Calculus of Variations: Minimizing a Family of Functions

Two famous optimization problems in mathematics are The Minimal Surface problem and The Brachistochrone Problem. In Calculus of variations, each can be related to a functional which is a family of functions. Solutions to such problems are the functions that minimize the functional. The theoretical solution to each of these problems can be found with methods of Calculus of Variations. In other words, we will derive the Euler-Lagrange equation from the Gateaux Variation of the functional set equal to zero, and the resulting partial differential equation will describe the true solution. The objective of this research is to numerically calculate the solution by selecting data points near the true solution and running them through the steepest descent algorithm until the points are aligned with the true solution. The curve that is mathematically described is the theoretical solution. This method of numerically calculating the minimum of a functional can be applied to anisotropic smoothing models in image processing such as restoring regions of interest in medical images.