmitted with distortions, since the sources are usually removed from its surface and the temperature distribution on the surface of the object transmits the image as blurred due to the processes of thermal conductivity and heat exchange. In this paper, the continuation of the temperature function as a harmonic function from the surface deep into the object under study in order to obtain a temperature distribution function near sources is considered as a correction principle. This distribution is considered as an adjusted thermogram. The continuation is carried out on the basis of solving the Cauchy problem for the Laplace equation - an ill-posed problem. The solution is constructed using the Tikhonov regularization method. The main part of the constructed approximate solution is presented as a Fourier series by the eigenfunctions of the Laplace operator. Discretization of the problem leads to discrete Fourier series. A modification of the Hamming method for summing Fourier series and calculating their coefficients is proposed.

The work is devoted to the construction of computational algorithms implementing the method of correction of thermographic images. The correction is carried out on the basis of solving some ill-posed mixed problem for the Laplace equation in a cylindrical region of rectangular cross-section. This problem corresponds to the problem of the analytical continuation of the stationary temperature distribution as a harmonic function from the surface of the object under study towards the heat sources. The cylindrical region is bounded by an arbitrary surface and plane. On an arbitrary surface, a temperature distribution is measured (and thus is known). It is called a thermogram and reproduces an image of the internal heat-generating structure. On this surface, which is the boundary of the object under study, convective heat exchange with the external environment of a given temperature takes place, which is described by Newton’s law. This is the third boundary condition, which together with the first boundary condition corresponds to the Cauchy conditions - the boundary values of the desired function and its normal derivative. The problem is ill-posed. In this paper, using the Tikhonov regularization method, an approximate solution of the problem was obtained, stable with respect to the error in the Cauchy data, and which can be used to build effective computational algorithms. The paper considers algorithms that can significantly reduce the amount of calculations.

This study led to the proposal of a new numerical scheme for nondestructive evaluation using electrostatics. A system of equations was constructed to describe the relationship between the charge density on the defect and the electrostatic field observed in space. The coefficient matrix of the simultaneous equations was derived using the Fredholm integral equation of the first kind. This ill-posed problem was solved using the Tikhonov regularization method. The proposed method uses a matrix that reflects the decay effect of the electric field for the regularization term. The accuracy of the proposed method was verified through numerical experiments in which the observed electric field was derived using forward analysis computation. A formula was defined to quantitatively evaluate the error. Our results confirmed that the proposed method provides solutions that are more accurate than the standard Tikhonov regularization. [ABSTRACT FROM AUTHOR]

We use piecewise polynomial basis functions to obtain the stable approximation solution of the Tikhonov regularized equation of the Fredholm integral equation of the first kind by utilizing multi-projection (multi-Galerkin and multi-collocation) methods. We evaluate the error bounds for the approximate solution with the exact solution in infinity norm. We provide an a priori parameter choice strategy under infinity norm. In addition to determining the regularization parameter, we discuss Arcangeli's discrepancy principle and calculate the convergence rates in infinity norm. We give test examples to validate the theoretical results. [ABSTRACT FROM AUTHOR]

Context. In this article, a generalized parametric identification procedure for linear nonstationary systems is proposed, which uses spline functions and orthogonal expansion in a series according to the Walsh function system, which makes it possible to find estimates of the desired parameters by minimizing the integral quadratic criterion of discrepancy based on solving a system of linear algebraic equations for a wide class of linear dynamical systems. The accuracy of parameter estimation is ensured by constructing a spline with a given accuracy and choosing the number of terms of the Walsh series expansion when solving systems of linear algebraic equations by the A. N. Tikhonov regularization method. To improve the accuracy of the assessment, an algorithm for adaptive partitioning of the observation interval is proposed. The partitioning criterion is the weighted square of the discrepancy between the state variables of the control object and the state variables of the model. The choice of the number of terms of the expansion into the Walsh series is carried out on the basis of adaptive approximation of non-stationary parameters in the observation interval, based on the specified accuracy of their estimates. The quality of the management of objects with variable parameters is largely determined by the accuracy of the evaluation of their parameters. Hence, obtaining reliable information about the actual nature of parameter changes is undoubtedly an urgent task. Objective. Improving the accuracy of parameter estimation of a wide class of linear dynamical systems through the joint use of spline functions and Walsh functions. Method. A generalized parametric identification procedure for a wide class of linear dynamical systems is proposed. The choice of the number of terms of the expansion into the Walsh series is made on the basis of the proposed algorithm for adaptive partitioning of the observation interval. Results. The results of modeling of specific linear non-stationary systems confirm the effectiveness of using the proposed approaches to estimating non-stationary parameters. Conclusions. The joint use of spline functions and Walsh functions makes it possible, based on the proposed generalized parametric identification procedure, to obtain analytically estimated parameters, which is very convenient for subsequent use in the synthesis of optimal controls of real technical objects. This procedure is applicable to a wide class of linear dynamical systems with concentrated and distributed parameters. [ABSTRACT FROM AUTHOR]

The paper aims to introduce inverse tensor variational inequalities and analyze their application to an economic control equilibrium model. More precisely, some existence and uniqueness results are established and the well-posedness analysis is investigated. Moreover, the Tikhonov regularization method is extended to tensor inverse problems to study them when they are ill-posed. Lastly, the policymaker's point of view for the oligopolistic market equilibrium problem is introduced. The equivalence between the equilibrium conditions and a suitable inverse tensor variational inequality is established. [ABSTRACT FROM AUTHOR]

The paper considers the method of correction of thermographic images (thermograms) obtained by recording in the infrared range of radiation from the surface of the object under study using a thermal imager. A thermogram with a certain degree of reliability transmits an image of the heat-generating structure inside the body. In this paper, the mathematical correction of images on a thermogram is performed based on an analytical continuation of the stationary temperature distribution as a harmonic function from the surface of the object under study towards the heat sources. The continuation is carried out by solving an ill-posed mixed problem for the Laplace equation in a cylindrical region of rectangular cross-section. To construct a stable solution to the problem, the principle of the minimum of the Tikhonov smoothing functional we used.

In this paper, we discuss the Legendre spectral projection method for solving Fredholm integral equations of the first kind using Tikhonov regularization. First, we discuss the convergence analysis under an a priori parameter strategy for the Tikhonov regularization using Legendre polynomial basis functions, and we obtain the optimal convergence rates in the uniform norm. Next, we discuss Arcangeli's discrepancy principle to find a suitable regularization parameter and obtain the optimal order of convergence in uniform norm. We present numerical examples to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an inverse problem of recovering the unknown boundary conditions in the time fractional coupled Burgers' equations under appropriate initial and boundary conditions and overdetermination conditions. For simplicity, the main problem has been studied in the one-dimensional case, however, the proposed method can be applied for high dimensional setting. We have applied the shifted Chebyshev wavelets for discretizing the space derivatives and the finite difference approximations for the discretization of time fractional derivatives, in this process, we use the Taylor expansion to linearize the nonlinear terms in the equations. The proposed algorithm along with the collocation approach reduces the main problem to the solution of ill-conditioned linear algebraic equations. To alleviate the difficulties arising from solving the resultant ill-conditioned linear system, a type of regularization technique is utilized to obtain a stable solution. The convergence analysis and the accuracy of the method have been discussed. Two examples will be taken into account to demonstrate the efficiency and applicability of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, a numerical method based on the finite element method and the least square scheme with the Tikhonov regularization method for nonlinear inverse diffusion problem is presented. For this propose, first finite element method and basis functions will be used to discretize the variational form of the problem; then the least square scheme and Tikhonov regularization method are proposed to correct diffusion. It is assumed that no prior information is available on the functional form of the unknown diffusion coefficient in the present study, and so, it is classified as the function estimation in inverse calculation. Numerical result shows that a good estimation on the unknown functions of the inverse problem can be obtained../files/site1/files/72/9Abstract.pdf

In the development of promising ULIS scaling technologies, one of the key roles is played by porous dielectrics with a low permittivity used to isolate interconnects in a metallization system. Condensation of gaseous products in the pores of such films makes it possible to solve the most important problem that prevents the integration of such films, to carry out low-damage plasma etching. However, methods for studying porosity are also based on the study of the adsorption isotherm during condensation in film pores. Therefore, the study of adsorption in pores is one of the most important practical problems arising in the creation of dielectrics with a low permittivity and the study of low-damaging methods for their structuring. The method of ellipsometric porosimetry is an easy-to-implement and accurate approach for obtaining an adsorption isotherm; however, its further analysis and determination of the pore size distribution are reduced to solving an integral equation and is an ill-posed problem. In this paper, we propose to apply Tikhonov's regularization method to solve it. The method is verified on model data and used to study a low-k dielectric sample with an initial thickness of 202 nm and a permittivity of 2.3 based on organosilicate glass. [ABSTRACT FROM AUTHOR]

In this paper, two numerical methods are presented to solve a nonlinear inverse parabolic problem of determining the unknown reaction term in the scalar reactiondiffusion equation. In the first method, the finite element method will be used to discretize the variational form of the problem and in the second method, we use the radial basis functions (RBFs) method for spatial discretization and finite-difference for time discretization. Usually, the matrices obtained from the discretization of the equations are ill-conditioned, especially in higher-dimensional problems. To overcome such difficulties, we use Tikhonov regularization method. In fact, this work considers a comparative study between the finite element method and radial basis functions method. As we will see, these methods are very useful and convenient tools for approximation problems and they are stable with respect to small perturbation in the input data. The effectiveness of the proposed methods are illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, a numerical method is proposed for the numerical solution of a linear wave equation with initial and boundary conditions by using the cubic B-spline method to determine the unknown boundary condition. We apply the cubic B-spline for the spatial variable and the derivatives, which generate an ill-posed linear system of equations. In this regard, to overcome, this drawback, we employ the Tikhonov regularization (TR) method for solving the resulting linear system. It is proved that the proposed method has the order of convergence O ( (∆t) ² + h ² ) . Also, the conditional stability by using the Von-Neumann method is established under suitable assumptions. Finally, some numerical experiments are reported to show the efficiency and capability of the proposed method for solving inverse problems. [ABSTRACT FROM AUTHOR]

The Tibetan Plateau is a high-elevation area in Asia and contains the largest volumes of glaciers outside the polar regions. Reconstruction of the glacier surface temperature history in this area is crucial for better understanding of the process of climate change in the Tibetan Plateau. The Tikhonov regularization method has been used on borehole temperatures measured at Malan Glacier, located on the north Tibetan Plateau, to reconstruct the surface temperature history in the twentieth century. We found that the glacier surface temperature, which rose significantly after the 1960s, increased about 1.1°C over the last century. The warmest period occurred in the 1980s to the 1990s and the highest temperature variation could be 1.5°C to 1.6°C. The results were also compared with those of nearby instrumental temperatures by Wudaoliang meteorological station and the stable oxygen isotope ($${\delta ^{18}}O$$) from the Malan ice core.

Abstract We study the inhomogeneous nonlinear time-fractional Schrödinger equation for linear potential, where the order of fractional time derivative parameter α varies between 0 < α < 1 $0 < \alpha < 1$ . First, we begin from the original Schrödinger equation, and then by the Caputo fractional derivative method in natural units, we introduce the fractional time-derivative Schrödinger equation. Moreover, by applying a finite-difference formula to time discretization and cubic B-splines for the spatial variable, we approximate the inhomogeneous nonlinear time-fractional Schrödinger equation; the simplicity of implementation and less computational cost can be mentioned as the main advantages of this method. In addition, we prove the convergence of the method and compute the order of the mentioned equations by getting an upper bound and using some theorems. Finally, having solved some examples by using the cubic B-splines for the spatial variable, we show the plots of approximate and exact solutions with the noisy data in figures.

In the paper, a new mathematical model for calculating the spectral characteristics of biological particles imitating formed elements of blood, as well as their aggregates has been put forward. The model takes into account the aggregate structure and multiple light scattering effect on them. The methods and algorithms based the T-matrix technique for calculating the laser radiation scattering on a biological cluster were considered. A particle size distribution function was determined on a basis of simulated in vitro experiment on light scattering by particle aggregates. A discussion of the obtained results was presented.

This paper is devoted to solving the inverse problem of restoring the hydrometeor distribution function from radar measurements. We use the Tikhonov regularization algorithm based on the minimization of the residual functional by the conjugate gradient method. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study an inverse source problem for the Rayleigh–Stokes problem for a generalized second-grade fluid with a fractional derivative model. The problem is severely ill-posed in the sense of Hadamard. To regularize the unstable solution, we apply the Tikhonov method regularization solution and obtain an a priori error estimate between the exact solution and regularized solutions. We also propose methods for both a priori and a posteriori parameter choice rules. In addition, we verify the proposed regularized methods by numerical experiments to estimate the errors between the regularized and exact solutions.

An accuracy analysis of the numerical implementation of the frequency method for solving the integral equation in the problem of interpreting technical observations using the spline approximation of the scanned function is presented. The algorithm for solving the integral equation of the interpretation problem, which is based on the application of the Tikhonov regularization method with the search for a solution in the frequency domain with a truncation of the frequency spectrum is investigated. To increase the accuracy of the interpretation results, the use of spline approximation of the values of the scanned function, i.e., the right-hand side of the integral equation, is proposed. The accuracy of the solution of the integral equation is estimated using the regularization method and taking into account the error accompanied by the inaccuracy of the right-hand side, as well as the error in calculating the kernel values. A method for calculating the accuracy-optimal smoothing spline approximation of the scanned function is proposed. [ABSTRACT FROM AUTHOR]

Copyright of Cybernetics & Systems Analysis / Kibernetiki i Sistemnyj Analiz is the property of V.M. Glushkov Institute of Cybernetics of NAS of Ukraine 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.)

When studying the temperature dependences of the acoustic absorption and the modulus of elasticity, absorption peaks are often observed, which correspond to the characteristic step on the temperature dependence of the modulus of elasticity. Such features are called relaxation resonances. It is believed that the occurrence of such relaxation resonances is due to the presence in the structure of the material of elementary microscopic relaxors that interact with the studied vibrational mode of mechanical vibrations of the sample. In a sufficiently perfect material, such a process is characterized by a relaxation time τ, and in a real defective material by a relaxation time spectrum P(τ). Most often such relaxation processes have a thermally activated character and the relaxation time τ(T) is determined by the Arrhenius ratio τ(T)=τ0exp(U0/kT), and the characteristics of the process will be U0 - activation energy, τ0 - period of attempts, Δ0 - characteristic elementary contribution of a single relaxator to the dynamic response of the material and their spectra. In the low temperatures region the statistical distribution of parameters τ0 and Δ0 can be neglected with exponential accuracy, and the relaxation contribution to the temperature dependences of absorption and the dynamic elasticity modulus of the material will be determined only by the activation energy spectrum P(U) of microscopic relaxors. The main task of mechanical spectroscopy in the analysis of such relaxation resonances is to determine U0, τ0, Δ0 and P(U). It is shown, that the problem of recovering of spectral function P(U) of acoustic relaxation of a real crystal can be reduced to the solving of the Fredholm integral equation of the first kind with an approximately known right part and concerns to a class of ill-posed problems. The method based on Tikhonov regularizing algorithm for recovering P(U) from experimental temperature dependences of absorption or elasticity module is offered. It is established, that acoustic relaxation in high-purity iron single crystal in the temperature range 5-100 K is characterized by two-modes spectral function P(U) with maxima at 0.037 eV and 0.015 eV, which correspond to the a-peak and its a' satellite.