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Journal articles on the topic 'Ill-posed Helmholtz equation'

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Published: 3 June 2025

Last updated: 31 July 2025

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1

Benedict, Barnes, O. Boateng F., K. Amponsah S., and Osei-Frimpong E. "On the Notes of Quasi-Boundary Value Method for Solving both Cauchy-Dirichlet Problem of the Helmholtz Equation." British Journal of Mathematics & Computer Science 22, no. 2 (2017): 1–10. https://doi.org/10.9734/BJMCS/2017/32727.

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The Cauchy-Dirichlet problem of the Helmholtz equation yields unstable solution, which when solved with the Quasi-Boundary Value Method (Q-BVM) for a regularization parameter = 0. At this point of regularization parameter, the solution of the Helmholtz equation with both Cauchy and Dirichlet boundary conditions is unstable when solved with the Q-BVM. Thus, the quasi-boundary value method is insufficient and inefficient for regularizing ill-posed Helmholtz equation with both Cauchy and Dirichlet boundary conditions. In this paper, we introduce an expression 1 (1+ 2) ; ∈ R, where is the regulari

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2

Kabanikhin, Sergey Igorevich, M. A. Shishlenin, D. B. Nurseitov, A. T. Nurseitova, and S. E. Kasenov. "Comparative Analysis of Methods for Regularizing an Initial Boundary Value Problem for the Helmholtz Equation." Journal of Applied Mathematics 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/786326.

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We consider an ill-posed initial boundary value problem for the Helmholtz equation. This problem is reduced to the inverse continuation problem for the Helmholtz equation. We prove the well-posedness of the direct problem and obtain a stability estimate of its solution. We solve numerically the inverse problem using the Tikhonov regularization, Godunov approach, and the Landweber iteration. Comparative analysis of these methods is presented.

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3

Barnes, Benedict, Anthony Y. Aidoo, and Joseph Ackora-Prah. "Using a Divergence Regularization Method to Solve an Ill-Posed Cauchy Problem for the Helmholtz Equation." Abstract and Applied Analysis 2022 (March 29, 2022): 1–10. http://dx.doi.org/10.1155/2022/4628634.

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The ill-posed Helmholtz equation with inhomogeneous boundary deflection in a Hilbert space is regularized using the divergence regularization method (DRM). The DRM includes a positive integer scaler that homogenizes the inhomogeneous boundary deflection in the Helmholtz equation’s Cauchy issue. This guarantees the existence and uniqueness of the equation’s solution. To reestablish the stability of the regularized Helmholtz equation and regularized Cauchy boundary conditions, the DRM uses its regularization term 1 + α 2 m e m , where α > 0 is the regularization parameter. As a result, DRM re

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4

Chen, Yong-Gang, Fan Yang, and Qian Ding. "The Landweber Iterative Regularization Method for Solving the Cauchy Problem of the Modified Helmholtz Equation." Symmetry 14, no. 6 (2022): 1209. http://dx.doi.org/10.3390/sym14061209.

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In this manuscript, the Cauchy problem of the modified Helmholtz equation is researched. This inverse problem is a serious ill-posed problem. The classical Landweber iterative regularization method is designed to find the regularized solution of this inverse problem. The error estimations between the exact solution and the regularization solution are all obtained under the a priori and the a posteriori regularization parameter selection rule. The Landweber iterative regularization method can also be applied to solve the Cauchy problem of the modified Helmholtz equation on the spherically symme

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5

Kashirin, A. A., and S. I. Smagin. "On the solvability on the spectrum of Fredholm boundary integral equations of the first kind for the three-dimensional transmission problem." Дифференциальные уравнения 60, no. 2 (2024): 211–23. http://dx.doi.org/10.31857/s0374064124020054.

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The paper considers two weakly singular Fredholm boundary integral equations of the first kind, to each of which the three-dimensional Helmholtz transmission problem can be reduced. The properties of these equations are studied on spectra, where they are ill-posed. For the first equation, it is shown that if its solution exists on the spectrum, it allows us to find a solution to the transmission problem. The second equation in this case always has infinitely many solutions, only one of which gives a solution to the transmission problem. The interpolation method for finding approximate solution

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6

Juraev, Davron Aslonqulovich, and Samad Noeiaghdam. "Regularization of the Ill-Posed Cauchy Problem for Matrix Factorizations of the Helmholtz Equation on the Plane." Axioms 10, no. 2 (2021): 82. http://dx.doi.org/10.3390/axioms10020082.

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In this paper, we present an explicit formula for the approximate solution of the Cauchy problem for the matrix factorizations of the Helmholtz equation in a bounded domain on the plane. Our formula for an approximate solution also includes the construction of a family of fundamental solutions for the Helmholtz operator on the plane. This family is parameterized by function K(w) which depends on the space dimension. In this paper, based on the results of previous works, the better results can be obtained by choosing the function K(w).

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7

Dou, Fang-Fang, and Chu-Li Fu. "A Wavelet Method for the Cauchy Problem for the Helmholtz Equation." ISRN Applied Mathematics 2012 (January 4, 2012): 1–18. http://dx.doi.org/10.5402/2012/435468.

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We consider a Cauchy problem for the Helmholtz equation at a fixed frequency. The problem is severely ill posed in the sense that the solution (if it exists) does not depend continuously on the data. We present a wavelet method to stabilize the problem. Some error estimates between the exact solution and its approximation are given, and numerical tests verify the efficiency and accuracy of the proposed method.

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8

He, Shangqin, and Xiufang Feng. "A mollification method with Dirichlet kernel to solve Cauchy problem for two-dimensional Helmholtz equation." International Journal of Wavelets, Multiresolution and Information Processing 17, no. 05 (2019): 1950029. http://dx.doi.org/10.1142/s0219691319500292.

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In this paper, the ill-posed Cauchy problem for the Helmholtz equation is investigated in a strip domain. To obtain stable numerical solution, a mollification regularization method with Dirichlet kernel is proposed. Error estimate between the exact solution and its approximation is given. A numerical experiment of interest shows that our procedure is effective and stable with respect to perturbations of noise in the data.

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9

Yang, Fan, Ping Fan, and Xiao-Xiao Li. "Fourier Truncation Regularization Method for a Three-Dimensional Cauchy Problem of the Modified Helmholtz Equation with Perturbed Wave Number." Mathematics 7, no. 8 (2019): 705. http://dx.doi.org/10.3390/math7080705.

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In this paper, the Cauchy problem of the modified Helmholtz equation (CPMHE) with perturbed wave number is considered. In the sense of Hadamard, this problem is severely ill-posed. The Fourier truncation regularization method is used to solve this Cauchy problem. Meanwhile, the corresponding error estimate between the exact solution and the regularized solution is obtained. A numerical example is presented to illustrate the validity and effectiveness of our methods.

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10

Li, Xiao-Xiao, Fan Yang, Jie Liu, and Lan Wang. "The Quasireversibility Regularization Method for Identifying the Unknown Source for the Modified Helmholtz Equation." Journal of Applied Mathematics 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/245963.

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This paper discusses the problem of determining an unknown source which depends only on one variable for the modified Helmholtz equation. This problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. The regularization solution is obtained by the quasireversibility regularization method. Convergence estimate is presented between the exact solution and the regularization solution. Moreover, numerical results are presented to illustrate the accuracy and efficiency of this method.

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11

He, Shangqin, and Xiufang Feng. "A Regularization Method to Solve a Cauchy Problem for the Two-Dimensional Modified Helmholtz Equation." Mathematics 7, no. 4 (2019): 360. http://dx.doi.org/10.3390/math7040360.

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In this paper, the ill-posed problem of the two-dimensional modified Helmholtz equation is investigated in a strip domain. For obtaining a stable numerical approximation solution, a mollification regularization method with the de la Vallée Poussin kernel is proposed. An error estimate between the exact solution and approximation solution is given under suitable choices of the regularization parameter. Two numerical experiments show that our procedure is effective and stable with respect to perturbations in the data.

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12

Yang, Fan, HengZhen Guo, and XiaoXiao Li. "The Simplified Tikhonov Regularization Method for Identifying the Unknown Source for the Modified Helmholtz Equation." Mathematical Problems in Engineering 2011 (2011): 1–14. http://dx.doi.org/10.1155/2011/953492.

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This paper discusses the problem of determining an unknown source which depends only on one variable for the modified Helmholtz equation. This problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. The regularization solution is obtained by the simplified Tikhonov regularization method. Convergence estimate is presented between the exact solution and the regularization solution. Moreover, numerical results are presented to illustrate the accuracy and efficiency of this method.

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13

Wang, R., J. Chen, and GM Dong. "A new data extrapolation method based on the modified Helmholtz equation least-squares method." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 227, no. 6 (2012): 1252–65. http://dx.doi.org/10.1177/0954406212458385.

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Fast Fourier transform-based near-field acoustic holography requires that the measurement aperture should completely enclose the source, which is impractical for large-scale sound sources. Helmholtz equation least-squares method can reconstruct the acoustic field with fewer measurements than fast Fourier transform-based near-field acoustic holography. However, it is not suitable for reconstructing acoustic radiation from multiple sources or a complicated source which consists of several separated parts. To circumvent this difficulty and enhance the reconstruction accuracy, a new data extrapola

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14

Daoudi, Jamal, and Chakir Tajani. "Solving the Cauchy Problem Related to the Helmholtz Equation through a Genetic Algorithm." WSEAS TRANSACTIONS ON MATHEMATICS 22 (October 9, 2023): 719–29. http://dx.doi.org/10.37394/23206.2023.22.79.

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The Cauchy problem associated with the Helmholtz equation is an ill-posed inverse problem that is challenging to solve due to its instability and sensitivity to noise. In this paper, we propose a metaheuristic approach to solve this problem using Genetic Algorithms in conjunction with Tikhonov regularization. Our approach is able to produce stable, convergent, and accurate solutions for the Cauchy problem, even in the presence of noise. Numerical results on both regular and irregular domains show the effectiveness and accuracy of our approach.

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15

Xu, Huilin, Baoxia Wang, and Duanmei Zhou. "A General Mollification Regularization Method to Solve a Cauchy Problem for the Multi-Dimensional Modified Helmholtz Equation." Symmetry 16, no. 11 (2024): 1549. http://dx.doi.org/10.3390/sym16111549.

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This paper considers a Cauchy problem for the multi-dimensional modified Helmholtz equation with inhomogeneous Dirichlet and Neumann data. The Cauchy problem is severely ill-posed, and a general mollification method is introduced to solve the problem. Both the a priori and a posteriori choice strategies of the regularization parameter are proposed, and error estimations of the corresponding regularization solutions are also presented. Finally, two numerical examples are introduced to show the effectiveness of the general mollification regularization method.

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16

Lobato, Thiago, Roland Sottek, and Michael Vorländer. "Using learned priors to regularize the Helmholtz equation least-squares method." Journal of the Acoustical Society of America 155, no. 2 (2024): 971–83. http://dx.doi.org/10.1121/10.0024726.

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The Helmholtz equation least-squares (HELS) method is a valuable tool for estimating equivalent sound sources of a radiating object. It solves an inverse problem by mapping measured pressures to a set of basis functions satisfying the Helmholtz equation in spherical coordinates. However, this problem is often ill-posed, necessitating additional regularization methods, in which often variations of Ridge or Lasso are used. These conventional methods do not explicitly consider the distribution underlying the source radiations (besides sparsity) and are often used in the context of obtaining only

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17

Zhang, Hongwu, and Xiaoju Zhang. "Generalized Tikhonov Method and Convergence Estimate for the Cauchy Problem of Modified Helmholtz Equation with Nonhomogeneous Dirichlet and Neumann Datum." Mathematics 7, no. 8 (2019): 667. http://dx.doi.org/10.3390/math7080667.

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We investigate a Cauchy problem of the modified Helmholtz equation with nonhomogeneous Dirichlet and Neumann datum, this problem is ill-posed and some regularization techniques are required to stabilize numerical computation. We established the result of conditional stability under an a priori assumption for an exact solution. A generalized Tikhonov method is proposed to solve this problem, we select the regularization parameter by a priori and a posteriori rules and derive the convergence results of sharp type for this method. The corresponding numerical experiments are implemented to verify

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18

Kuo, Chung-Lun, Weichung Yeih, Chein-Shan Liu, and Jiang-Ren Chang. "Solving Helmholtz equation with high wave number and ill-posed inverse problem using the multiple scales Trefftz collocation method." Engineering Analysis with Boundary Elements 61 (December 2015): 145–52. http://dx.doi.org/10.1016/j.enganabound.2015.07.015.

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19

Xiao, Youhong, Qingqing Song, Shaowei Li, Guoxue Lv, and Zhenlin Ji. "Comparisons of regularization methods and regularization parameter selection methods in sound source identification using inverse boundary element method." Noise Control Engineering Journal 67, no. 3 (2019): 219–27. http://dx.doi.org/10.3397/1/376720.

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In noise source identification based on the inverse boundary element method (IBEM), the boundary vibration velocity is predicted based on the field pressure through a transfer matrix of the vibration velocity and field pressure established on the Helmholtz integral equation. Because the matrix is often ill-posed, it needs to be regularized before reconstructing the vibration velocity. Two regularization methods and two methods of selecting the regularization parameter are investigated through the simulation analysis of a pulsating sphere. The result of transfer matrix regularization is further

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20

He, Shangqin, Congna Di, and Li Yang. "The mollification method based on a modified operator to the ill-posed problem for 3D Helmholtz equation with mixed boundary." Applied Numerical Mathematics 160 (February 2021): 422–35. http://dx.doi.org/10.1016/j.apnum.2020.10.012.

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21

BONNET-BEN DHIA, ANNE-SOPHIE, LUCAS CHESNEL, and XAVIER CLAEYS. "RADIATION CONDITION FOR A NON-SMOOTH INTERFACE BETWEEN A DIELECTRIC AND A METAMATERIAL." Mathematical Models and Methods in Applied Sciences 23, no. 09 (2013): 1629–62. http://dx.doi.org/10.1142/s0218202513500188.

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We study a 2D scalar harmonic wave transmission problem between a classical dielectric and a medium with a real-valued negative permittivity/permeability which models an ideal metamaterial. When the interface between the two media has a corner, according to the value of the contrast (ratio) of the physical constants, this non-coercive problem can be ill-posed (not Fredholm) in H 1. This is due to the degeneration of the two dual singularities which then behave like r±iη = e±iη ln r with η ∈ ℝ*. This apparition of propagative singularities is very similar to the apparition of propagative modes

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22

Ambrose, David M. "Regularization of the Kelvin–Helmholtz instability by surface tension." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 365, no. 1858 (2007): 2253–66. http://dx.doi.org/10.1098/rsta.2007.2006.

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The Kelvin–Helmholtz instability is present in the motion of a vortex sheet without surface tension. This can be seen from the linearization of the equations of motion, and there have also been proofs of ill-posedness for the full nonlinear equations. In the presence of surface tension, the linearized equations no longer exhibit an instability, and it has been believed that the full equations should then be well-posed. In this paper, I sketch a proof that the vortex sheet with surface tension is well-posed in the case of both two- and three-dimensional fluids. The proof in the case of three-di

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23

Stewart, Andrew L., and Paul J. Dellar. "Multilayer shallow water equations with complete Coriolis force. Part 3. Hyperbolicity and stability under shear." Journal of Fluid Mechanics 723 (April 16, 2013): 289–317. http://dx.doi.org/10.1017/jfm.2013.121.

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AbstractWe analyse the hyperbolicity of our multilayer shallow water equations that include the complete Coriolis force due to the Earth’s rotation. Shallow water theory represents flows in which the vertical shear is concentrated into vortex sheets between layers of uniform velocity. Such configurations are subject to Kelvin–Helmholtz instabilities, with arbitrarily large growth rates for sufficiently short-wavelength disturbances. These instabilities manifest themselves through a loss of hyperbolicity in the shallow water equations, rendering them ill-posed for the solution of initial value

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24

Nechita, Mihai. "Solving ill-posed Helmholtz problems with physics-informed neural networks." Journal of Numerical Analysis and Approximation Theory, July 10, 2023. http://dx.doi.org/10.33993/jnaat521-1305.

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We consider the unique continuation (data assimilation) problem for the Helmholtz equation and study its numerical approximation based on physics-informed neural networks (PINNs). Exploiting the conditional stability of the problem, we first give a bound on the generalization error of PINNs. We then present numerical experiments in 2d for different frequencies and for geometric configurations with different stability bounds for the continuation problem. The results show that vanilla PINNs provide good approximations even for noisy data in configurations with robust stability (both low and mode

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25

Klumpp, Maximilian, and Guido Schneider. "A note on the validity of the Schrödinger approximation for the Helmholtz equation." Journal of Applied Analysis, September 3, 2021. http://dx.doi.org/10.1515/jaa-2021-2058.

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Abstract Time-harmonic electromagnetic waves in vacuum are described by the Helmholtz equation Δ ⁢ u + ω 2 ⁢ u = 0 for ⁢ ( x , y , z ) ∈ ℝ 3 . \Delta u+\omega^{2}u=0\quad\text{for }(x,y,z)\in{\mathbb{R}}^{3}. For the evolution of such waves along the z-axis, a Schrödinger equation can be derived through a multiple scaling ansatz. It is the purpose of this paper to justify this formal approximation by proving bounds between this formal approximation and true solutions of the original system. The challenge of the presented validity analysis is the fact that the Helmholtz equation is ill-posed as

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26

Arendt, W., and T. Regińska. "An ill-posed boundary value problem for the Helmholtz equation on Lipschitz domains." Journal of Inverse and Ill-posed Problems 17, no. 7 (2009). http://dx.doi.org/10.1515/jiip.2009.041.

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27

Regińska, Teresa, and Kazimierz Regiński. "Regularization strategy for determining laser beam quality parameters." Journal of Inverse and Ill-posed Problems 23, no. 6 (2015). http://dx.doi.org/10.1515/jiip-2014-0084.

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AbstractA simplified model of a laser beam leads to an ill-posed Cauchy problem for the Helmholtz equation on an infinite strip. In the case of large wave numbers, the problem corresponds to an operator equation with an unbounded operator. The first problem considered concerns optimality of a spectral type regularization method for reconstructing the radiation field from measurements given only on a part of the boundary. The optimal order of convergence, previously known for particular cases, is proved for an arbitrary wave number and for nonzero Dirichlet and Neumann conditions under

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28

Achieng, Pauline, Fredrik Berntsson, and Vladimir Kozlov. "Reconstruction of the Radiation Condition and Solution for the Helmholtz Equation in a Semi-infinite Strip from Cauchy Data on an Interior Segment." Computational Methods in Applied Mathematics, July 25, 2023. http://dx.doi.org/10.1515/cmam-2022-0244.

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Abstract We consider an inverse problem for the Helmholtz equation of reconstructing a solution from measurements taken on a segment inside a semi-infinite strip. Homogeneous Neumann conditions are prescribed on both side boundaries of the strip and an unknown Dirichlet condition on the remaining part of the boundary. Additional complexity is that the radiation condition at infinity is unknown. Our aim is to find the unknown function in the Dirichlet boundary condition and the radiation condition. Such problems appear in acoustics to determine acoustical sources and surface vibrations from aco

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29

Jday, Fadhel, and Haithem Omri. "Adaptive Runge–Kutta regularization for a Cauchy problem of a modified Helmholtz equation." Journal of Inverse and Ill-posed Problems, March 18, 2023. http://dx.doi.org/10.1515/jiip-2020-0014.

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Abstract In this paper, we investigate the Cauchy problem for the modified Helmholtz equation. We consider the data completion problem in a bounded cylindrical domain on which the Neumann and the Dirichlet conditions are given in a part of the boundary. Since this problem is ill-posed, we reformulate it as an optimal control problem with an appropriate cost function. The method of factorization of boundary value problems is used to immediately obtain an approximation of the missing boundary data. In order to regularize this problem, we firstly scrutinize two classical regularizations for the c

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30

Chen, Lingguang, and Sean F. Wu. "A Modified Helmholtz Equation Least Squares Method for Reconstructing Vibroacoustic Quantities on an Arbitrarily Shaped Vibrating Structure." Journal of Theoretical and Computational Acoustics, June 24, 2021, 2150006. http://dx.doi.org/10.1142/s2591728521500067.

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A modified Helmholtz equation least-square (HELS) method is developed to reconstruct vibroacoustic quantities on an arbitrarily shaped vibrating structure. Unlike the traditional nearfield acoustical holography that relies on the acoustic pressures collected on a hologram surface at a short stand-off distance to a target structure, this modified HELS method takes the partial normal surface velocities and partial acoustic pressures as the input data. The advantages of this approach include but not limited to: (1) The normal surface velocities that represent the nearfield effects are collected d

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31

Kow, Pu-Zhao, and Jenn-Nan Wang. "Refined instability estimates for some inverse problems." Inverse Problems and Imaging, 2022, 0. http://dx.doi.org/10.3934/ipi.2022017.

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<p style='text-indent:20px;'>Many inverse problems are known to be ill-posed. The ill-posedness can be manifested by an instability estimate of exponential type, first derived by Mandache [<xref ref-type="bibr" rid="b29">29</xref>]. In this work, based on Mandache's idea, we refine the instability estimates for two inverse problems, including the inverse inclusion problem and the inverse scattering problem. Our aim is to derive explicitly the dependence of the instability estimates on key parameters.</p><p style='text-indent:20px;'>The first result of this work is

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32

Katsiavria, Anna, and Demetrios T. Papageorgiou. "Stability analysis of viscous multi-layer shear flows with interfacial slip." IMA Journal of Applied Mathematics, June 13, 2024. http://dx.doi.org/10.1093/imamat/hxae012.

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Abstract One of the most fundamental interfacial instabilities in ideal, immiscible, incompressible multifluid flows is the celebrated Kelvin-Helmholtz (KH) instability. It predicts short-wave instabilities that in the absence of other mollifying physical mechanisms (e.g. surface tension, viscosity) render the nonlinear problem ill-posed and lead to finite-time singularities. The crucial driving mechanism is the jump in tangential velocity across the liquid-liquid interface, i.e. interfacial slip, that can occur since viscosity is absent. The purpose of the present work is to analyse analogous

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33

Wu, Ziku, Xiaoming Han, and GuoFeng Li. "Learning solutions to a Cauchy problem for the modified Helmholtz equations using LS-SVM." Engineering Computations ahead-of-print, ahead-of-print (2020). http://dx.doi.org/10.1108/ec-04-2019-0168.

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Purpose The purpose of this paper is to develop a mesh-free algorithm based on the least square support vector machines method for numerical simulation of the modified Helmholtz equations. Design/methodology/approach The proposed method deals with a Cauchy problem for the modified Helmholtz equations. The algorithm converts the problem into a quadratic programming. It can be divided into three steps. First, some training points are allocated. Then, an approximate function is constructed. Finally, the shape parameters are estimated. Findings The proposed method's stability is discussed. Numeric

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Abdalkaleg Atia Idris Hamad. "Numerical Experimental for an Inverse Source Problem Method." Libyan Journal of Basic Sciences, August 25, 2021, 125–39. http://dx.doi.org/10.36811/ljbs.2021.110074.

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This paper examines extensions of an iterative method for inverse evaluation of the source function for two elliptic systems. The method begins with a starting value for the undetermined source. Next, a background field and equations for the error field are obtained. 2-D domains are considered. This method is suitable for Helmholtz and Poisson operators. In the presence of finite-difference grid resolution, a varying amount of boundary data, and methods of filtering the noise in the boundary data and the noise intensity of the boundary data, the performance, accuracy, and iteration count of th

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