An inverse source problem for the Monge--Ampere equation from large boundary data

We study an inverse source problem for the Monge–Ampere equation $$\det D^{2}u=f(x)$$ on a bounded smooth uniformly convex domain. In the smooth classical regime, we prove that the Dirichlet-to-Neumann map associated with convex solutions determines the positive source uniquely. The proof uses a family of large boundary values and reduces the inverse source problem to the injectivity of the Euclidean X-ray transform.