Applications of Numerical Methods

This book explores the application of numerical methods in finance and environmental sciences.In the financial field it investigates the estimation of local volatility from option market data and the modeling of stochastic volatility. To address the ill-posed nature of volatility calibration it employs Tikhonov regularization and efficient numerical schemes such as the Discontinuous Galerkin Euler and Runge-Kutta methods.In environmental sciences it studies saltwater intrusion by identifying an unknown source relevant to sustainable groundwater management and introduces a mixed formulation linking interface depth and hydraulic head through a flow variable. The book proves both formulations well-posed and provides optimal error estimates.