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Appropriate regularization parameter specification is the linchpin for solving ill-posed inverse problems when regularization method is applied. This paper presents a novel technique to determine cut off singular values in the truncated singular value decomposition (TSVD) methods. Simple formulae are presented to calculate the index number of the singular value, beyond which all the smaller singular values and the corresponding vectors are truncated. The determination method of optimal truncation threshold is firstly theoretically inferred. Two-dimensional inverse problems processing Radon transform are then exemplified. Formulae to solve the problem with insufficient image resolution and projection angle number are derived by the currently proposed method. The results show that accuracy of the current method is similar to that of TSVD but with much superior efficiency. On the other hand, insufficiency in input data affects the output accuracy of the inverse solution, a least square method can be engaged to establish formulae calculating the truncation threshold. For an insufficient set of input data, the percentage difference between inversely reconstructed signal and TSVD reconstructed signal is about 3%. The current formulae offer reliable and more efficient approach to calculate the truncation threshold when TSVD is applied to solve inverse problems with known system characteristics. [ABSTRACT FROM AUTHOR]

This article is devoted to the study of nonstationary Iterated Tikhonov (nIT) type methods (Hanke M, Groetsch CW. Nonstationary iterated Tikhonov regularization. J Optim Theory Appl. 1998;98(1):37–53; Engl HW, Hanke M, Neubauer A. Regularization of inverse problems. Vol. 375, Mathematics and its Applications. Dordrecht: Kluwer Academic Publishers Group; 1996. MR 1408680) for obtaining stable approximations to linear ill-posed problems modelled by operators mapping between Banach spaces. Here we propose and analyse an a posteriori strategy for choosing the sequence of regularization parameters for the nIT method, aiming to obtain a pre-defined decay rate of the residual. Convergence analysis of the proposed nIT type method is provided (convergence, stability and semi-convergence results). Moreover, in order to test the method's efficiency, numerical experiments for three distinct applications are conducted: (i) a 1D convolution problem (smooth Tikhonov functional and Banach parameter-space); (ii) a 2D deblurring problem (nonsmooth Tikhonov functional and Hilbert parameter-space); (iii) a 2D elliptic inverse potential problem. [ABSTRACT FROM AUTHOR]

This paper is devoted to the problem of determining the initial data for the backward non-homogeneous time fractional heat conduction problem by the Fourier truncation method. The exact solution for the forward and backward fractional heat problems is expressed in terms of eigen function expansion and Mittag–Leffler function. Due to the instability of determining initial data, a regularized truncated solution is considered. Further, the stability estimate for the exact solution and the convergence estimates for the regularized solution using an á-priori choice rule and an á-posteriori choice rule are derived. [ABSTRACT FROM AUTHOR]

We devise a fast and reliable computational framework for the elasticity imaging inverse problem of detecting cancerous tumors in the human body using an output least-squares (OLS) approach. From a mathematical standpoint, this inverse problem requires identifying a parameter in a mixed variational problem. We develop, in a continuous setting, a first-order adjoint method and two second-order adjoint methods. The continuous formulae are then used to devise a scheme for an efficient computation of the gradient and the Hessian of the OLS objective. We give detailed numerical examples. [ABSTRACT FROM PUBLISHER]

We study the problem of determining the initial condition in parabolic equations with time-dependent coefficients from integral observations which can be regarded as generalizations of point-wise interior observations. Our approach is new in the sense that for determining the initial condition we do not assume that the data available in the whole space domain at the final moment or in a subset of the space domain during a certain time interval, but some integral observations during a time interval. We propose a variational method in combination with Tikhonov regularization for solving the problem and then discretize it by finite difference splitting methods. The discretized minimization problem is solved by the conjugate gradient method and tested on computer to show its efficiency. Also as a by-product of the variational method, we propose a numerical scheme for estimating the degree of ill-posedness of the problem. [ABSTRACT FROM AUTHOR]

There are cases in engineering applications where the involved impact forces are impractical or difficult to be directly measured. In such cases, the impact forces may inversely be reconstructed on the basis of the associated-induced elastic responses. However, the inverse impact force reconstruction is an ill-posed problem. So far, various methods have been proposed. Many require complex mathematical treatments; thus, they are impractical for experimental engineers. In this article, we adopt the Wiener filter method and introduce a regularization technique so that the method can provide an optimum solution which balances the aspects of the minimum noise and the minimum residual. The proposed method is evaluated using three examples of the impact problems. The evaluation reveals that the regularized Wiener filter method is less sensitive to the regularization parameter unlike many of the existing methods. Therefore, the method is more suitable for practical applications. [ABSTRACT FROM PUBLISHER]

The determination of space-dependent functions from boundary measurements or inner pointwise measurements is ill-posed inverse problems that require regularization tools to be stabilized. Among numerous regularization strategies, the Tikhonov penalization is one of the most used in the field of space-dependent function estimation. Its efficient use relies on the Tikhonov parameter value for which search is time consuming although necessary, especially in the field of nonlinear inversion. Other strategies, such as appropriate parameterization, have recently proven to be very efficient to cope with the ill-posedness of such problems. This paper shows that the optimal Tikhonov parameter is almost independent of the mesh used to project the functions to be retrieved. As a consequence, this value should be seeked using a coarse mesh even though reconstructions could further be done on finer meshes. This conclusion is validated by numerical means. [ABSTRACT FROM PUBLISHER]

Second discrepancy principle was found as an effective rule for choosing the regularization parameter in Tikhonov’s method applied for solving linear operator equations of the first kind. The rule is formulated as an additional non-linear equation involving second discrepancy to be satisfied by the regularized solution and regularization parameter together. Two constant parameters appearing in the definition of this criterion may be chosen freely within some limits, preserving asymptotic convergence properties of the method when errors in data tend to zero. This work presents a heuristic approach to determining both constants: they are found on the basis of numerical simulations exploiting realistic model representations with inexact data of problems which are to be solved and accounting for appropriate error levels. The approach is expressed as an approximate optimization problem defined on a class of inverse problems studied. The formulation of the method is given together with a numerical illustration. A set of Fredholm’s integral equations of the first kind with convolution-type operators is taken as an example. Inexact data imitate peaks from X-ray diffraction patterns realistically. The effectiveness of Tikhonov’s regularization with this version of second discrepancy principle is demonstrated on further examples. Weak sensitivity of the method to over- or under-estimation of errors in data is observed. It is concluded that only random errors in data (while discretization errors are omitted) can be accounted for solving the analysed inverse problems effectively. [ABSTRACT FROM PUBLISHER]

We consider an ill-posed equation in a Hilbert space with a noisy operator and a noisy right-hand side. The noise level information is given in a general form, as a norm of a certain operator applied to the noise. We derive the monotone error rule (ME-rule) for the choice of the regularization parameter in many methods, giving parameter such that the error is monotonically increasing for larger parameters in the Tikhonov method and for smaller stopping indices in iteration methods. Regularization methods considered include-scale regularization in (iterated) Tikhonov method and in iteration methods (Landweber method, CG type methods, semi-iterative methods). We also consider modifications of the ME-rule and show in numerical experiments (test problems from Hansen’s Regularization Toolbox, including the sideways heat equation) their advantages over the discrepancy principle. [ABSTRACT FROM PUBLISHER]

This study explores an application of the quasi-solution and the Fourier methods used to solve a mixed initial-boundary problem involving Laplace’s equation in a rectangle. The problem is reduced to a system consisting of mutually dependent direct and adjoint problems. The explicit formula for the quasi-solution is obtained using the Fourier method. Computational experiments are performed for different types of synthetic data, and admissible parameter ranges are established. [ABSTRACT FROM PUBLISHER]

In this article, we investigate some Cauchy problems associated with Helmholtz-type equations in rectangle using the method of truncation. Not alike the previous literature, we propose to choose the regularization parameter by ana posteriorirule, with which we can obtain error estimates of Hölder-type between the exact solution and their regularized approximation. Numerical simulations show that the proposed scheme is effective and stable. [ABSTRACT FROM PUBLISHER]

In this paper, we present results of reconstructions using real data from a new planar electrical impedance tomography device developed at the Institut für Physik, Johannes Gutenberg Universität, Mainz, Germany. The prototype consists of a planar sensing head of circular geometry, and it was designed mainly for breast cancer detection. There are 12 large outer electrodes arranged on a ring of radius cm where the external currents are injected, and a set of 54 point-like high-impedance inner electrodes where the induced voltages are measured. Two direct (i.e. non-iterative) reconstruction algorithms are considered: one is based on a discrete resistor model, and the other one is an integral equation approach for smooth conductivity distributions. [ABSTRACT FROM AUTHOR]

We describe a new hybrid-optimization method for solving the full-regularization problem of computing both the regularization parameter and the corresponding regularized solution in 1-norm and 2-norm Tikhonov regularizations with additional non-negativity constraints. The approach combines the simulated annealing technique for global optimization and the low-cost spectral projected gradient method for the minimization of large-scale smooth functions on convex sets. The new method is matrix-free in the sense that it relies on matrix–vector multiplications only. We describe the method and discuss some of its properties. Numerical results indicate that the new approach is a promising tool for solving large-scale image restoration problems. [ABSTRACT FROM AUTHOR]

We investigate level-set-type approaches for solving ill-posed inverse problems, under the assumption that the solution is a piecewise constant function. Our goal is to identify the level sets as well as the level values of the unknown parameter function. Two distinct level-set frameworks are proposed for solving the inverse problem. Among both of them, the level-set function is assumed to be in L 2. Corresponding Tikhonov regularization approaches are derived and analysed. Existence of minimizers for the Tikhonov functionals is proven. Moreover, convergence and stability results of the variational approaches are established, characterizing the Tikhonov approaches as regularization methods. [ABSTRACT FROM AUTHOR]

In this article, a new regularization method is proposed and applied to the identification of multi-source dynamic loads acting on a surface of composite laminated cylindrical shell. In general terms, regularization is the approximation of an ill-posed problem by a family of neighbouring well-posed problems. Based on the construction of a new regularization operator, corresponding regularization method is established. We propose a regularization operator, thus construct a new regularization method, and prove its regular property. The present method is applied to load identification of composite laminated cylindrical shell. The transient displacement response can be obtained by the finite element method. The multi-source dynamic loads on a surface of composite laminated cylindrical shell are successfully identified, which demonstrates the efficiency and robustness of the present method. [ABSTRACT FROM AUTHOR]

In this article, we present two reconstruction methods intended to be used for conductivity imaging with data obtained from a planar electrical impedance tomography device for breast cancer detection. The inverse problem to solve is different from the classical inverse conductivity problem. We reconstruct the electrical conductivity of a two-dimensional domain from boundary measurements of currents and interior measurements of the potential. One reconstruction algorithm is based on a discrete resistor model; the other one is an integral equation approach for smooth conductivity distributions. [ABSTRACT FROM AUTHOR]

In this article, we consider a classical ill-posed problem which is the backward problem for parabolic equation with time-dependent coefficient [image omitted] Applying the method of Fourier transform and the modified quasi-boundary value method, we shall give a regularization for the problem. A sharp error estimate between the regularized solution and the exact solution is given in this article. A numerical experiment is provided illustrating the main results. [ABSTRACT FROM AUTHOR]

A standard variational formulation for solving the linear operator equation Au = z is to compute the function u that minimizes |Au - z|L2(Ω). This is often an ill-posed problem that requires regularization, which is to say that there may exist infinitely many minimizers or that the solution may not be stable with respect to measurement errors. In many cases, however, the measured function z is contaminated by the noise of both Gaussian and Poisson types. If the Poisson noise is mathematically significant but not of high values, then solving the operator equation via the ubiquitous least-squares minimization problem may not produce an appropriate reconstruction, and the Poisson negative log-likelihood estimator may lead to a more accurate result. The Poisson likelihood minimization is also generally an ill-posed problem, and we require a regularization term to impose stability or to pick out which of the infinitely many minimizers is appropriate. Many possible regularizations have been analysed, including Tikhonov and total-variation regularizations, but in this work we perform the theoretical analysis to show that choosing the Poisson minimizer of smallest L1 norm leads to a theoretically well-posed problem. [ABSTRACT FROM AUTHOR]

In this article, we present a reconstruction method for the inverse conductivity problem suitable for smooth conductivity distributions. The inverse problem is reformulated in terms of a pair of coupled integral equations, one of which is of the first kind which we regularize using mollifier methods. An interesting feature of this method is that the kernel of this integral equation is not given, but can be modified for the choice of mollifier. We are able to obtain conductivity reconstructions rapidly and without relying on accurate a priori information. [ABSTRACT FROM AUTHOR]

The noise contained in images collected by a charge coupled device (CCD) camera is predominantly of Poisson type. This motivates the use of the negative-log Poisson likelihood function in place of the least-squares fit-to-data function. However, if the underlying mathematical model is assumed to have the form z = Au + γ, where z is the data and A is a compact operator and γ is the background light intensity, minimizing the negative-log Poisson likelihood function is an ill-posed problem, and hence some form of regularization is required. In previous work, the authors have performed theoretical analyses of two approaches for regularization in this setting: standard Tikhonov and total variation regularization. In this article, we consider a class of regularization functionals defined by differential operators of diffusion type, and our main results constitute a theoretical justification of this approach. However, in order to demonstrate that the approach is effective in practice, we follow our theoretical analysis with a numerical experiment. [ABSTRACT FROM AUTHOR]

The computation of Grunwald-Letnikov fractional derivatives from noisy data is considered as an ill-posed problem and treated by mollification techniques. It is shown that, with the appropriate choice of the radius of mollification, the method is a regularizing algorithm. Next, the recovery of the boundary temperature and heat flux functions from one measured transient temperature data at some interior point of a one-dimensional semi-infinite conductor when the governing diffusion equation is of fractional type is discussed. A simple algorithm based on space marching mollification techniques and Grunwald-Letnikov fractional derivatives is introduced for the numerical solution of this inverse ill-posed problem. In all cases, stability and error estimates are included together with numerical examples of interest. [ABSTRACT FROM AUTHOR]

An iterative scheme for the dynamical systems method (DSM) is given, such that one does not have to solve the Cauchy problem occuring in the application of the DSM for solving ill-conditioned Linear Algebraic Systems (LAS). The novelty of the algorithm is that the algorithm does not have to find the regularization parameter α by solving a nonlinear equation. Numerical experiments show that DSM competes favorably with the variational regularization. [ABSTRACT FROM AUTHOR]

The noise contained in images collected by a charge coupled device camera is predominantly of Poisson type. This motivates the use of the negative logarithm of the Poisson likelihood in place of the ubiquitous least squares fit-to-data. However if the underlying mathematical model is assumed to have the form z = Au, where A is a linear, compact operator, Poisson likelihood estimation is ill-posed, and hence some form of regularization is required. In Bardsley, J.M. and Vogel, C.R., 2004, A nonnegatively constrained convex programming method for image reconstruction. SIAM Journal on Scientific Computing, 25(4), pp. 1326-1343, a numerical method is presented and analyzed for Tikhonov regularized Poisson likelihood estimation, but no theoretical justification of the approach is given. Our primary objective in this article is to provide such a theoretical justification. We then briefly present the computational method of Bardsley, J.M. and Vogel, C.R., 2004, A nonnegatively constrained convex programming method for image reconstruction. SIAM Journal on Scientific Computing, 25(4), pp. 1326-1343, which is very effective and computationally efficient for this problem. The practical validity of the approach is then demonstrated on a synthetic example from astronomical imaging. [ABSTRACT FROM AUTHOR]