MODERN APPROACHES TO OPTIMIZATION AND APPROXIMATION IN APPLIED MATHEMATICS: DEVELOPING EFFICIENT MATHEMATICAL TOOLS FOR REAL-WORLD ENGINEERING AND SCIENTIFIC CHALLENGES

Abstract

Applied mathematics centers on optimization and approximation methods and through these methods engineers and scientists can model, analyse and solve complex real-world problems efficiently. Recent technologies of increased computational power, designs of algorithms, and mathematical modeling have resulted in the creation of modern optimization techniques, including metaheuristic algorithms, convex and non-convex optimization and machine learning-assisted approximation. These techniques can provide quicker convergence, increased accuracy and also to solve large and high dimensional problems that were not previously solvable. The following paper describes an in-depth research on the modern method of optimization and approximation, their relevance to engineering design, control systems, data analysis, and scientific computation. Gradient-based methodologies, evolutionary algorithms, surrogate modelling approach, and adaptive approximation are mentioned, and their performance has greatly been compared. Findings suggest that a combination of classical mathematical methods with contemporary concepts in computing algorithm techniques is marked by high performance and reliability in most cases. Issues such as computational cost, sensitivity to initial conditions as well as issues in real-time use are discussed. The future highlights the combination of the artificial intelligence, parallel computing, and quantification of uncertainties to increase the strength and usability of these mathematical instruments.