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In this paper, we focus on a backward problem for an inhomogeneous time-fractional diffusion equation on a spherical symmetric domain. This problem is ill-posed in the sense of Hadamard. We propose an iterated fractional Tikhonov regularization method in both cases: the deterministic case and random noise case. The a-priori and the a-posteriori choice rules for regularization parameters are discussed and both rules yield the corresponding convergence rates. The iterated fractional Tikhonov regularization method goes beyond the saturation results of classical Tikhonov method, and iterative fractional Tikhonov regularization method is superior to classical iterative Tikhonov regularization method under the a-priori parameter choice rule. A numerical example is conducted for showing the validity of the proposed method.

In this paper, we consider the problem of analytic continuation of the analytic function $g(z) = g(x+iy)$ on a strip domain Ω = $\{z = x+iy\in \mathbb{C}|\, x\in\mathbb{R}, 0 < y < y_0\}$, where the data is given only on the line $y = 0$. This problem is a severely ill-posed problem. We propose the fraction Landweber iterative regularization method to deal with this problem. Under the a priori and a posteriori regularization parameter choice rule, we all obtain the error estimates between the regularization solution and the exact solution. Some numerical examples are given to verify the efficiency and accuracy of the proposed methods.

In this paper, we are concerned with an inverse source problem for the time-fractional diffusion equation with variable coefficients in a general bounded domain. The problem is mildly ill-posed. A new fractional Tikhonov method is proposed. We discuss the a-priori regularization parameter choice rule and the a-posteriori regularization parameter choice rule, and prove the corresponding convergence estimates. Numerical experiments are conducted for illustrating effectiveness of the proposed method.

One of the main tasks for sparse recovery is to develop and analyze computationally tractable algorithms for obtaining sparse solutions of underdetermined linear systems. Jiao et al. (2015) proposed a primal-dual active set with a continuation (PDASC) algorithm for the $\ell^0$-regularized least-squares problem and established finite step global convergence and obtained a sharp estimation error under a certain restricted isometry property (RIP) condition. In this paper, we relax the condition on the RIP constant to be independent of the sparsity level $T$ for a class of signals with a strong decay property. Furthermore, we propose a novel data-driven regularization parameter selection rule. Several numerical examples are presented to verify the efficiency and accuracy of the PDASC algorithm and the data-driven parameter choice rule.

Inverse problems arise in many branches of science and engineering. In order to get a good approximation of the solution of this kind of problems, the use of regularization methods is required. Tikhonov regularization is one of the most popular methods for estimating the solutions of inverse problems. This method needs a regularization parameter and the quality of the approximate solution depends on how good the regularization parameter is. The L-curve method is a convenient parameter choice strategy for selecting the Tikhonov regularization parameter and it works well most of the time. There are some problems in which the L-curve criterion does not perform properly. Multiplicative regularization is a method for solving inverse problems and does not require any parameter selection strategies. However, it turns out that there is a close connection between multiplicative regularization and Tikhonov regularization; in fact, multiplicative regularization can be regarded as defining a parameter choice rule for Tikhonov regularization. In this work, we have analyzed multiplicative regularization for finite-dimensional

In this paper, an identical approximate regularization method is extended to the Cauchy problem of two-dimensional heat conduction equation, this kind of problem is severely ill-posed. The convergence rates are obtained under a priori regularization parameter choice rule. Numerical results are presented for two examples with smooth and continuous but not smooth boundaries, and compared the identical approximate regularization solutions which are displayed in paper. The numerical results show that our method is effective, accurate and stable to solve the ill-posed Cauchy problems.

Nous considerons le probleme inverse stochastique de reconstruire $f$ a partir de donnees bruitees $Y=Tf+\sigma \xi $ ou $\xi $ est un bruite blanc et $T$ un operateur compact entre espaces de Hilbert. Considerant des methodes de reconstruction generales de la forme $\hat{f}_{\alpha }=q_{\alpha }(T^{*}T)T^{*}Y$ avec un filtre ordonne $q_{\alpha }$, nous examinons le choix du parametre de regularisation $\alpha $ en minimisant un estimateur non biaise du risque predictif $\mathbb{E}[\Vert Tf-T\hat{f}_{\alpha }\Vert^{2}]$. Le parametre correspondant $\alpha_{\mathrm{pred}}$ et son utilisation sont bien connus dans la litterature mais les inegalites oracles et les resultats d’optimalite dans ce cadre general sont inconnus. Nous prouvons une inegalite oracle (generalisee), qui relie le risque direct $\mathbb{E}[\Vert f-\hat{f}_{\alpha_{\mathrm{pred}}}\Vert^{2}]$ au risque de l’oracle de prediction $\inf_{\alpha >0}\mathbb{E}[\Vert Tf-T\hat{f}_{\alpha }\Vert^{2}]$. A partir de cette inegalite oracle nous sommes alors capable de conclure que la regle de choix du parametre examine est d’ordre optimale au sens minimax. Finalement nous presentons aussi des simulations numeriques qui confirment l’optimalite de l’ordre de la methode et la qualite du choix du parametre dans des situations discretes.

Multiplicative regularization solves a linear inverse problem by minimizing the product of the norm of the data misfit and the norm of the solution. This technique is related to Tikhonov regularization with the parameter chosen to make the data misfit and regularization terms (of the Tikhonov objective function) equal. This suggests a heuristic parameter choice method, equivalent to the rule previously proposed by Reginska. Reginska's rule is well defined provided the data is sufficiently close to exact data and does not lie in the range of the operator. If a sufficiently large portion of the data error lies outside the range of the operator, then the solution defined by Reginska's rule converges weakly to the exact solution as the data error converges to zero. The regularization parameter converges to zero like the square of the norm of the data noise, leading to under-regularization for small noise levels. Nevertheless, the method performs well on a suite of test problems, as shown by comparison with th...

In this paper, we consider the regularization of the backward problem of diffusion process with time-fractional derivative. Since the equation under consideration involves the time-fractional derivative, we introduce a new perturbation which is related to the time-fractional derivative into the original equation. This leads to a fractional-order quasi-reversibility method. In theory, we give the regularity of the regularized solution and the corresponding convergence rate is also proved under the appropriate regularization parameter choice rule. In numerics, some numerical experiments are presented to illustrate the effectiveness of our method and some numerical comparison with the existing quasi-reversibility method is conducted. Both theoretical and numerical results show the advantage of the new method.

In this paper we investigate the convergence of Chebyshev type regularization strategy combined with the Morozov discrepancy principle, which is an a posteriori parameter choice rule and independent of the a priori information of exact solution. Compared with the standard Tikhonov regularization method, the Chebyshev type regularization method has no saturation restriction so that its convergence order holds for any υ ≥ 1 2 , where p = 2 υ + 1 2 υ in the H o l d e r type estimate.

In this paper, we consider a class of severely ill-posed backward problems for linear parabolic equations. We use a convolution regularization method to obtain a stable approximate initial data from the noisy final data. The convergence rates are obtained under an a priori and an a posteriori regularization parameter choice rule in which the a posteriori parameter choice is a new generalized discrepancy principle based on a modified version of Morozov's discrepancy principle. The log-type convergence order under the a priori regularization parameter choice rule and log ź log -type order under the a posteriori regularization parameter choice rule are obtained. Two numerical examples are tested to support our theoretical results.

In this paper, we consider a backward space-fractional diffusion problem. We propose an a posteriori parameter choice rule for the regularization method given in Zheng and Wei (2010), where the authors proposed a regularization method called convolution regularization method, and gave an a priori parameter choice strategy. In this paper, we study the same problem but give a new a posteriori parameter choice based on a modified version of the discrepancy principle, and obtain a log -type error estimate under an additional source condition. Numerical results show that our method is feasible.

The inverse problem of determining the unknown source for a fractional diffusion equation is studied. This problem is ill-posed in the sense of Hadamard, i.e., small changes in the measured data can blow up the solution. The quasi-reversibility regularization method is applied to solve it. Convergence estimates are presented under an a priori parameter choice rule and an a posteriori parameter choice rule, respectively. Numerical examples are given to show that the regularization method is effective and stable.

In this paper, we mainly consider the inverse problem for identifying the unknown heat source in spherical symmetric domain. We propose a truncation regularization method combined with an a posteriori regularization parameter choice rule to deal with this problem. The Hölder type convergence estimate is obtained. Numerical results are presented to illustrate the accuracy and efficiency of this method.

In the present paper, we consider an inverse source problem for a fractional diffusion equation. This problem is ill-posed, i.e. the solution (if it exists) does not depend continuously on the data. Based on an a priori assumption, we give the optimal error bound analysis and a conditional stability result. Moreover, we use the Fourier regularization method to deal with this problem. An a priori error estimate between the exact solution and its regularized approximation is obtained. Meanwhile, a new a posteriori parameter choice rule is also proposed. For the a priori and the a posteriori regularization parameters choice rules, we all obtain the convergence error estimates which are all order optimal. Numerical examples are presented to illustrate the validity and effectiveness of this method.

Sporadic machine-to-machine communication requires a signaling efficient medium access strategy, which can be achieved by compressed sensing based multi-user detection (CS-MUD). This novel application for Compressed Sensing requires adapted reconstruction algorithms considering typical communications assumptions. In order to get fast signal reconstruction, we utilize a smoothed version of the l 1 -regularization, namely, the elastic-net. Further, we connect this approach with a final projection step to incorporate finite alphabets and introduce a new parameter choice strategy for the elastic-net: the so-called C-curve.

The aim of this work is to solve the backward problem for a time-fractional diffusion equation with variable coefficients in a general bounded domain. The problem is ill-posed in L-2 norm sense. An iteration scheme is proposed to obtain a regularized solution. Two kinds of convergence rates are obtained using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed methods. (C) 2014 Wiley Periodicals, Inc.

In the present paper we consider the problem of numerical analytic continuation for an analytic function f ( z ) = f ( x + iy ) on a strip domain Ω + = { z = x + iy ∈ C | x ∈ R , 0 y y 0 } . The key difference with the known methods is that a novel iteration regularization method with a priori and a posteriori parameter choice rule is presented. The comparison in numerical aspect with other methods is also discussed.

In this article, we consider a class of ill-posed problems in two-dimensional setting which can be formulated as the problem of solving ill-posed operator equation. The spectral method is a simple and effective regularization method for such a class of ill-posed problems. For the spectral method, we obtain the error estimates with optimal order under an adaptive a posteriori parameter choice rule called balancing principle. Two ill-posed examples including an inverse diffraction problem and a deblurring problem are provided.

In the present paper, we consider an inverse source problem for a fractional diffusion equation. This problem is ill-posed, i.e. the solution (if it exists) does not depend continuously on the data. We give the mollification regularization method to solve this problem. An a priori error estimate between the exact solution and its regularized approximation is obtained. Moreover, a new a posteriori parameter choice rule is also proposed and a good error estimate is also obtained. Numerical examples are presented to illustrate the validity and effectiveness of this method.

The goal of this paper is to investigate a backward parabolic equation with time-dependent coefficient by the truncation method (TM). The key point to our paper is that we give an a posteriori choice of regularization parameter for the TM, with which the convergence estimates between the exact solution and the regularized approximation are obtained. Numerical implementation sheds light on the accuracy and efficiency of the proposed method.

The widespread applicability of the multipenalty regularization is limited by the fact that theoretically optimal rate of reconstruction for a given problem can be realized by a one-parameter counterpart, provided that relevant information on the problem is available and taken into account in the regularization. In this paper, we explore the situation where no such information is given, but still accuracy of optimal order can be guaranteed by employing multipenalty regularization. Our focus is on the analysis and the justification of an a posteriori parameter choice rule for such a regularization scheme. First we present a modified version of the discrepancy principle within the multipenalty regularization framework. As a consequence we provide a theoretical justification to the multipenalty regularization scheme equipped with the a posteriori parameter choice rule. We then establish a fast numerical realization of the proposed discrepancy principle based on a model function approximation. Finally, we pro...

In this paper, we consider the inverse problem of determining a heat source in a parabolic equation where data are given at a fixed time. This problem is ill-posed, i.e., the solution (if it exists) does not depend continuously on the data. The mollification method with Gauss kernel is proposed to solve this problem. An a priori error estimate between the exact solution and its regularized approximation is obtained. Moreover, we also propose a new a posteriori parameter choice rule and get a good error estimate. Numerical results for several benchmark test problems indicate that the method is an accurate and flexible method to determine the unknown spatial-dependent heat source.

We consider the problem of determining an unknown source which depends only on one variable in two-dimensional Poisson equation. We prove a conditional stability for this problem. Moreover, we propose a truncation regularization method combined with an a posteriori regularization parameter choice rule to deal with this problem and give the corresponding convergence estimate. Numerical results are presented to illustrate the accuracy and efficiency of this method.

In this paper, we consider a backward problem for a time-fractional diffusion equation with variable coefficients in a general bounded domain. That is to determine the initial data from a noisy final data. We propose a quasi-boundary value regularization method combined with an a posteriori regularization parameter choice rule to deal with the backward problem and give the corresponding convergence estimate.

A crucial problem concerning Tikhonov regularization is the proper choice of the regularization parameter. This paper deals with a generalization of a parameter choice rule due to Reginska (1996) [31] , analyzed and algorithmically realized through a fast fixed-point method in Bazan (2008) [3] , which results in a fixed-point method for multi-parameter Tikhonov regularization called MFP. Like the single-parameter case, the algorithm does not require any information on the noise level. Further, combining projection over the Krylov subspace generated by the Golub–Kahan bidiagonalization (GKB) algorithm and the MFP method at each iteration, we derive a new algorithm for large-scale multi-parameter Tikhonov regularization problems. The performance of MFP when applied to well known discrete ill-posed problems is evaluated and compared with results obtained by the discrepancy principle. The results indicate that MFP is efficient and competitive. The efficiency of the new algorithm on a super-resolution problem is also illustrated.

In this paper, we consider an analytic continuation problem on a strip domain with the data given approximately only on the real axis. The Gauss mollification method is proposed to solve this problem. An a priori error estimate between the exact solution and its regularized approximation is obtained. Moreover, we also propose a new a posteriori parameter choice rule and get a good error estimate. Several numerical examples are provided, which show the method works effectively.

We introduce three spectral regularization methods for solving a backward heat conduction problem (BHCP). For the three spectral regularization methods, we give the stability error estimates with optimal order under an a-priori and an a-posteriori regularization parameter choice rule. Numerical results show that our theoretical results are effective.

The numerical solution of inverse problems using Tikhonov’s regularization methods requires a huge amount of computations in iterative processes. It can employ extrapolation techniques to accelerate the convergence process or to improve accuracy of the regularized solution. This chapter aims to introduce some main extrapolation methods that have been studied for solving linear inverse problems in detail. Our emphasis is to discuss related technical problems, to propose a new extrapolation algorithm based on the Hermitian interpolation and to present results of numerical experiments for showing the merits of extrapolated regularization methods.

In this paper, we present a Newton-type method with double regularization parameters for nonlinear ill-pose problems. The key step in the process that a reasonable choice rule to determine these two regularization parameters is presented. And the convergence and the stability of the method are discussed. Numerical experiment shows the effectiveness of the method.

In standard inverse problems, the task is to solve an operator equation from given noisy data. However, sometimes also the operator is not known exactly. Therefore we propose a method that allows errors both in the operator and the data. In particular, we consider operator equations where the operator can be characterized by a function. For the stable reconstruction we propose the use of a Tikhonov-type functional with a generalized misfit term and an additional penalty term which promotes sparsity. Using an appropriate parameter choice rule for the two regularization parameters, we prove convergence and convergence rates for the method, and provide a first numerical example.

In this paper we deal with Morozov's discrepancy principle as an a posteriori parameter choice rule for Tikhonov regularization with general convex penalty terms Ψ for nonlinear inverse problems. It is shown that a regularization parameter α fulfilling the discprepancy principle exists, whenever the operator F satisfies some basic conditions, and that for suitable penalty terms the regularized solutions converge to the true solution in the topology induced by Ψ. It is illustrated that for this parameter choice rule it holds α → 0, δq/α → 0 as the noise level δ goes to 0. Finally, we establish convergence rates with respect to the generalized Bregman distance and a numerical example is presented.

This paper deals with the well-known discrepancy principle of Morozov. We show that the principle can be used as an a posteriori choice rule for determining the regularization parameter of Tikhonov regularization considering more general penalty terms than the classical quadratic one. We show regularization properties as well as convergence rates.