Double Well Potential Function and Its Optimization in The n-dimensional Real Space -- Part I

TL;DR

This paper studies the mathematical structure of a special quartic polynomial called the double well potential function in n-dimensional space, focusing on its local minima, maxima, and saddle points, with a duality approach and numerical illustrations.

Contribution

It categorizes all possible configurations of the double well potential functions and introduces a duality framework linking the original problem to convex minimization.

Findings

The dual of the double well problem is a linearly constrained convex problem.

The function exhibits multiple local minima separated by maxima or saddle points.

Numerical examples demonstrate the problem's key features and the duality mapping.

Abstract

A special type of multi-variate polynomial of degree 4, called the double well potential function, is studied. When the function is bounded from below, it has a very unique property that two or more local minimum solutions are separated by one local maximum solution, or one saddle point. Our intension in this paper is to categorize all possible configurations of the double well potential functions mathematically. In part I, we begin the study with deriving the double well potential function from a numerical estimation of the generalized Ginzburg-Landau functional. Then, we solve the global minimum solution from the dual side by introducing a geometrically nonlinear measure which is a type of Cauchy-Green strain. We show that the dual of the dual problem is a linearly constrained convex minimization problem, which is mapped equivalently to a portion of the original double well problem subject to additional linear constraints. Numerical examples are provided to illustrate the important features of the problem and the mapping in between.