Enhancing curve smoothness with whale optimization algorithm in positivity and monotonicity-preserving interpolation

Salwa Syazwani Mahzir; Md Yushalify Misro; Salwa Syazwani Mahzir; Md Yushalify Misro

doi:10.3934/math.2025316

AIMS Mathematics

2025, Volume 10, Issue 3: 6910-6933. doi: 10.3934/math.2025316

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Enhancing curve smoothness with whale optimization algorithm in positivity and monotonicity-preserving interpolation

Salwa Syazwani Mahzir ,
Md Yushalify Misro ^,

School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Gelugor, Pulau Pinang, Malaysia

Received: 24 December 2024 Revised: 13 March 2025 Accepted: 17 March 2025 Published: 26 March 2025
MSC : 65D05, 65D07, 65D17, 68U07

This study integrated a metaheuristic optimization method, specifically the whale optimization algorithm (WOA), to enhance the positivity and monotonicity-preserving interpolation methods by optimizing the free shape parameter. While this free parameter offers flexibility in modifying the shape of the curve, improper selection can lead to visually unpleasing results. By applying an optimization method, this study introduced a more efficient approach to determine the optimal parameter value. To achieve an optimally smooth curve, three different smoothness metrics, including arc length, strain energy, and curvature variation energy, were minimized as objective functions. The resulting curves were then compared to identify the most effective smoothness metric. Results demonstrated that WOA effectively optimized the free shape parameters, and the curvature variation energy was proven to be the best smoothness metric as it produced the smoothest interpolation. Applications of this technique were demonstrated in preserving positivity for COVID-19 death cases and ensuring monotonicity of cumulative rainfall measurements.
Citation: Salwa Syazwani Mahzir, Md Yushalify Misro. Enhancing curve smoothness with whale optimization algorithm in positivity and monotonicity-preserving interpolation[J]. AIMS Mathematics, 2025, 10(3): 6910-6933. doi: 10.3934/math.2025316

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Abstract

This study integrated a metaheuristic optimization method, specifically the whale optimization algorithm (WOA), to enhance the positivity and monotonicity-preserving interpolation methods by optimizing the free shape parameter. While this free parameter offers flexibility in modifying the shape of the curve, improper selection can lead to visually unpleasing results. By applying an optimization method, this study introduced a more efficient approach to determine the optimal parameter value. To achieve an optimally smooth curve, three different smoothness metrics, including arc length, strain energy, and curvature variation energy, were minimized as objective functions. The resulting curves were then compared to identify the most effective smoothness metric. Results demonstrated that WOA effectively optimized the free shape parameters, and the curvature variation energy was proven to be the best smoothness metric as it produced the smoothest interpolation. Applications of this technique were demonstrated in preserving positivity for COVID-19 death cases and ensuring monotonicity of cumulative rainfall measurements.

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