# A variational approach to the inverse imaging of composite elastic materials

## TL;DR

This paper presents a variational framework for inverse imaging of composite elastic materials using surface acoustic wave data, combining mathematical modeling, numerical methods, and gradient flow analysis to recover interior composition.

## Contribution

It introduces a novel variational approach with gradient-based optimization for reconstructing composite material interiors from boundary acoustic data.

## Key findings

- Gradient flow effectively recovers interior composition
- Numerical methods successfully solve the inverse problem
- Framework demonstrates accurate reconstruction in simulations

## Abstract

We introduce a framework for performing the inverse imaging of composite elastic materials. Our technique uses surface acoustic wave (SAW) boundary observations within a minimization problem to express the interior composition of the composite elastic materials. We have approached our target problem by developing mathematical and computational methods for investigating the numerical solution of the corresponding inverse problem. We also discuss a mathematical model for expressing the propagation of elastic waves through composite elastic bodies, and develop approximation schemes for investigating its numerical solutions. Using these methods, we define a cost functional for measuring the difference between simulated and given SAW data. Then, using a Lagrangian approach, we are able to determine the gradient of the cost functional and analyze the inverse imaging problem's solution as a gradient flow. The cost functional's gradient is composed of solutions to a state equation, as well as of solutions to related adjoint problems. We thus developed numerical methods for solving these problems and investigated the gradient flow of the cost functional. Our results show that the gradient flow is able to recover the interior composition of the composite, and we illustrate this fact using the numerical realization of our proposed framework.