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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]

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]

Copyright of Iraqi Journal of Science is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) 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.)

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]

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.)

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]

Copyright of Iraqi Journal of Science is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) 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.)

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,. .. ,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]

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]

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, 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 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]

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. [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]

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]

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]

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.)

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]

A dozen of inversion methods are applied and tested to estimate the permeability of the area where gas-fingering event has taken place in an Iranian carbonate reservoir located southwest of Iran. In a previous work, the gas-fingering event was detected by inverting the 3D seismic data and in this study the permeability model in that area is estimated. Because the lateral area of the gas-fingering event is narrow, the whole system conducting the injected gas can be considered as one rock unit system and therefore the assumption of horizontal linear steady-state flow can be applied. Inversion methods are exploited to determine the permeability in the interval of interest. The interval of interest is located at the crest and involves four wells among which one is the gas-injection well. To investigate the feasibility of such an approach and select the best possible inversion method, first a controlled experiment for the system is designed and studied. The porosity values of the system are known from seismic data inversion and the permeability values are the desired parameters. The permeability values at well locations are known via well-test data and are used as constraints in the inversion procedure. The interval of interest is discretized and a simulator is used to simulate the fluid flow in the controlled system in order to apply and validate the inversion methods. All calculations are performed in the MATLAB environment. According to the results from the controlled experiment, the Maximum Entropy and Total Variation methods were found to be the best two inversion methods which were successful in retrieving the true permeability model. Similar comparative study using different inversion methods is performed for the real case for which the results retrieved by the Total Variation method is most reliable as it suggests the best recovery of the permeability value for the check-well. An estimation of the fracture permeabilities for the area under study also indicated that the inverted permeability values are most representing the fracture permeabilities rather than the matrix. The results of this study will be used to tune the field simulation model in terms of rock and fluid properties, consider the inverted permeability model as further constraints for the reservoir history-matching of the oil field, reconsider the factors involving the gas injection plan for the oil field, and obtain insights for further field development plans in other nearby oil fields. [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 ( δ 18 O) from the Malan ice core. [ABSTRACT FROM AUTHOR]

In the present paper, we study one-dimensional inverse source problem of identifying a time-dependent source function from the over-specification condition. The unknown source term is recovered by solving an operator integral equation of the first kind. Due to the ill-posedness of this operator integral equation, the Tikhonov regularization approach of the 1st order is applied in order to acquire a stable approximation. The stable solution is defined by minimization of the Tikhonov functional. The value of the regularization parameter in this method is obtained as a function of error in input data. The finite element method is applied for numerical simulation based on obtained results. The illustrative example is presented to show the accuracy and applicability of the proposed method. [ABSTRACT FROM AUTHOR]

Tikhonov regularization is an usual method to solve an ill-posed problem, recommended when the input data are contaminated with noise. However, in some cases, the use of this technique is not sufficient to provide good solutions. In this work, an improvement of the Tikhonov regularization method was proposed and tested in the inverse black body radiation problem. The method proposed consisted in including the norm of the fractional-order derivative of the solution in the original functional proposed by Tikhonov. In the present framework, the regularized solution depends on the regularization parameter, λ, and on the fractional derivative order, α. For α assuming real value between 0 and 2, the solution obtained is more precise than those from the usual Tikhonov regularization method. [ABSTRACT FROM AUTHOR]

We introduce the augmented Tikhonov regularization method motivated by Bayesian principle to improve the load identification accuracy in seriously ill-posed problems. Firstly, the Green kernel function of a structural dynamic response is established; then, the unknown external loads are identified. In order to reduce the identification error, the augmented Tikhonov regularization method is combined with the Green kernel function. It should be also noted that we propose a novel algorithm to determine the initial values of the regularization parameters. The initial value is selected by finding a local minimum value of the slope of the residual norm. To verify the effectiveness and the accuracy of the proposed method, three experiments are performed, and then the proposed algorithm is used to reproduce the experimental results numerically. Numerical comparisons with the standard Tikhonov regularization method show the advantages of the proposed method. Furthermore, the presented results show clear advantages when dealing with ill-posedness of the problem. [ABSTRACT FROM AUTHOR]

This paper develops a new method to deal with the problem of identifying the unknown source in the Poisson equation. We obtain the regularization solution by the Tikhonov regularization method with a super-order penalty term. The order optimal error bounds can be obtained for various smooth conditions when we choose the regularization parameter by a discrepancy principle and the solution process of the new method is uniform. Numerical examples show that the proposed method is effective and stable. [ABSTRACT FROM AUTHOR]

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 . 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. [ABSTRACT FROM AUTHOR]

In this paper, we study the inverse nodal problem and the eigenvalue gap for the one-dimensional sloshing problem with the p-Laplacian operator. By applying the Prüfer substitution, we first derive the reconstruction formula of the depth function by using the information of the nodal data. Furthermore, we employ the Tikhonov regularization method to consider how to reconstruct the depth function using only zeros of one eigenfunction. Finally, we investigate the eigenvalue gap under the restriction of symmetric single-well depth functions. We show the gap attains its minimum when the depth function is constant. [ABSTRACT FROM AUTHOR]

Copyright of China Mechanical Engineering is the property of Editorial Board of China Mechanical Engineering 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.)

We consider the questions of correction of improper convex programs, first of all, problems with inconsistent systems of constraints. Such problems often arise in the practice of mathematical simulation of specific applied settings in operations research. Since improper problems are rather frequent, it is important to develop methods of their correction, i.e., methods of construction of solvable models that are close to the original problems in a certain sense. Solutions of these models are taken as generalized (approximation) solutions of the original problems. We construct the correcting problems using a variation of the right-hand sides of the constraints with respect to the minimum of a certain penalty function, which, in particular, can be taken as some norm of the vector of constraints. As a result, we obtain optimal correction methods that are modifications of the (Tikhonov) regularized method of penalty functions. Special attention is paid to the application of the exact penalty method. Convergence conditions are formulated for the proposed methods and convergence rates are established. [ABSTRACT FROM AUTHOR]

As this paper is concerned with a class of multistep numerical difference techniques to solve one-dimensional parabolic inverse problems with source control parameter , we apply the linear multistep method combining with Lagrange interpolation to develop three different numerical difference schemes. The problem of numerical differentiation with noisy scattered data is mildly ill-posed, the smoothing spline model based on Tikhonov regularization method is developed to compute numerical derivative contaminated by noise error. Simultaneously, the truncation error estimations and the convergence conclusions are proposed for the above difference methods respectively. The results of numerical tests with different noise levels are given to show that the presented algorithms are accurate and effective. [ABSTRACT FROM AUTHOR]

We consider the questions of correction of improper convex programs, first of all, problems with inconsistent systems of constraints. Such problems often arise in the practice of mathematical simulation of specific applied settings in operations research. Since improper problems are rather frequent, it is important to develop methods of their correction, i.e., methods of construction of solvable models that are close to the original problems in a certain sense. Solutions of these models are taken as generalized (approximation) solutions of the original problems. We construct the correcting problems using a variation of the right-hand sides of the constraints with respect to the minimum of a certain penalty function, which, in particular, can be taken as some norm of the vector of constraints. As a result, we obtain optimal correction methods that are modifications of the (Tikhonov) regularized method of penalty functions. Special attention is paid to the application of the exact penalty method. Convergence conditions are formulated for the proposed methods and convergence rates are established. [ABSTRACT FROM AUTHOR]

Ionospheric tomography based on the observed total electron content along different satellite-receiver rays is a typically ill-posed inverse problem. Incorporating the electron density profiles data from COSMIC radio occultation technique and ground ionosondes, the Tikhonov regularization method is adopted to reconstruct the 3-D ionospheric electron density, and a regularization parameter is used to balance the weights between the prior (or background) information and real measurements. To determine the optimal regularization parameter, the model function in the modified L-curve method is used. This new method combines the advantages of the model function approach and L-curve criterion, and it has not only high accuracy, but also rapid convergence. To validate the effectiveness of this reconstruction algorithm, both the ideal test and real measurements test are carried out, and the results show that this algorithm can significantly improve the background model outputs. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the problem of finding minimum-norm solutions of a split convex feasibility problem in real Hilbert spaces. We study and analyze the convergence of a new self-adaptive CQ algorithm. The main advantage of the algorithm is that there is no need to calculate the norm of the involved operator. [ABSTRACT FROM AUTHOR]

We consider convex programming problems with a possibly inconsistent constraint system. Such problems constitute an important class of improper models of convex optimization and often arise in the mathematical modeling of real-life operations research statements. Since improper problems arise rather frequently, the theory and methods of their numerical approximation (correction) should be developed, which would allow to design objective procedures that resolve inconsistent constraints, turn an improper model into a family of feasible problems, and choose an optimal correction among them. In the present paper, an approximating problem is constructed by the variation of the right-hand sides of the constraints with respect to the minimum of some vector norm. The type of the norm defines the form of a penalty function, and the minimization of the penalty function together with a stabilizing term is the core of each specific method of optimal correction of improper problems. The Euclidean norm implies the application of a quadratic penalty, whereas a piecewise linear (Chebyshev or octahedral) norm is concerned with the use of an exact penalty function. The proposed algorithms may also be interpreted as (Tikhonov) regularization methods for convex programming problems with inaccurate input information. Convergence conditions are formulated for the methods under consideration and convergence bounds are established. [ABSTRACT FROM AUTHOR]

The application of Tikhonov regularization method for dealing with the ill-conditioned problems in the regional gravity field modeling by Poisson wavelets is studied. In particular, the choices of the regularization matrices as well as the approaches for estimating the regularization parameters are investigated in details. The numerical results show that the regularized solutions derived from the first-order regularization are better than the ones obtained from zero-order regularization. For cross validation, the optimal regularization parameters are estimated from L-curve, variance component estimation (VCE) and minimum standard deviation (MSTD) approach, respectively, and the results show that the derived regularization parameters from different methods are consistent with each other. Together with the first-order Tikhonov regularization and VCE method, the optimal network of Poisson wavelets is derived, based on which the local gravimetric geoid is computed. The accuracy of the corresponding gravimetric geoid reaches 1.1 cm in Netherlands, which validates the reliability of using Tikhonov regularization method in tackling the ill-conditioned problem for regional gravity field modeling. [ABSTRACT FROM AUTHOR]

An iterative algorithm for narrowing the definition domain of the distribution function of the hyperfine interaction parameter is proposed. The use of this algorithm increases the resolving power of the Tikhonov regularization method for solving the inverse problem of Mössbauer spectroscopy. [ABSTRACT FROM AUTHOR]

In this paper, based on the cubic B-spline finite element (CBSFE) and the radial basis functions (RBFs) methods, the inverse problems of finding the nonlinear source term for system of reaction-diffusion equations are studied. The approach of the proposed methods are to approximate unknown coefficients by a polynomial function whose coefficients are determined from the solution of minimization problem based on the overspecified data. In fact, this work considers a comparative study between the cubic B-spline finite element method and radial basis functions method. The stability and convergence analysis for these problems are investigated and some examples are given to illustrate the results. [ABSTRACT FROM AUTHOR]

Fractional (nonlocal) diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogs and they are used to model anomalous diffusion, especially in physics. In this paper, we study a backward problem for an inhomogeneous time-fractional diffusion equation with variable coefficients in a general bounded domain. Such a backward problem is of practically great importance because we often do not know the initial density of substance, but we can observe the density at a positive moment. The backward problem is ill-posed and we propose a regularizing scheme by using Tikhonov regularization method. We also prove the convergence rate for the regularized solution by using an a priori regularization parameter choice rule. Numerical examples illustrate applicability and high accuracy of the proposed method. [ABSTRACT FROM AUTHOR]