A UNIFIED SPECTRAL FILTER FRAMEWORK FOR ILL-POSED LINEAR OPERATOR EQUATIONS IN HILBERT SPACES

B.bhaskar Reddy; Raja Kumari Chilukuri; Anil Chowdary Tummala; Venkateswara Rao Musala; Hari Kishore Kakarla; Kavitha Manoharan

A UNIFIED SPECTRAL FILTER FRAMEWORK FOR ILL-POSED LINEAR OPERATOR EQUATIONS IN HILBERT SPACES

Abstract

Regularization is useful for stable recovery in inverse problems with ill-posed linear operator equations in Hilbert spaces because small perturbations in data can make problems highly unstable. Tikhonov regularization, truncated singular value decomposition, and iterative polynomial filtering, classical methods, have been understood from singular value decay and the Picard condition. However, most literature analyses convergence, parameter choice, and saturation from separate perspectives. This study fills the gap by constructing a unified spectral filter framework that integrates bias--variance decomposition, polynomial and exponential decay, convergence rate analysis, and stability-consistent parameter choice frameworks, including the discrepancy principle and the L-curve criterion. To enhance saturation control and qualification, we propose extensions to fractional and generalized spectral filters. In the severely ill-posed setting, we identify logarithmic convergence barriers with the inductive method, thereby exposing accuracy limits that exist independently from filter design. The findings are directly applicable to stable inversion and are operator theoretically sound for real-world applications, including medical imaging, geophysical reconstruction, signal processing, and data-driven recovery of ill-conditioned systems.

Keywords

References

[1] Clason, C. (2020). Regularization of inverse problems. arXiv preprint arXiv:2001.00617.
[2] Zhang, Y., Hofmann, B. (2020). Two new non-negativity preserving iterative regularization methods for ill-posed inverse problems. arXiv preprint arXiv:2002.04474.
[3] Bortolotti, V., Landi, G., Zama, F. (2025). Uniform multipenalty regularization for linear ill-posed inverse problems. SIAM Journal on Scientific Computing, 47(2), A790-A810.
[4] Jin, Q. (2024). On saturation of the discrepancy principle for nonlinear Tikhonov regularization in Hilbert spaces. Applied Mathematics Letters, 156, 109150.
[5] Kavitha, M. (2025). Hybrid AI-mathematical modeling approach for predictive maintenance in rotating machinery systems. Journal of Applied Mathematical Models in Engineering, 1(1), 1–8.
[6] Yu, S., Yang, H. (2024). Regularization with two differential operators and its application to inverse problems. Applied Numerical Mathematics, 201, 129-144.
[7] Hauptmann, A., Mukherjee, S., Sch¨onlieb, C. B., & Sherry, F. (2025). Convergent regularization in inverse problems and linear plug-and-play denoisers. Foundations of Computational Mathematics, 25(4), 1087-1120.
[8] Namrata Mishra. (2026). Advanced Spectral and Statistical Learning Methods for Intelligent Interpretation of Non-Stationary Signals. Transactions on Advanced Signal Processing and Analytics, 1(1), 38–45.

[9] Mathé, P., Hofmann, B. (2025). Comparing the ill-posedness for linear operators in Hilbert spaces. Advances in Operator Theory, 10(2), 36.
[10] Surendar, A. (2025). Hybrid LLM-algorithm optimization: iterated fine-tuning for combinatorial problems. Frontiers in Mathematical and Computational Research, 32-39.
[11] T. Senthil, ”Translating Theory into Practice: A Multidisciplinary Framework for Addressing Complex Real-World Challenges”, Bridge: Journal of Multidisciplinary Explorations, vol. 1, no. 3, pp. 34–40.
[12] Dykes, L., Kuian, M., Mach, T., Noschese, S., Reichel, L. (2025). Solution of linear ill-posed operator equations by modified truncated singular value expansion. Journal of Computational and Applied Mathematics, 467, 116621.
[13] George, S., Sadananda, R., Padikkal, J., Kunnarath, A., Argyros, I. K. (2024). New Trends in Applying LRM to Nonlinear Ill-Posed Equations. Mathematics, 12(15), 2377.
[14] N. Arvinth, & David Gichoya. (2025). A Scalable Reconfigurable Processor Architecture for Heterogeneous Edge Computing Applications.SCCTS Transactions on Reconfigurable Computing, 3(2), 40-48.
[15] Hurault, S. (2023). Convergent plug-and-play methods for image inverse problems with explicit and nonconvex deep regularization (Doctoral dissertation, Universit´e de Bordeaux).
[16] Beraldo, R. G., Ferreira, L. A., Suyama, R. (2025, September). Regularization for Inverse Problems and Machine Learning: A Tutorial. In 2025 38th SIBGRAPI Conference on Graphics, Patterns and Images (SIBGRAPI) (pp. 1-6). IEEE.
[17] Nair, M. T. (2021). Regularization of ill-posed operator equations: an overview. The Journal of Analysis, 29(2), 519-541.
[18] Sadaghiyanfam, S., Kamberaj, H., Isler, Y. (2025). Predicting Ionic Liquid Toxicity with Graph Attention Networks. Akıllı sistemler ve uygulamaları dergisi, 25-35.
[19] Singh, H. (2022). Regularizing An Ill-Posed Problem with Tikhonov’s Regularization.
[20] Nisha Milind Shrirao, Nidhi Mishra. (2025). Terabit-per-Second Photonic Nanoelectronic Devices for Data Center Interconnects. tProgress in Electronics and Communication Engineering, 3(1), 73-79.
[21] Bakushinsky, A. B., Kokurin, M. M., Kokurin, M. Y. (2018). Regularization algorithms for ill-posed problems (Vol. 61). Walter de Gruyter GmbH and Co KG.
[22] Krnst Beken, Hardley Caddwine. (2026). Data-Efficient Learning-Assisted Predictive Control for Real-Time Trajectory Planning Under Dynamic Constraints. Journal of Scalable Data Engineering and Intelligent Computing, 17-23.
[23] Kim, S., Rim, B., Choi, S., Lee, A., Min, S., Hong, M. (2022). Deep learning in multi-class lung diseases’ classification on chest X-ray images. Diagnostics, 12(4), 915.
[24] Vimala, D. (2026). Integrating mathematical literacy and language development: A curriculum innovation framework for secondary education. Journal of Mathematics, Language and Curriculum Innovation, 2(1), 1–6.
[25] K. Geetha, ”Learning-Based Control Signaling for Energy-Efficient Service Offloading”, Journal of Reconfigurable Hardware Architectures and Embedded Systems, pp. 18–26.
[26] S. Praveen Kumar. (2026). Algorithm–Architecture Co-Design of High-Throughput DSP Hardware Accelerators for Next-Generation Embedded Systems. Journal of Integrated VLSI and Signal Processing, 1(1), 26–34.

Details

Primary Language

English

Subjects

Operator Algebras and Functional Analysis

Journal Section

Research Article

Authors

B.bhaskar Reddy ^* This is me
0000-0002-3200-0182
India

Raja Kumari Chilukuri
0000-0002-0793-133X
India

Anil Chowdary Tummala This is me
0000-0002-7115-8214
India

Venkateswara Rao Musala This is me
0000-0002-9404-8202
India

Hari Kishore Kakarla This is me
0000-0003-2622-3483
India

Kavitha Manoharan This is me
0000-0001-9054-1449
India

Publication Date

June 9, 2026

Submission Date

May 20, 2025

Acceptance Date

December 15, 2025

Published in Issue

Year 2026 Volume: 16 Number: 6

IZ

https://izlik.org/JA72EW73MX

Cite

RIS / Bibtex

APA

Reddy, B., Chilukuri, R. K., Tummala, A. C., Musala, V. R., Kakarla, H. K., & Manoharan, K. (2026). A UNIFIED SPECTRAL FILTER FRAMEWORK FOR ILL-POSED LINEAR OPERATOR EQUATIONS IN HILBERT SPACES. TWMS Journal of Applied and Engineering Mathematics, 16(6), 808-823. https://izlik.org/JA72EW73MX

AMA

1.Reddy B, Chilukuri RK, Tummala AC, Musala VR, Kakarla HK, Manoharan K. A UNIFIED SPECTRAL FILTER FRAMEWORK FOR ILL-POSED LINEAR OPERATOR EQUATIONS IN HILBERT SPACES. JAEM. 2026;16(6):808-823. https://izlik.org/JA72EW73MX

Chicago

Reddy, B.bhaskar, Raja Kumari Chilukuri, Anil Chowdary Tummala, Venkateswara Rao Musala, Hari Kishore Kakarla, and Kavitha Manoharan. 2026. “A UNIFIED SPECTRAL FILTER FRAMEWORK FOR ILL-POSED LINEAR OPERATOR EQUATIONS IN HILBERT SPACES”. TWMS Journal of Applied and Engineering Mathematics 16 (6): 808-23. https://izlik.org/JA72EW73MX.

EndNote

Reddy B, Chilukuri RK, Tummala AC, Musala VR, Kakarla HK, Manoharan K (June 1, 2026) A UNIFIED SPECTRAL FILTER FRAMEWORK FOR ILL-POSED LINEAR OPERATOR EQUATIONS IN HILBERT SPACES. TWMS Journal of Applied and Engineering Mathematics 16 6 808–823.

IEEE

[1]B. Reddy, R. K. Chilukuri, A. C. Tummala, V. R. Musala, H. K. Kakarla, and K. Manoharan, “A UNIFIED SPECTRAL FILTER FRAMEWORK FOR ILL-POSED LINEAR OPERATOR EQUATIONS IN HILBERT SPACES”, JAEM, vol. 16, no. 6, pp. 808–823, June 2026, [Online]. Available: https://izlik.org/JA72EW73MX

ISNAD

Reddy, B.bhaskar - Chilukuri, Raja Kumari - Tummala, Anil Chowdary - Musala, Venkateswara Rao - Kakarla, Hari Kishore - Manoharan, Kavitha. “A UNIFIED SPECTRAL FILTER FRAMEWORK FOR ILL-POSED LINEAR OPERATOR EQUATIONS IN HILBERT SPACES”. TWMS Journal of Applied and Engineering Mathematics 16/6 (June 1, 2026): 808-823. https://izlik.org/JA72EW73MX.

JAMA

1.Reddy B, Chilukuri RK, Tummala AC, Musala VR, Kakarla HK, Manoharan K. A UNIFIED SPECTRAL FILTER FRAMEWORK FOR ILL-POSED LINEAR OPERATOR EQUATIONS IN HILBERT SPACES. JAEM. 2026;16:808–823.

MLA

Reddy, B.bhaskar, et al. “A UNIFIED SPECTRAL FILTER FRAMEWORK FOR ILL-POSED LINEAR OPERATOR EQUATIONS IN HILBERT SPACES”. TWMS Journal of Applied and Engineering Mathematics, vol. 16, no. 6, June 2026, pp. 808-23, https://izlik.org/JA72EW73MX.

Vancouver

1.B.bhaskar Reddy, Raja Kumari Chilukuri, Anil Chowdary Tummala, Venkateswara Rao Musala, Hari Kishore Kakarla, Kavitha Manoharan. A UNIFIED SPECTRAL FILTER FRAMEWORK FOR ILL-POSED LINEAR OPERATOR EQUATIONS IN HILBERT SPACES. JAEM [Internet]. 2026 Jun. 1;16(6):808-23. Available from: https://izlik.org/JA72EW73MX