Botond Tibor Szabo, Bocconi University, Italy
We consider recovering an unknown function f from a noisy observation of the solution uf to a partial differential equation of the type Luf = c(f, uf ) for a differential operator L, and invertible function c, i.e. f = e(Luf ). Examples include amongst others the time-independent Schrödinger equation 1/2 ∆uf = uff and the heat equation with absorption term duf /dt − 1/2 ∆uf = f . We transform this problem into the linear inverse problem of recovering Luf under Dirichlet boundary condition, and show that Bayesian methods (with priors placed either on uf or Luf ) for this problem may yield optimal recovery rates not only for uf , but also for f. We also derive frequentist coverage guarantees for the corresponding Bayesian credible sets. Adaptive priors are shown to yield adaptive contraction rates for f, thus eliminating the need to know the smoothness of this function. The results are illustrated by several numerical analysis on synthetic data sets.
This is a joint work with Aad van der Vaart (Delft) and Geerten Koers (Delft).
