Optimal Propagating Fronts Using Hamilton-Jacobi Equations

Angelo Alessandri,Mauro Gaggero,Patrizia Bagnerini,Roberto Cianci

doi:10.3390/math7111122

Abstract

The optimal handling of level sets associated to the solution of Hamilton-Jacobi equations such as the normal flow equation is investigated. The goal is to find the normal velocity minimizing a suitable cost functional that accounts for a desired behavior of level sets over time. Sufficient conditions of optimality are derived that require the solution of a system of nonlinear Hamilton-Jacobi equations. Since finding analytic solutions is difficult in general, the use of numerical methods to obtain approximate solutions is addressed by dealing with some case studies in two and three dimensions.

Highlights

Hamilton-Jacobi equations are well-know partial differential equations (PDEs) that have been successfully used to model moving interfaces
A number of points is positioned along the front, and such points are moved by using ordinary differential equations (ODEs) [13,14]
The solution of (6) with (7) was found by the approach presented in [27], where the minimization in (7) was obtained by using the fmincon routine of the Matlab Optimization Toolbox. We refer to such a solution procedure as Guo-Sun numerical optimization (GSNO) method

Summary

Introduction

Hamilton-Jacobi equations are well-know partial differential equations (PDEs) that have been successfully used to model moving interfaces. They are the basis of a widely-studied family of methods usually referred to as level set methods [1]. Level set methods are an important family of Eulerian approaches that overcome the aforementioned limits of Lagrangian techniques. In such methods, the front is described by the zero level set of a multidimensional function [9].

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Conclusion