Application of Ill-Posed Problems in Mathematical Modeling, Data Analysis, and Business Mathematics

Abstract

This paper explores the theoretical foundations and practical applications of ill-posed problems in mathematical modeling, data analysis, and business mathematics. Ill-posed problems, which lack existence, uniqueness, or stability of solutions, frequently emerge in real-world scenarios such as inverse problems, machine learning, and optimization. The study reviews regularization techniques like Tikhonov and LASSO methods, which are essential for stabilizing solutions and ensuring reliable outcomes in fields ranging from medical imaging and geophysics to financial forecasting and risk modeling. Through detailed case studies and mathematical formulations, the authors highlight the crucial role of handling ill-posedness in extracting meaningful insights and making informed decisions under uncertainty.