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All Journal Proximal: Jurnal Penelitian Matematika dan Pendidikan Matematika
Salma Ari Nastiti
Universitas Nahdlatul Ulama Sunan Giri
Mathematics

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Lipschitz Based Polynomial Interpolation for Stable Incomplete Data Reconstruction M. Husen Al Farisy; M. Ivan Ariful Fathoni; Salma Ari Nastiti
Proximal: Jurnal Penelitian Matematika dan Pendidikan Matematika Vol. 9 No. 3 (2026): Volume 9 Nomor 3 Tahun 2026
Publisher : Universitas Cokroaminoto Palopo

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.30605/proximal.v9i3.9607

Abstract

Incomplete data reconstruction may experience excessive fluctuations when the available observations become sparse or irregular and are contaminated by noise, making global polynomial interpolation sensitive to local variations. This study proposes a Lipschitz-based polynomial interpolation approach that combines classical Lagrange interpolation with a gradient-limiting mechanism, which constrains local changes between consecutive points within a prescribed Lipschitz bound. Unlike reconstruction approaches based on optimization or iterative regularization, the proposed mechanism is applied as a simple post-interpolation stabilization process without requiring model training or repeated optimization. The method was evaluated using synthetic nonlinear data with Gaussian noise under missing-data rates of 30%, 40%, and 50%. The results reveal a trade-off between reconstruction accuracy and smoothness. While classical interpolation achieved lower RMSE and MAE values, the Lipschitz-based method produced lower Total Variation (TV) values in most scenarios. These results indicate that gradient limiting effectively controls local fluctuations and produces smoother reconstructions at the expense of reduced pointwise accuracy. Therefore, the proposed approach provides a simple and interpretable stabilization mechanism for incomplete and noisy data reconstruction when oscillation control is a primary concern.
Co-Authors M. Husen Al Farisy M. Ivan Ariful Fathoni