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In this paper, a meshless numerical scheme for solving an inverse source problem is considered. The proposed scheme is based on approximating the solution employing the thin plate spline (TPS) radial basis function (RBF). Applying this radial basis function results in a badly ill-condition system of equations. The Tikhonov regularization method is employed for solving this system of equations. Determination of regularization parameter is based on generalized cross-validation (GCV) technique. Some numerical examples are presented to demonstrate the accuracy and ability of this method. [ABSTRACT FROM AUTHOR]

In this paper, we study the inverse problem reconstruction approach for parallel-beam projection in-scheme short-scan SPECT image and obtain approximate reconstruction algorithms if the attenuation is real constant and complex-valued angle-dependent cases and the parallel-beam projection data functions are acquired from-scheme short-scan single-photon emission computed tomography. Finally, we also give an numerical implementation. [ABSTRACT FROM AUTHOR]

We consider towed electrical sensing for detecting and localizing small inhomogeneities in the marine environment. Assuming the domain to be homogeneous apart from a few dispersed inclusions, the low-frequency electrical measurements can be modeled using a single-layer potential formulation for a source function defined at the boundaries of the inclusions. A key component of these measurements is the potential induced by the polarization of the inclusions, which at the far field can be shown to be equivalent to the potential of dipole sources centered at the inclusions. Under this approximation, we formulate an inverse problem for localizing the inclusions and then enforce some regularization in the form of an a priori assumption on the shape of the inclusions. In this context, solving the inverse problem requires tracing some coordinates where the polarization potential at the current injecting electrodes becomes zero since these define a set of lines intersecting at the center of the targeted inclusions. This methodology is implemented by a simple algorithm, whose computational complexity mounts to solving a small number of low-dimensional linear systems. Analysis indicates fair robustness of the algorithm to measurement noise and model inaccuracies, and this is also supported by numerical simulation experiments. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper we propose a variation of the soft-thresholding algorithm for finding sparse approximate solutions of the equation , where as the sparsity of the iterate increases the penalty function changes. In this approach, sufficiently large entries in a sparse iterate are left untouched. The advantage of this approach is that a higher regularization constant can be used, leading to a significant reduction of the total number of iterations. Numerical experiments for sparse recovery problems, also with noisy data, are included. [Copyright &y& Elsevier]

Abstract: In this paper we present a technique for shape reconstruction based on the topological and shape gradients. The shape in consideration is a solution of an inverse conductivity problem. To solve such a problem numerically, we compute the topological gradient of a Kohn–Vogelius-type cost function when the domain under consideration is perturbed by the introduction of a small inclusion instead of a hole. The reconstruction is done by considering the shape as a superposition of very thin elliptic inclusions to get a first approximation. Then, we use a gradient-type algorithm to perform a good reconstruction. Various numerical experiments of single and multiple inclusions demonstrate the viability of the designed algorithm. [Copyright &y& Elsevier]

Abstract: Image deblurring is a basic and important task of image processing. Traditional filtering based image deblurring methods, e.g. enhancement filters, partial differential equation (PDE) and etc., are limited by the hypothesis that natural images and noise are with low and high frequency terms, respectively. Noise removal and edge protection are always the dilemma for traditional models. In this paper, we study image deblurring problem from a brand new perspective—classification. And we also generalize the traditional PDE model to a more general case, using the theories of calculus of variations. Furthermore, inspired by the theories of approximation of functions, we transform the operator-learning problem into a coefficient-learning problem by means of selecting a group of basis, and build a filter-learning model. Based on extreme learning machine (ELM) , an algorithm is designed and a group of filters are learned effectively. Then a generalized image deblurring model, learned filtering PDE (LF-PDE), is built. The experiments verify the effectiveness of our models and the corresponding learned filters. It is shown that our model can overcome many drawbacks of the traditional models and achieve much better results. [Copyright &y& Elsevier]

Among inverse problems for partial differential equations, we distinguish coefficient inverse problems, which are associated with the identification of coefficients and/or the right-hand side of an equation using some additional information. When considering time-dependent problems, the identification of the coefficient dependences on space and on time is usually separated into individual problems. In some cases, we have linear inverse problems (e.g. identification problems for the right-hand side of an equation); this situation essentially simplify their study. This work deals with the problem of determining in a multidimensional parabolic equation the lower coefficient that depends on time only. To solve numerically a non-linear inverse problem, linearized approximations in time are constructed using standard finite difference approximations in space. The computational algorithm is based on a special decomposition, where the transition to a new time level is implemented via solving two standard elliptic problems. [ABSTRACT FROM PUBLISHER]

A numerical method for an inverse problem for an elliptic equation with the running source at multiple positions is presented. This algorithm does not rely on a good first guess for the solution. The so-called ‘approximate global convergence’ property of this method is shown here. The performance of the algorithm is verified on real data for Diffusion Optical Tomography. Direct applications are in near-infrared laser imaging technology for stroke detection in brains of small animals. [ABSTRACT FROM PUBLISHER]

The article contains an exposition of the basic idea of construction of numerical algorithms based upon the several local approximations by linear polynomials for every sought-for function of dynamic problems of solids. We discuss also some problems of determination of physical and geometrical characteristics of layered inhomogeneous medium. [ABSTRACT FROM AUTHOR]