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This paper is concerned with the Cauchy problem connected with the Helmholtz equation. We investigate the denseness of Fourier–Bessel functions and propose a numerical method for approximating the solution to the problem. The convergence and stability are analyzed with truncated singular value decomposition method. Numerical experiments are also presented to show the effectiveness of our method. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Cauchy problem connected with the Helmholtz equation in a smooth-bounded domain. The Fourier–Bessel method with Tikhonov regularization is applied to achieve a regularized solution to the problem with noisy data. The convergence and stability are obtained with a suitable choice of the regularization parameter. Numerical experiments are also presented to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Cauchy problem connected with the Laplace equation in a smooth bounded domain. The harmonic polynomial method with Tikhonov regularization is applied to achieve a regularized solution to the problem with noisy data. The convergence and stability are obtained with a suitable choice of the regularization parameter. Numerical experiments are also presented to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper is concerned with the scattering problem of a polygonal-line arc. We solve this polygonal-line arc-scattering problem by a least-squares finite element method. In the method, Fourier–Bessel functions is used to capture the singularities around tips and corners. A combination of fundamental solutions is used to represent the scattered field towards infinity. We also analyse the convergence and give an error estimate of the method. Numerical experiments are also presented to show the effectiveness of our method. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Cauchy problem connected with the Laplace equation. We investigate the denseness of harmonic polynomials and propose a numerical method for approximating the solution to the problem. The convergence and stability are analyzed with a suitable choice of the regularization method. Numerical experiments are also presented to show the effectiveness of our method. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the Cauchy problem connected with the Helmholtz equation. On the basis of the denseness of Herglotz wavefunctions, we propose a numerical method for obtaining an approximate solution to the problem. We analyze the convergence and stability with a suitable choice of regularization method. Numerical experiments are also presented to show the effectiveness of our method. [Copyright &y& Elsevier]

The inverse problem considered in this article is to determine the shape of a two-dimensional time-harmonic acoustic scatterer with Dirichlet boundary conditions from the knowledge of some far field patterns. Based on the optimization method due to Kirsch and Kress for the inverse scattering problem, we propose a new scheme by reformulating the cost functional via a technique of piecewise integration with respect to incident directions. Convergence analysis of this method is given. Numerical experiments show that our method accelerates the computations without losing the accuracy of the reconstructions for the full-aperture problems. The method is extended to the limited-aperture case by weighting the total fields with special factors. Numerical examples for limited-aperture problems are also presented which show that our method produces satisfactory results efficiently in the illuminated regions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Fourier–Bessel method for the boundary value problems of the Helmholtz equation in a smooth simply connected domain. Based on the denseness of Fourier–Bessel functions, the problem can be approximated by determining the unknown coefficients in the linear combination. By the boundary conditions, an operator equation can be obtained. We derive a lower bound for the smallest singular value of the operator, and obtain a stability and convergence result for the regularized solution with a suitable choice of the regularization parameter. Numerical experiments are also presented to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In the context of Industry 4.0, the most popular way to identify and track objects is to add tags, and currently most companies still use cheap quick response (QR) tags, which can be positioned by computer vision (CV) technology. In CV, instance segmentation (IS) can detect the position of tags while also segmenting each instance. Currently, the mask region-based convolutional neural network (Mask R-CNN) method is used to realize IS, but the completeness of the instance mask cannot be guaranteed. Furthermore, due to the rich texture of QR tags, low-quality images can lower intersection-over-union (IoU) significantly, disabling it from accurately measuring the completeness of the instance mask. In order to optimize the IoU of the instance mask, a QR tag IS method named the mask UNet region-based convolutional neural network (MU R-CNN) is proposed. We utilize the UNet branch to reduce the impact of low image quality on IoU through texture segmentation. The UNet branch does not depend on the features of the Mask R-CNN branch so its training process can be carried out independently. The pre-trained optimal UNet model can ensure that the loss of MU R-CNN is accurate from the beginning of the end-to-end training. Experimental results show that the proposed MU R-CNN is applicable to both high- and low-quality images, and thus more suitable for Industry 4.0. [ABSTRACT FROM AUTHOR]