Statistical Inverse Problems on Manifolds

This article examines statistical inverse problems on compact Riemannian manifolds. The approach is to use aspects of spectral geometry associated with the Laplace-Beltrami operator on compact Riemannian manifolds. Optimality in terms of upper and lower rates of convergence is established. It turns out that if the operator is polynomially bounded, then optimal convergence is polynomial, while if the operator is exponentially bounded, then optimal convergence proceeds logarithmically. Application to estimating the initial heat distribution is analyzed.