Comparative Analysis Of Numerical Efficiency In Modern Approximation Methods: Computational Performance Metrics And Application Domains - Sangeet; Dr. Jogender (Revista Electronica de Veterinaria, 2024)
This paper presents a comprehensive evaluation of the numerical efficiency of various approximation methods commonly employed in computational mathematics and scientific computing. We systematically compare polynomial approximation, spectral methods, finite element methods, and machine learning-based approximation techniques across multiple dimensions of numerical efficiency, including computational complexity, memory requirements, convergence rates, and error propagation characteristics. Our analysis employs a unified framework of performance metrics to evaluate these methods across diverse application domains, including fluid dynamics, structural mechanics, and signal processing. Extensive numerical experiments demonstrate that spectral methods exhibit superior convergence rates for smooth functions, while adaptive finite element approaches offer better efficiency for problems with singularities or sharp transitions. Machine learning-based approximations show promising performance for high-dimensional problems when sufficient training data is available. We provide quantitative benchmarks for practitioners to select appropriate approximation techniques based on problem characteristics and computational constraints. This comparative framework offers valuable insights for optimizing numerical algorithms in scientific computing applications and highlights emerging directions for hybrid approximation strategies.