  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.
