Second Order Shape Optimization for an Interface Identification Problem constrained by Nonlocal Models

TL;DR

This paper extends shape optimization techniques to identify interfaces in nonlocal PDE models by deriving the second shape derivative and exploring a second order optimization algorithm.

Contribution

It introduces the derivation of the second shape derivative for nonlocal models and applies a second order optimization method to interface identification.

Findings

Derived the second shape derivative for nonlocal Dirichlet problems.

Developed a second order optimization algorithm for interface identification.

Demonstrated the applicability of shape optimization in nonlocal PDE contexts.

Abstract

Since shape optimization methods have been proven useful for identifying interfaces in models governed by partial differential equations, we show how shape optimization techniques can also be applied to an interface identification problem constrained by a nonlocal Dirichlet problem. Here, we focus on deriving the second shape derivative of the corresponding reduced functional and we further investigate a second order optimization algorithm.