Search

The Tikhonov method is studied as applied to ill-posed problems of minimizing a smooth nonconvex functional. Assuming that the sought solution satisfies the source condition, an accuracy estimate for the Tikhonov method is obtained in terms of the regularization parameter. Previously, such an estimate was obtained only under the assumption that the functional is convex or under a structural condition imposed on its nonlinearity. Additionally, a new accuracy estimate for the Tikhonov method is obtained in the case of an approximately specified functional. [ABSTRACT FROM AUTHOR]

We study nested variational inequalities, which are variational inequalities whose feasible set is the solution set of another variational inequality. We present a projected averaging Tikhonov algorithm requiring the weakest conditions in the literature to guarantee the convergence to solutions of the nested variational inequality. Specifically, we only need monotonicity of the upper- and the lower-level variational inequalities. Also, we provide the first complexity analysis for nested variational inequalities considering optimality of both the upper- and lower-level. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the reconstruction problem of aperture in the plane from their diffraction patterns. The problem is severely ill-posed. The reconstruction solutions of classical Tikhonov method and Fourier truncated method are usually over-smoothing. To overcome this disadvantage of the classical methods, we introduce a quasi-boundary regularization method for stabilizing the problem by adding a-priori assumption on the exact solution. The corresponding error estimate is derived. At the continuation boundary z=0, the error estimate under the a-priori assumption is also proved. In theory without noise, the proposed method has better approximation than the classical Tikhonov method. For illustration, two numerical examples are constructed to demonstrate the feasibility and efficiency of the proposed method.

This paper is concerned with the reconstruction of relaxation time distributions in Nuclear Magnetic Resonance (NMR) relaxometry. This is a large-scale and ill-posed inverse problem with many potential applications in biology, medicine, chemistry, and other disciplines. However, the large amount of data and the consequently long inversion times, together with the high sensitivity of the solution to the value of the regularization parameter, still represent a major issue in the applicability of the NMR relaxometry. We present a method for two-dimensional data inversion (2DNMR) which combines Truncated Singular Value Decomposition and Tikhonov regularization in order to accelerate the inversion time and to reduce the sensitivity to the value of the regularization parameter. The Discrete Picard condition is used to jointly select the SVD truncation and Tikhonov regularization parameters. We evaluate the performance of the proposed method on both simulated and real NMR measurements. [ABSTRACT FROM AUTHOR]

We consider nested variational inequalities consisting in a (upper-level) variational inequality whose feasible set is given by the solution set of another (lower-level) variational inequality. Purely hierarchical convex bilevel optimization problems and certain multi-follower games are particular instances of nested variational inequalities. We present an explicit and ready-to-implement Tikhonov-type solution method for such problems. We give conditions that guarantee the convergence of the proposed method. Moreover, inspired by recent works in the literature, we provide a convergence rate analysis. In particular, for the simple bilevel instance, we are able to obtain enhanced convergence results. [ABSTRACT FROM AUTHOR]

We consider Fredholm integral equation of the first kind with noisy data and use Landweber-type iterative methods as an iterative solver. We compare regularization property of Tikhonov, truncated singular value decomposition and the iterative methods. Furthermore, we present a necessary and sufficient condition for the convergence analysis of the iterative method. The performance of the iterative method is shown and compared with modulus-based iterative methods for the constrained Tikhonov regularization. [ABSTRACT FROM AUTHOR]

We consider numerical methods for stable approximation of solutions to irregular nonlinear equations with general smooth operators in Hilbert space. Generally, the known variational procedures and iterative regularization methods deliver approximations with accuracy estimates greater in order than error levels in the input data. In the paper for certain components of the desired solution we establish the possibility of obtaining approximations with linear accuracy estimates relative to the error level. These components correspond to the projections of the solution onto proper subspaces of the symmetrized derivative for the operator of the problem. [ABSTRACT FROM AUTHOR]

This paper is concerned with the reconstruction of relaxation time distributions in Nuclear Magnetic Resonance (NMR) relaxometry. This is a large-scale and ill-posed inverse problem with many potential applications in biology, medicine, chemistry, and other disciplines. However, the large amount of data and the consequently long inversion times, together with the high sensitivity of the solution to the value of the regularization parameter, still represent a major issue in the applicability of the NMR relaxometry. We present a method for two-dimensional data inversion (2DNMR) which combines Truncated Singular Value Decomposition and Tikhonov regularization in order to accelerate the inversion time and to reduce the sensitivity to the value of the regularization parameter. The Discrete Picard condition is used to jointly select the SVD truncation and Tikhonov regularization parameters. We evaluate the performance of the proposed method on both simulated and real NMR measurements.

In traditional plane near-field acoustic holography reconstruction process,regardless of wave number domain filter or regularization method,they cannot solve their own ill-posed problem,and thus cannot improve the reconstruction accuracy. Plane near-field acoustic holography based on Compressive Sampling,translates traditional solved particle velocity into solving sparse coefficient,by effective using of the particle velocity sparsity,avoiding the reconstruction process of ill-posed problem. Combined with plane near-field acoustic holography based on Tikhonov regularization method, we analyze the reconstruction accuracy of numerical simulation. The results show that, plane near-field acoustic holography based on Compressive Sampling has a good accuracy in reconstruction process,especially in the border region of reconstructive surface.Finally,this method can effectively identify the source of the noise of the refrigerator,to provide an effective basis for noise control.

A linear operator equation of the first kind is investigated. The solution of this equation contains singularities of various types; namely, along with a smooth background, the solution has sharp bends and jump discontinuities. For the construction of a stable approximated solution, a modified Tikhonov method with a stabilizer in the form of the sum of three functionals is proposed. Each of the functionals accounts for the specific character of the corresponding component of the solution. Convergence theorems are formulated, a general discrete approximation scheme of the regularizing algorithm is justified, and results of numerical experiments are discussed. [ABSTRACT FROM AUTHOR]

We consider regularization of linear ill‐posed problem Au = f with noisy data fδ, ¦fδ - f¦≤ δ . The approximate solution is computed as the extrapolated Tikhonov approximation, which is a linear combination of n ≥ 2 Tikhonov approximations with different parameters. If the solution u* belongs to R((A*A) n ), then the maximal guaranteed accuracy of Tikhonov approximation is O(δ 2/3) versus accuracy O(δ 2n/(2n+1)) of corresponding extrapolated approximation. We propose several rules for choice of the regularization parameter, some of these are also good in case of moderate over‐ and underestimation of the noise level. Numerical examples are given. First published online: 09 Jun 2011

This article presents a direct application of a Tikhonov inversion with a quadratic constraint applied in the case of a corrosion diagnosis. The main originality of this method is to inject physical information during the inversion to automatically restrict the Tikhonov parameter space. This application is then tested on a real case of corrosion diagnosis from electrical measurements in the water. [ABSTRACT FROM PUBLISHER]

Linear Fredholm integral equations of the first kind over surfaces are less familiar than those of the second kind, although they arise in many applications like computer tomography, heat conduction and inverse scattering. This article emphasizes their numerical treatment, since discretization usually leads to ill-conditioned linear systems. Strictly speaking, the matrix is nearly singular and ordinary numerical methods fail. However, there exists a numerical regularization method - the Tikhonov method - to deal with this ill-conditioning and to obtain accurate numerical results. [ABSTRACT FROM AUTHOR]

Abstract: During the past few decades, computerized tomography (CT) was widely used for non-destructive testing (NDT) and non-destructive examination (NDE) in the industrial area because of its characteristics of non-invasiveness and visibility. Recently, CT technology has been applied to multi-phase flow measurement. Using the principle of radiation attenuation measurements along different directions through the investigated object with a special reconstruction algorithm, cross-sectional information of the scanned object can be worked out. It is a typical inverse problem and has always been a challenge for its nonlinearity and ill-conditions. The Tikhonov regulation method is widely used for similar ill-posed problems. However, the conventional Tikhonov method does not provide reconstructions with qualities good enough, the relative errors between the reconstructed images and the real distribution should be further reduced. In this paper, a modified conjugate gradient (CG) method is applied to a Tikhonov system (MCGT method) for reconstructing CT images. The computational load is dominated by the number of independent measurements , and a preconditioner is imported to lower the condition number of the Tikhonov system. Both simulation and experiment results indicate that the proposed method can reduce the computational time and improve the quality of image reconstruction. [Copyright &y& Elsevier]

We consider linear ill-posed problems in Hilbert space with noisy data. The noise level may be given exactly or approximately or there may be no information about the noise level. We regularize the problem using the Landweber method, the Tikhonov method or the extrapolated version of the Tikhonov method. For all three cases of noise information we propose rules for choice of the regularization parameter. Extensive numerical experiments show the advantage of the proposed rules over known rules, including the discrepancy principle, the quasioptimality criterion, the Hanke-Raus rule, the Brezinski-Rodriguez-Seatzu rule and others. Numerical comparison also shows at which information about the noise level our rules for approximately given noise level should be preferred to other rules. [ABSTRACT FROM AUTHOR]

We consider regularization of linear ill-posed problem Au = f with noisy data fδ, ∥fδ - f∥ ≤ δ. The approximate solution is computed as the extrapolated Tikhonov approximation, which is a linear combination of n ≥ 2 Tikhonov approximations with different parameters. If the solution u∗ belongs to R((A* A)n), then the maximal guaranteed accuracy of Tikhonov approximation is O(δ2/3) versus accuracy O(δ2n/(2n+1)) of corresponding extrapolated approximation. We propose several rules for choice of the regularization parameter, some of these are also good in case of moderate over- and underestimation of the noise level. Numerical examples are given. [ABSTRACT FROM AUTHOR]

The desingularized meshless method (DMM) has been successfully used to solve boundary-value problems with specified boundary conditions (a direct problem) numerically. In this paper, the DMM is applied to deal with the problems with over-specified boundary conditions. The accompanied ill-posed problem in the inverse problem is remedied by using the Tikhonov regularization method and the truncated singular value decomposition method. The numerical evidences are given to verify the accuracy of the solutions after comparing with the results of analytical solutions through several numerical examples. The comparisons of results using Tikhonov method and truncated singular value decomposition method are also discussed in the examples. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of this paper is to develop an inverse procedure, which is used to obtain the history of the Coulomb''s friction coefficient on the contact surfaces between the workpiece and the die during the AA1050 aluminum upsetting process. This procedure is based on the experimental upsetting normal loadings and combines the matrix-presentation linear least-squares errors method of inverse elastic–plastic large deformation finite element model developed in this paper with the regularization of Tikhonov method. The square of the difference of the friction coefficients between successive stages is adopted as the stabilizing function of the regularization of Tikhonov method. In this paper, the Coulomb''s friction model is adopted before the sticking effect occurs. After the sticking effect occurrence, the shear strength friction model is adopted. From the results of this paper, it can be shown and demonstrated that the inverse procedure proposed in this paper can make the history of the Coulomb''s friction coefficient more stable and reasonable. [Copyright &y& Elsevier]

Abstract: Radiant enclosures used in industrial processes have traditionally been designed by trial-and-error, a technique that usually demands considerable time to find a solution of limited quality. As an alternative, designers have recently adopted optimization and inverse methodologies to solve design problems involving radiative transfer; the optimization methodology solves the inverse problem implicitly by transforming it into a multivariable minimization problem, while the inverse design methodology solves the problem explicitly using regularization. This paper presents the details of both methodologies, and demonstrates them by solving for the optimal heater settings in an industrially relevant radiant enclosure design problem. [Copyright &y& Elsevier]

Theoretical and numerical results presented in this article show advantages of the method of multicriteria optimization in comparison with the Tikhonov method for solving ill-posed problems of signal restoration by the following properties, sensitivity of the solution relative to accuracy of determination of the regularization parameter, stability relative to errors of measured signal, convergence of the numerical process. [ABSTRACT FROM AUTHOR]