Search

In this paper, for solving a broad class of large-scale nonconvex and nonsmooth optimization problems, we propose a stochastic two-step inertial Bregman proximal alternating linearized minimization (STiBPALM) algorithm with variance-reduced stochastic gradient estimators. And we show that SAGA and SARAH are variance-reduced gradient estimators. Under expectation conditions with the Kurdyka–Łojasiewicz property and some suitable conditions on the parameters, we obtain that the sequence generated by the proposed algorithm converges to a critical point. And the general convergence rate is also provided. Numerical experiments on sparse nonnegative matrix factorization and blind image-deblurring are presented to demonstrate the performance of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we consider an inverse problem related to the fractional pseudo-parabolic equation with a nonlinear source term. Our investigation reveals the ill-posedness of the problem according to Hadamard’s definition. We present two improved variations of the optimal filtering method introduced by Seidman (SIAM J Numer Anal 33:162–170, 1996) to establish some optimal estimates under some an a priori assumptions on the regularity of the exact solution. Finally, the effectiveness of our algorithm is demonstrated through numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we formulate the modified iteratively regularized Landweber iteration method in Banach spaces to solve the inverse problems for which the forward operator may be smooth or non-smooth. We study the convergence analysis of the modified method for both the perturbed as well as unperturbed data by utilizing the Hölder stability estimates. In the presence of perturbed data, we terminate the method via a discrepancy principle and show that it is in fact a convergence regularization method that terminates within a few iterations. In the presence of unperturbed data, we show that the iterates converge to the exact solution. Additionally, we deduce the convergence rates in the presence of perturbed as well as unperturbed data. Finally, we discuss two inverse problems on which the method is applicable. [ABSTRACT FROM AUTHOR]

In 2021, Z. Fu, Y. Chen and B. Han introduced an inexact Newton regularization (REGINN-IT) using an idea involving the non-stationary iterated Tikhonov regularization scheme for solving nonlinear ill-posed operator equations. In this paper, we suggest a simplified version of the REGINN-IT scheme by using the Bregman distance, duality mapping and a suitable parameter choice strategy to produce an approximate solution. The method is comprised of inner and outer iteration steps. The outer iterates are stopped by a Morozov-type stopping rule, while the inner iterate is executed by making use of the non-stationary iterated Tikhonov scheme. We have studied convergence of the proposed method under some standard assumptions and utilizing tools from convex analysis. The novelty of the method is that it requires computation of the Fréchet derivative only at an initial guess of an exact solution and hence can be identified as more efficient compared to the method given by Z. Fu, Y. Chen and B. Han. Further, in the last section of the paper, we discuss test examples to inspect the proficiency of the method. [ABSTRACT FROM AUTHOR]

In this paper, we consider the simplified Levenberg–Marquardt method for nonlinear ill-posed inverse problems in Hilbert spaces for obtaining stable approximations of solutions to the ill-posed nonlinear equations of the form F ⁢ ( u ) = y {F(u)=y} , where F : 풟 ⁢ ( F ) ⊂ 햴 → 햸 {F:\mathcal{D}(F)\subset\mathsf{U}\to\mathsf{Y}} is a nonlinear operator between Hilbert spaces 햴 {\mathsf{U}} and 햸 {\mathsf{Y}} . The method is defined as follows: u n + 1 δ = u n δ - ( T 0 ∗ ⁢ T 0 + α n ⁢ I ) - 1 ⁢ T 0 ∗ ⁢ ( F ⁢ ( u n δ ) - y δ ) , u_{n+1}^{\delta}=u_{n}^{\delta}-(T_{0}^{\ast}T_{0}+\alpha_{n}I)^{-1}T_{0}^{% \ast}(F(u_{n}^{\delta})-y^{\delta}), where T 0 = F ′ ⁢ ( u 0 ) {T_{0}=F^{\prime}(u_{0})} and T 0 ∗ = F ′ ⁢ ( u 0 ) ∗ {T_{0}^{\ast}=F^{\prime}(u_{0})^{\ast}} . Here F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} denotes the Frèchet derivative of F at an initial guess u 0 ∈ 풟 ⁢ ( F ) {u_{0}\in\mathcal{D}(F)} for the exact solution u † {u^{\dagger}} , F ′ ⁢ ( u 0 ) ∗ {F^{\prime}(u_{0})^{\ast}} is the adjoint of F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} and { α n } {\{\alpha_{n}\}} is an a priori chosen sequence of non-negative real numbers satisfying suitable properties. We use Morozov-type stopping rule to terminate the iterations. Under suitable non-linearity conditions on operator F, we show convergence of the method and also obtain a convergence rate result under a Hölder-type source condition on the element u 0 - u † {u_{0}-u^{\dagger}} . Furthermore, we derive convergence of the method for the case when no source conditions are used and the study concludes with numerical examples which validate the theoretical conclusions. [ABSTRACT FROM AUTHOR]

Jin and Wang (Inverse Probl.34(3) (2018):035001) used a heuristic rule as a stopping criterion to terminate the iterations for the iteratively regularized Gauss–Newton method for getting stable approximate solution for nonlinear ill-posed operator equations. The advantage of using heuristic rule over the existing stopping rules is that, it does not require exact information about the noise level. In this paper, we do convergence analysis of the simplified generalized Gauss–Newton method under heuristic rule. We obtain rate of convergence results using the general form of source condition. We will also prove convergence of the method without using any source condition. We show the validity of the proposed algorithm using numerical examples. [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]

In this article, we first study the inverse source problem for parabolic with memory term. We show that our problem is ill-posed in the sense of Hadamard. Then, we construct the convergence result when the parameter tends to zero. We also investigate the regularized solution using the Fourier truncation method. The error estimate between the regularized solution and the exact solution is obtained.

In this work, we propose an iterative method for finding a common solution of a variational inclusion problem involving a maximally monotone operator and a fixed point problem for a pseudocontractive mapping in real Hilbert space. Under some standard and easy-to-verify conditions, we establish that the sequence generated by the proposed method converges strongly to a solution of the considered problem. Numerical illustrations and application in image recovery suggest that the proposed method is easy to implement and efficient. [ABSTRACT FROM AUTHOR]

A variation method for solving the inverse problems of determining of the complex quantum potential in a nonlinear non-stationary Schrödinger-type equation with final and boundary observations is considered. The existence and uniqueness theorem of the solution of the variation formulation of the inverse problem is proved, the continuity and continuous differentiability of the quality criterion are established, the formula for its gradient is found, the necessary condition of optimality is proved and iterative regularization of the solution is indicated. [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]

Lanweber-type methods are a well-known regularization methods to solve linear and nonlinear ill-posed problems. In this article, we consider a simplified form of Landweber method, say, an iteratively-regularized simplified Landweber iteration for solving nonlinear ill-posed problems. We study a detailed convergence analysis of the method under standard conditions on the nonlinearity and the rate of convergence under a Hölder-type source condition. Finally, numerical simulations are performed to validate the performance of the method. [ABSTRACT FROM AUTHOR]

In this article we propose and analyze a Levenberg–Marquardt–Kaczmarz-type (LMK) method for obtaining stable approximate solutions to systems of ill-posed equations modeled by non-linear operators acting between Hilbert spaces. We extend to the LMK iteration the strategy proposed in [A. Leitão, F. Margotti and B. F. Svaiter, Range-relaxed criteria for choosing the Lagrange multipliers in the Levenberg–Marquardt method, IMA J. Numer. Anal. 41 2021, 4, 2962–2989] for choosing the Lagrange multipliers in the Levenberg–Marquardt (LM) method. Our main goal is to devise a simple (and easy to implement) strategy for computing the multiplier in each iterative step, such that the resulting LMK iteration is both stable and numerically efficient. Convergence analysis for the proposed LMK type method is provided, including convergence for exact data, stability and semi-convergence. Numerical experiments using real data are presented for a 2D parameter identification problem, namely the Electrical Impedance Tomography (EIT) problem. The mathematical model known as complete electrode model (EIT-CEM) is considered. The obtained numerical results validate the efficiency of the proposed LMK-type method. [ABSTRACT FROM AUTHOR]

In this paper, an orthogonal basis for the space of linear splines based on linear B-splines is introduced. Some properties and modifications of this basis are investigated, then operational matrices of integration in collocation points are obtained using stable formula. Theoretical considerations are discussed. Applications to the numerical solutions for some linear and nonlinear inverse problems are given, in which the approximations are obtained using the first and second integrals of orthogonal linear splines that lead to an efficient solution procedure. For solving linear problems, the proposed method is combined with the Tikhonov regularization. Also, the trust-region-dogleg method is used for nonlinear equations. Numerical results validate the effectiveness of the orthogonal linear spline basis for linear and nonlinear inverse problems. [ABSTRACT FROM AUTHOR]

In this paper, we study secant-type iteration for nonlinear ill-posed equations involving 푚-accretive mappings in Banach spaces. We prove that the proposed iterative scheme has a convergence order at least 2.20557 using assumptions only on the first Fréchet derivative of the operator. Further, using a general Hölder-type source condition, we obtain an optimal error estimate. We also use the adaptive parameter choice strategy proposed by Pereverzev and Schock (2005) for choosing the regularization parameter. [ABSTRACT FROM AUTHOR]

In 2011, Jin Qinian has proposed a Levenberg–Marquardt method, by making use of duality mapping and the Bregman distance, to get an approximate solution of a nonlinear ill-posed operator equation in Banach space using an a posteriori parameter choice strategy and Morozov-type stopping rule. The method considered by Jin Qinian was an extension of the method proposed by M. Hanke in 1997 for the Hilbert space case. In this paper, we suggest a modified variant of the method, namely, the simplified Levenberg–Marquardt scheme in Banach spaces. The advantage of the method considered in the paper is that, it is also applicable for the operator equation with non-smooth operator. We establish convergence of the method under a modified a posteriori parameter choice strategy which is more feasible than the one considered by Jin Qinian (2011). Numerical example to demonstrate the validity of the considered method is discussed. [ABSTRACT FROM AUTHOR]

In 2010, Qinian Jin considered a regularized Levenberg–Marquardt method in Hilbert spaces for getting stable approximate solution for nonlinear ill-posed operator equation F ⁢ (x) = y , where F : D ⁢ (F) ⊂ X → Y is a nonlinear operator between Hilbert spaces X and Y and obtained rate of convergence results under an appropriate source condition. In this paper, we propose a simplified Levenberg–Marquardt method in Hilbert spaces for solving nonlinear ill-posed equations in which sequence of iteration { x n δ } is defined as x n + 1 δ = x n δ - ( α n ⁢ I + F ′ ⁢ (x 0) * ⁢ F ′ ⁢ (x 0) ) - 1 ⁢ F ′ ⁢ (x 0) * ⁢ (F ⁢ (x n δ) - y δ) . Here { α n } is a decreasing sequence of nonnegative numbers which converges to zero, F ′ ⁢ (x 0) denotes the Fréchet derivative of F at an initial guess x 0 ∈ D ⁢ (F) for the exact solution x † and (F ′ ⁢ (x 0)) * denote the adjoint of F ′ ⁢ (x 0) . In our proposed method, we need to calculate Fréchet derivative of F only at an initial guess x 0 . Hence, it is more economic to use in numerical computations than the Levenberg–Marquardt method used in the literature. We have proved convergence of the method under Morozov-type stopping rule using a general tangential cone condition. In the last section of the paper, numerical examples are presented to demonstrate advantages of the proposed method. [ABSTRACT FROM AUTHOR]

A novel optimization algorithm for stable parameter estimation and forecasting from limited incidence data for an emerging outbreak is proposed. The algorithm combines a compartmental model of disease progression with iteratively regularized predictor-corrector numerical scheme aimed at the reconstruction of case reporting ratio, transmission rate, and effective reproduction number. The algorithm is illustrated with real data on COVID-19 pandemic in the states of Georgia and New York, USA. The techniques of functional data analysis are applied for uncertainty quantification in extracted parameters and in future projections of new cases. [ABSTRACT FROM AUTHOR]

This paper is devoted to identifying an unknown source for a time-fractional diffusion equation in a general bounded domain. First, we prove the problem is non-well posed and the stability of the source function. Second, by using the Modified Fractional Landweber method, we present regularization solutions and show the convergence rate between regularization solutions and sought solution are given under a priori and a posteriori choice rules of the regularization parameter, respectively. Finally, we present an illustrative numerical example to test the results of our theory. [ABSTRACT FROM AUTHOR]

In this paper, we propose an algorithm combining Bregman alternating minimization algorithm with two-step inertial force for solving a minimization problem composed of two nonsmooth functions with a smooth one in the absence of convexity. For solving nonconvex and nonsmooth problems, we give an abstract convergence theorem for general descent methods satisfying a sufficient decrease assumption, and allowing a relative error tolerance. Our result holds under the assumption that the objective function satisfies the Kurdyka–Łojasiewicz inequality. The proposed algorithm is shown to satisfy the requirements of our abstract convergence theorem. The convergence is obtained provided an appropriate regularization of the objective function satisfies the Kurdyka–Łojasiewicz inequality. Finally, numerical results are reported to show the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

We consider a class of conditionally well-posed inverse problems characterized by a Hölder estimate of conditional stability on a convex compact in a Hilbert space. The input data and the operator of the forward problem are available with errors. We investigate the discretized residual functional constructed according to a general scheme of finite dimensional approximation. We prove that each its stationary point that is not too far from the finite dimensional approximation of the solution to the original inverse problem, generates an approximation from a small neighborhood of this solution. The diameter of the specified neighborhood is estimated in terms of characteristics of the approximation scheme. This partially removes iterating over local minimizers of the residual functional when implementing the discrete quasi-solution method for solving the inverse problem. The developed theory is illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the problem of recovering the historical distribution for diffusion equations in which the diffusion operators are described by the coupling of local and nonlocal type. The problem essentially arises in many real-world circumstances including the biological population dynamic where a population competes for the resources and diffuses by a combination of the Brownian and Lévy processes. We first design a typical example to illustrate the ill-posed nature of the problem. A fractional filter method is then proposed to achieve reliable approximations of the problem. The stability and convergence of the proposed method are gingerly analyzed. Four numerical examples, with the support from the finite difference method and the fast Fourier transform, are implemented to validate the theoretical results including the ill-posedness and the effect of regularization. The numerical results agree with the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we consider an inverse source problem for a fractional pseudo-parabolic equation. We show that the problem is severely ill-posed (in the sense of Hadamard) and the Tikhonov regularization method is proposed to solve the problem. In addition, we present numerical examples to illustrate applicability and accuracy of the proposed method to some extent. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in finding the function u(t,x),(t,x)∈[0,T)×Ω from the final data u(T,x)=ϕ(x), satisfies a nonhomogeneous fractional pseudo-parabolic equation. The problem is stable for the cases σν, the problem is ill-posed (in the sense of Hadamard). We propose the general filtering method to regularize this problem. The well-posedness of the regularized problem and some regularity estimates of the regularized solution are obtained. Moreover, error estimates are established under some a priori conditions of the sought solution. The numerical illustrations are given to show the convergence of our method.

Tomb murals are the special kind of murals that are buried underground. Due to the narrow exit of the tomb passage, the tomb murals were excavated by dividing the whole mural into blocks, which made lots of information missing between the blocks. The digital restoration technology Image inpainting uses the edge information around the missing parts to spread the information inside of the defect area and fills the information from the outside to the inside. But it is not suitable for filling the missing parts between the tomb mural blocks. Because these parts are large for exemplar-based inpainting which may make texture dislocation and for PDE which may make cartoon blur. It is a need to generate the information outwards to complete the information. The generative adversarial network uses deep learning training by the murals remains to generate the information from inside to outside, but the typical GAN doesn‘t have a good nonlinear feature. This paper provided a generating technology based on the deep convolution generative adversarial network to rebuild the missing information between the tomb mural blocks. It built the training data set of the simulation platform with Keras and designed a whole mural generation scheme based on DCGAN. In order to get better generated results to avoid the bad artifacts; it adds the nonlinear layers by choosing 13 layers convolution and 2 deconvolution layers of the generator and contained 5 layers convolution discriminator; it designed a new phased nonlinear loss function by using Pycharm pretreatment for Numpy array file data sets; finally, it completed the generate tomb mural information to obtain the good simulation effect.

We construct and study a class of numerically implementable iteratively regularized Gauss–Newton type methods for approximate solution of irregular nonlinear operator equations in Hilbert space. The methods include a general finite-dimensional approximation for equations under consideration and cover the projection, collocation and quadrature discretization schemes. Using an a posteriori stopping rule for the iterative processes and the standard source condition on the solution, we establish accuracy estimates for the approximations generated by the methods. We also investigate projected versions of the processes which take into account a priori information about a convex compact containing the solution. An iteratively regularized quadrature process is applied to an inverse 2D problem of gravimetry. [ABSTRACT FROM AUTHOR]

In this paper we study properties of the Laplace approximation of the posterior distribution arising in nonlinear Bayesian inverse problems. Our work is motivated by Schillings et al. (Numer Math 145:915–971, 2020. https://doi.org/10.1007/s00211-020-01131-1), where it is shown that in such a setting the Laplace approximation error in Hellinger distance converges to zero in the order of the noise level. Here, we prove novel error estimates for a given noise level that also quantify the effect due to the nonlinearity of the forward mapping and the dimension of the problem. In particular, we are interested in settings in which a linear forward mapping is perturbed by a small nonlinear mapping. Our results indicate that in this case, the Laplace approximation error is of the size of the perturbation. The paper provides insight into Bayesian inference in nonlinear inverse problems, where linearization of the forward mapping has suitable approximation properties. [ABSTRACT FROM AUTHOR]

A modified minimal error method for nonlinear integral equations is established. Such method, combined with the discrepancy principle as stopping rule, is a regularization method, which yields convergence to an exact solution when the nonlinear operator F satisfies certain conditions. Furthermore, the convergence of the modified minimal error method via multiscale Galerkin methods is also proved. Finally, numerical results show the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

We investigate a class of finite dimensional iteratively regularized Gauss–Newton methods for solving nonlinear irregular operator equations in a Hilbert space. The developed technique allows to investigate in a uniform style various discretization methods such as projection, quadrature and collocation schemes and to take into account available restrictions on the solution. We propose an a posteriori stopping rule for the iterative process and establish an accuracy estimate for obtained approximation. The regularized Gauss–Newton method combined with the quadrature discretization and the a posteriori iteration stopping is applied to a model ionospheric radiotomography problem. The problem is reduced to a nonlinear integral equation describing the phase shift of a sounding radio signal in dependence of the free electron concentration in the ionospheric plasma. We establish the unique solvability of the inverse problem in the class of analytic functions. [ABSTRACT FROM AUTHOR]

This article is devoted to the study of final value problems for biparabolic equation with discrete data in two cases as the linear source and the nonlinear source, respectively. In each of the cases, we show the instability of the solutions and then establish approximate solutions by applying some regularization methods. In addition, the convergence estimates and numerical experiments showing the flexibility of those methods are given. [ABSTRACT FROM AUTHOR]

In this paper, I am interested in the study of the inverse Cauchy boundary value problem for the time-fractional diffusion equation in two layers spherical domain. Given the data in the first layer, my goal is to recover the temperature distribution and the heat flux in the second layer. First, I prove that the problem is severely ill-posed in the Hadamard sense. After, I propose a truncation-type regularization approach enabling us to achieve Hölder-type error in L 2 -norm between the regularized and exact solutions. [ABSTRACT FROM AUTHOR]

In this paper, the problem of finding the source function for the Rayleigh–Stokes equation is considered. According to Hadamard's definition, the sought solution of this problem is both unstable and independent of continuous data. By using the fractional Tikhonov method, we give the regularized solutions and then deal with a priori error estimate between the exact solution and its regularized solutions. Finally, the proposed regularized methods have been verified by simple numerical experiments to check error estimate between the sought solution and the regularized solution. [ABSTRACT FROM AUTHOR]

The main target of this paper is to study a problem of recovering a spherically symmetric domain with fractional derivative from observed data of nonlocal type. This problem can be established as a new boundary value problem where a Cauchy condition is replaced with a prescribed time average of the solution. In this work, we set some of the results above existence and regularity of the mild solutions of the proposed problem in some suitable space. Next, we also show the ill-posedness of our problem in the sense of Hadamard. The regularized solution is given by the fractional Tikhonov method and convergence rate between the regularized solution and the exact solution under a priori parameter choice rule and under a posteriori parameter choice rule. [ABSTRACT FROM AUTHOR]

We consider the spectral truncation as the regularization for an ill-posed non-homogeneous parabolic final value problem, and obtain error estimates under a general source condition when the data, which consist of the non-homogeneous term as well as the final value, are noisy. The resulting error estimate is compared with the corresponding estimate under the Lavrentieve method, and showed that the truncation method has no index of saturation. [ABSTRACT FROM AUTHOR]

In the present paper, we devote our aspiration to some initial and final value problems for a class of space-fractional diffusion equation with time-dependent diffusivity factor. For the initial value problem (IVP), we investigate the stability of the solution concerning the data and the fractional order. For the final value problem, we prove the ill-posedness and suggest a filter method to regularize the problem. Explicit convergence rate of Hölder type is established. Finally, several numerical examples based on the finite difference approximation and the discrete Fourier transform are performed to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Fractional derivative is an important notion in the study of the contemporary mathematics not only because it is more mathematically general than the classical derivative but also it really has applications to understand many physical phenomena. In particular, fractional derivatives are related to long power-law particle jumps, which can be understood as transient anomalous sub-diffusion model (see Sabzikar F, Meerschaert M, Chen J. Tempered fractional calculus. J Comput Phys. 2015;293:14–28; Sokolov IM, Klafter J, Blumen A. Fractional kinetics. Phys Today. 2002;55:48–54; Sokolov IM, Klafter J. Anomalous diffusion spreads its wings. Phys World. 2005;18:19–22; Zhang Y, Meerschaert MM, Packman AI. Linking fluvial bed sediment transport across scales. Geophys Res Lett. 2012;39(20):20404. doi:10.1029/2012GL053476). Based on the models given in Scher H, Montroll EW. Anomalous transit-time dispersion in amorphous solids. Phys Rev B. 1975;12(6):2455–2477 and Zheng GH, Wei T. Spectral regularization method for solving a time-fractional inverse diffusion problem. Appl Math Comput. 2011;218:1972–1990, we study an inverse problem for the advection equation with a nonlinear reaction term in a two-dimensional semi-infinite domain for which we recover the initial distribution from the observation data provided at the final location x = 1. This problem is severely ill-posed in the sense of Hadamard. Thus, we propose a regularization method to construct an approximate solution for the problem. From that, convergence rate of the regularized solution is obtained under some a priori bound assumptions on the exact solution. Eventually, a numerical experiment is given to show the effectiveness of the proposed regularization methods. [ABSTRACT FROM AUTHOR]

A numerical algorithm for the solution of an inverse coefficient problem for nonstationary, nonlinear production-destruction type model is proposed and tested on an example of the Lorenz'63 system. With an ensemble of adjoint problem solutions, the inverse problem is transformed into a quasi-linear matrix problem and solved with Newton-type algorithm. Two different ways of the adjoint ensemble construction are compared. In the first one, a trigonometric basis is used. In the second one in situ measurements are taken into account. Local convergence properties of the algorithm are studied numerically to find out when the use of more data can lead to the degradation of the reconstruction results. [ABSTRACT FROM AUTHOR]

We propose three approximate methods using regularization techniques and finite Hartley transforms for solving first-kind integral equations involving linear Toeplitz plus Hankel parts. Numerical examples are given for illustrating these new algorithms. [ABSTRACT FROM AUTHOR]

The diffusion equation has many applications in fields such as physics, environment, and fluid mechanics. In this paper, we consider the problem of identifying an unknown source for a time-fractional diffusion equation in a general bounded domain from the nonlocal integral condition. The problem is non-well-posed in the sense of Hadamard, i.e, if the problem has only one solution, the solution will not depend continuously on the input data. To get a stable solution and approximation, we need to offer the regularization methods. The first contribution to the paper is to provide a regularized solution using the modified fractional Landweber method. Two choices are proposed including a priori and a posteriori parameter choice rules, to estimate the convergence rate of the regularized methods. The second new contribution is to use truncation to give an estimate of L p for the convergence rate. [ABSTRACT FROM AUTHOR]

In this paper, we deal with an inverse problem of giving the terminal condition for time-fractional diffusion equations. This problem is ill-posed, i.e. the solution does not depend continuously on the data. To regularize the instable solution, we use the spectral method (or called the Fourier truncation method) combined with some techniques in nonparametric regression for establishing a regularized solution. We also investigate the convergence rate between the sought solution and the regularized solution. [ABSTRACT FROM AUTHOR]

In 2012, Jin Qinian considered an inexact Newton–Landweber iterative method for solving nonlinear ill-posed operator equations in the Banach space setting by making use of duality mapping. The method consists of two steps; the first one is an inner iteration which gives increments by using Landweber iteration, and the second one is an outer iteration which provides increments by using Newton iteration. He has proved a convergence result for the exact data case, and for the perturbed data case, a weak convergence result has been obtained under a Morozov type stopping rule. However, no error bound has been given. In 2013, Kaltenbacher and Tomba have considered the modified version of the Newton–Landweber iterations, in which the combination of the outer Newton loop with an iteratively regularized Landweber iteration has been used. The convergence rate result has been obtained under a Hölder type source condition. In this paper, we study the modified version of inexact Newton–Landweber iteration under the approximate source condition and will obtain an order-optimal error estimate under a suitable choice of stopping rules for the inner and outer iterations. We will also show that the results proved in this paper are more general as compared to the results proved by Kaltenbacher and Tomba in 2013. Also, we will give a numerical example of a parameter identification problem to support our method. [ABSTRACT FROM AUTHOR]