# A numerical method for an inverse optimization problem through the generalized method of lines

## TL;DR

This paper introduces a numerical approach using the generalized method of lines to solve an inverse shape optimization problem governed by Laplace's equation with complex boundary conditions.

## Contribution

It applies the generalized method of lines to an inverse boundary shape optimization problem, a novel use of this numerical technique.

## Key findings

- Successfully computes internal boundary shapes satisfying boundary conditions
- Demonstrates effectiveness of the generalized method of lines in inverse problems
- Provides a new computational tool for boundary shape optimization

## Abstract

This article develops a solution for an inverse problem through the generalized method of lines. We consider a Laplace equation on a domain with internal and external boundaries with standard Dirichlet boundary conditions. Also, we specify a third non-homogeneous Newmann type boundary condition for the external boundary, and consider the problem of finding the optimal shape for the internal boundary such that all the prescribed boundary conditions are satisfied. The novelty here presented is the application of the generalized method of lines as a tool to compute a solution for such an inverse optimization problem.