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The Cauchy problem for the Helmholtz equation is considered. It was demonstrated in a previous paper by the authors that the alternating algorithm suggested by V.A. Kozlov and V.G. Maz’ya does not converge for large wavenumbers k in the Helmholtz equation. Here, we present some simple modifications of the algorithm which may restore the convergence. They consist of the replacement of the Neumann-Dirichlet iterations by the Robin-Dirichlet ones which repairs the convergence for less than the first Dirichlet-Laplacian eigenvalue. In order to treat large wavenumbers, we present an algorithm based on iterative solution of Robin-Dirichlet boundary value problems in a sufficiently narrow border strip. Numerical implementations obtained using the finite difference method are presented. The numerical results illustrate that the algorithms suggested in this paper, produce convergent iterative sequences. [ABSTRACT FROM AUTHOR]

To find 3D similarity transformation parameters (a scale, three rotational angles, and three translation elements) between two orthogonal coordinate systems in 3D is an ill-posed non-linear inverse problem by means of common points (their Cartesian components are known in the both systems). The problem can be solved via Linearized Least Squares (LLS) or Direct (non-iterative) Least Squares (DLS). Since the parameters in LLS take different quantities (and units) from each other, the condition problems can arise during the solution of normal equations. In this paper, we propose a combined solution to reducing ill-conditioning and to perform precision analysis and global outlier test in LLS accordingly. The way is based on column norming and uses the normalized unknowns instead of the original ones at the solution stage of the normal equations. While the global outlier test is fulfilled on the normalized unknowns, the original unknowns and their precisions obtained using the normalized matrix with linear transformation to the normalized unknowns and by the error propagation law to their variance-covariance matrix. A direct method (Direct Least Squares, DLS) is used to compute the initial values of LLS for reducing iteration number (to 1-3 cycles) with a 1.E-7 threshold in the paper. And, when the solution of 3D-ST having large rotational angles and uncertainties, it is also shown that DLS goes to a worse solution than LLS by means of a simulated transformation problem. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel class of iterative reconstruction methods for severely angular undersampled or/and limited-view tomographic problems with fan-beam scanning geometry. The proposed algorithms are based on a new analytical transform which generalizes Fourier-slice theorem to divergent-beam scanning geometries. Using a non-rigid coordinate transform, divergent rays can be reorganized into parallel ones. Therefore, one can employ a simpler parallel-beam projection model instead of more complicated divergent-beam geometries. Various existing iterative reconstruction techniques for divergent-beam geometries can be easily adapted to the proposed framework. The significant advantage of this formulation is the possibility of exploiting efficient Fourier-based recovery methods without rebinning of the projections. In case of highly sparse measurements (few-view data), rebinning methods are not suitable due to error-prone angular interpolation involved. In this work, three new methods based on the novel analytical framework for fan-beam geometry are presented: the Gerchberg-Papoulis algorithm, the Neumann decomposition method and its total variation regularized version. Presented numerical experiments demonstrate that the methods can be competitive in reconstructing from few-view noisy tomographic measurements. [ABSTRACT FROM AUTHOR]

In this paper, a non-iterative technique is proposed for the transverse load identification on Kirchhoff plates using approximate Green’s functions (AGFs). In this way, we firstly employ the recently introduced meshless method to construct the AGFs, as the combination of a series of Trefftz basis, i.e. Exponential basis functions (EBFs), and the fundamental solutions of the governing equation. As will be explained, using a proper set of EBFs, as well as a collocation technique, enables us to construct the AGFs for different types of domain shape and boundary conditions. In the second step, a set of artificially generated results, in the absence of realistic experimental results, are used to express the plate’s response field, i.e. deflection or velocity fields, as a series of AGFs through a collocation technique. It will be shown how the constant coefficients of the response series are related to the intensity of the reconstructed force at a set of selected points. The proposed method is capable of constructing both distributed and concentrated loads with desirable accuracy. This ability is shown in the solution of three sample problems of the static and time-harmonic force recovery. [ABSTRACT FROM PUBLISHER]

Dual-energy CT can be represented as the dual-energy equations by decomposing the linear attenuation coefficient of the X-ray scanned object into two material basis functions of photoelectric absorption and Compton scatter. To solve the dual-energy equations, in this paper, we apply the mean-value theorem for integrals and propose a new projection-based iterative algorithm. We discuss the convergence of the proposed algorithm and carry out various numerical simulations for demonstrating its feasibility. [ABSTRACT FROM PUBLISHER]

Iterative re-weighted least square (IRLS) algorithms for-minimization problems require to select proper value of regularization parameter, for which Stein’s unbiased risk estimate (SURE) – an unbiased estimate of prediction error – is often used as a criterion for this selection. In this paper, we propose a recursive SURE to estimate the prediction error during the IRLS iterations. Particularly, we derive the recursion of Jacobian matrix by incorporating matrix splitting scheme into IRLS algorithms. Numerical examples demonstrate that minimizing SURE consistently leads to the nearly optimal reconstructions in terms of prediction error. Theoretical derivations in this work related to the evaluation of Jacobian matrix can be extended, in principle, to other types of regularizers and regularized iterative reconstruction algorithms. [ABSTRACT FROM AUTHOR]

Inverse shape design in the context of fluid flow problems is commonly referred to the determination of the boundary shape corresponding to a given target surface pressure. Designers naturally turn to this class of problems whenever there are concerns regarding pressure-related phenomena such as cavitation, separation, shock waves, surface loading, etc. Numerical solution is often unavoidable and, therefore, three computational tools, i.e. a grid generator, a flow field solver and a shape updater, are required in an iterative solution procedure. In all existing iterative inverse design methods, the shape updater comes from separate mathematical or physical considerations that are not derived solely from the equations governing the flow field. In this paper, the currently used strategies to solve inverse shape design problems are categorized and reviewed, and then a truly physical-based iterative inverse shape design method is introduced in which the governing equations are not only used in the flow solver, but are also employed to update the shape. To explore the features of the proposed method, it is used to solve a number of shape design problems. The previously developed direct shape design method and a typical iterative solution approach are also explained and used for the solution of the same problems for the purpose of comparison. Computational results reveal that the proposed algorithm has a two to three times faster convergence rate as compared to the iterative algorithm. The convergence rate of the proposed algorithm usually stalls after three or two orders of magnitude reduction of the residual. For the test cases in this study, the direct design method is the fastest and most accurate method as compared to other algorithms. Both in-house developed computer codes and commercial software are used as a field solver to show the general applicability of the proposed method in its current state of development. [ABSTRACT FROM PUBLISHER]

Bregman iterative algorithms have been extensively used forand total variation regularization problems, allowing to obtain simple, fast and effective algorithms. In this paper, three already-available algorithms for geosounding inversion are modified by including them in a Bregman iterative procedure. The resulting algorithms are easy to implement and do not require any optimization package. Modelling results are presented for synthetic and field data, observing better convergence properties than the original versions, avoiding the need of any continuation descent procedure. [ABSTRACT FROM AUTHOR]

In this paper, we apply an iterative regularization technique based on Newton’s method to the ill-posed problem of parameter identification in a differential equation describing corneal surface. This equation is based on our previously derived model of corneal topography. We prove the convergence of the proposed method along with some stability estimates. When compared with real corneal data, this method gives reasonable results with error of the same order of magnitude as is introduced by measuring equipment in data acquisition. Moreover, the iteration converges very quickly to the parameters associated with the intra-ocular pressure and the elasticity of the cornea. [ABSTRACT FROM AUTHOR]

The paper presents the iterative solutions of two coefficient inverse problems (CIPs) arising in frequency sounding and electrical prospecting. An iterative algorithm is constructed to obtain such solutions. Exploiting the Beilina–Klibanov approach to CIPs, this algorithm possesses the new iterative and refinement procedures. These features enhance significantly both the spatial and contrast resolutions of reconstructed coefficients. The computational effectiveness of the proposed numerical technique is demonstrated in computational experiments with two applied CIPs: electromagnetic or acoustic frequency sounding and electrical prospecting of layered media. The Slichter–Langer–Tikhonov formulation is exploited as a mathematical model of the latter. [ABSTRACT FROM PUBLISHER]