Poisson Inverse Problems
Jérémie Bigot
Université Paul Sabatier, Toulouse
In this talk, we
focus on nonparametric estimators in inverse problems for Poisson
processes involving the use of wavelet decompositions. Adopting an
adaptive wavelet Galerkin discretization we find that our method
combines the well know theoretical advantages of wavelet-vaguelette
decompositions for inverse problems in terms of optimally adapting to
the unknown smoothness of the solution, together with the remarquably
simple closed form expressions of Galerkin inversion methods. Adapting
the results of Barron and Sheu to the context of log-intensity
functions approximated by wavelet series with the use of the
Kullback-Leibler distance between two point processes, we also present
an asymptotic analysis of convergence rates that justify our approach.
In order to shade some light on the theoretical results obtained and to
examine the accuracy of our estimates in finite samples we illustrate
our method by the analysis of some simulated examples.