A regularization technique to overcome the Ill-posedness arising in specific engineering models: formulation, implementation, error analysis, and some engineering applications

Hatamzadeh, Saeed; Allahviranloo, Tofigh; Masouri, Zahra

doi:10.22034/cmde.2025.67540.3223

A regularization technique to overcome the Ill-posedness arising in specific engineering models: formulation, implementation, error analysis, and some engineering applications

Document Type : Research Paper

Authors

¹ Department of Electrical and Electronics Enginnering, Faculty of Engineering and Natural Sciences, Istinye University, Istanbul, Turkey.

² Research Center of Performance and Productivity Analysis, Istinye University, Istanbul, Turkey.

³ Department of Mathematics, Isl.C., Islamic Azad University, Islamshahr, Iran.

10.22034/cmde.2025.67540.3223

Abstract

This article presents a regularization technique for the stable numerical solution of first-kind Fredholm integral equations, which frequently arise in the mathematical modeling of engineering and physical science problems. The proposed technique combines an approximation framework based on a special representation of the triangular function vector forms and their properties with a stabilization strategy to convert the original ill-posed problem into a well-posed algebraic system. Another notable advantage is the reduced computational cost of the proposed technique, as it eliminates the need for performing any integrations during the setup of the algebraic system. Detailed error analysis and convergence proofs are provided, offering rigorous theoretical guarantees for the method’s performance. Numerical experiments on test problems demonstrate the efficiency, stability, and high accuracy of the proposed technique, especially when compared with other regularization methods. Furthermore, the proposed technique is applied to analyze some engineering models, including electromagnetic scatterers and thin-wire antennas. In all cases, the results show excellent agreement with full-wave simulations performed using Altair FEKO software. These findings confirm the robustness, versatility, and computational effectiveness of the proposed regularization strategy for practical ill-posed problems.

Keywords

Main Subjects

Integral equations

References

[1] E. Babolian, Z. Masouri, and S. Hatamzadeh-Varmazyar, Numerical solution of nonlinear Volterra-Fredholm integro-differential equations via direct method using triangular functions, Computers & Mathematics with Applications, 58(2) (2009), 239–247.
[2] C. A. Balanis, Advanced Engineering Electromagnetics, Wiley, New York, 1989.
[3] C. A. Balanis, Antenna Theory: Analysis and Design, Wiley, New York, 1982.
[4] R. Bancroft, Understanding Electromagnetic Scattering Using the Moment Method, Artech House, London, 1996.
[5] R. Danesfahani, S. Hatamzadeh-Varmazyar, E. Babolian, and Z. Masouri, A scheme for RCS determination using wavelet basis, AEU-International Journal of Electronics and Communications, 64(8) (2010), 757–765.
[6] A. Deb, A. Dasgupta, and G. Sarkar, A new set of orthogonal functions and its application to the analysis of dynamic systems, Journal of the Franklin Institute, 343 (2006), 1–26.
[7] R. S. Elliot, Antenna Theory and Design, Englewood Cliffs, New Jersey, Prentice Hall, (1981).
[8] H. W. Engl and M. Hanke, A. Neubauer, Regularization of Inverse Problems, Springer, 1996.
[9] Z. Fang, Y. Wang, P. Xie, Z. Wang, and Y. Zhang, HisynSeg: Weakly-supervised histopathological image segmentation via image-mixing synthesis and consistency regularization, IEEE Transactions on Medical Imaging, 44(4) (2025), 1765–1782.
[10] G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed., Johns Hopkins University Press, 2013.
[11] X. Guo, L. Yu, J. Shi, H. Wang, J. Zhao, R. Zhang, H. Li, and N. Lei, Optimal transport and central moment consistency regularization for semi-supervised medical image segmentation, IEEE Transactions on Medical Imaging, in press.
[12] P. C. Hansen, Discrete Inverse Problems: Insight and Algorithms, SIAM, 2010.
[13] S. Hatamzadeh-Varmazyar and Z. Masouri, A numerical approach for calculating the radar cross-section of two-dimensional perfect electrically conducting structures, Journal of Electromagnetic Waves and Applications, 28(11) (2014), 1360–1375.
[14] S. Hatamzadeh-Varmazyar and Z. Masouri, Numerical expansion-iterative method for analysis of integral equation models arising in one-and two-dimensional electromagnetic scattering, Engineering Analysis with Boundary Elements, 36(3) (2012), 416–422.
[15] S. Hatamzadeh-Varmazyar and Z. Masouri, A fast numerical method for analysis of one-and two-dimensional electromagnetic scattering using a set of cardinal functions, Engineering Analysis with Boundary Elements, 36(11) (2012), 1631–1639.
[16] S. Hatamzadeh-Varmazyar and Z. Masouri, Numerical method for analysis of one-and two-dimensional electromagnetic scattering based on using linear Fredholm integral equation models, Mathematical and Computer Modelling, 54 (2011), 2199–2210.
[17] S. Hatamzadeh-Varmazyar, M. Naser-Moghadasi, E. Babolian, and Z. Masouri, Calculating the radar cross section of the resistive targets using the Haar wavelets, Progress In Electromagnetics Research, 83 (2008), 55–80.
[18] S. Hatamzadeh-Varmazyar, M. Naser-Moghadasi, E. Babolian, and Z. Masouri, Numerical approach to survey the problem of electromagnetic scattering from resistive strips based on using a set of orthogonal basis functions, Progress In Electromagnetics Research, 81 (2008), 393–412.
[19] S. Hatamzadeh-Varmazyar, M. Naser-Moghadasi, and Z. Masouri, A moment method simulation of electromagnetic scattering from conducting bodies, Progress In Electromagnetics Research, 81 (2008), 99–119.
[20] J. Jia, C. He , J. Wang, G. Cheung, and J. Zeng, Deep unrolled graph Laplacian regularization for robust Timeof-Flight depth denoising, IEEE Signal Processing Letters, 32 (2025), 821–825.
[21] R. Kress, Linear Integral Equations, Springer-Verlag, Berlin, (1989).
[22] B. G. Le and V. C. Ta, On the effectiveness of regularization methods for Soft Actor-Critic in discrete-action domains, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 55(2) (2025), 1425–1438.
[23] X. Luo and J. Lv, A novel truncated capped norm regularization method for hyperspectral image denoising, IEEE Geoscience and Remote Sensing Letters, 22 (2025), 5503905.
[24] Z. Masouri and S. Hatamzadeh, A Regularization-direct Method to Numerically Solve First Kind Fredholm Integral Equation, Kyungpook Mathematical Journal, 60(4) (2020), 869–881.
[25] N. Reza and H. Y. Jung, Re-calibrating network by refining initial features through generative gradient regularization, IEEE Access, 13 (2025), 20191–20202.
[26] A. K. Roonizi and R. Sassi, Bandpass filters: A penalized least-squares optimization with ℓ1-norm regularization design, IEEE Signal Processing Letters, 32 (2025), 1416–1419.
[27] W. L. Stutsman and G. A. Thiele, Antenna Theory and Design, John Wiley & Sons, (1998).
[28] Y. Zhang, D. Yu, X. Zhang, and Y. Liu, An autoregressive model-based differential framework with learnable regularization for CSI feedback in time-varying massive MIMO systems, IEEE Communications Letters, 29(1) (2025), 230–234.
[29] Y. Zhu, D. Li, C. Xi, and W. Pedrycz, Multi-view unsupervised representation learning via integration of Fuzzy rules and graph-based adaptive regularization, IEEE Transactions on Fuzzy Systems, 33(7) (2025), 2226–2237.