Linear methods for nonlinear inverse problems

We consider the recovery of an unknown function f from a noisy observation of the solution uf to a partial differential equation that can be written in the form Luf=c(f,uf), for a differential operator L that is rich enough to recover f from Luf. Examples include the time-independent Schrödinger equation Δuf=2uff, the heat equation with absorption term (∂t−Δx/2)uf=fuf and the Darcy problem ∇·(f∇uf)=h. We transform this problem into the linear inverse problem of recovering Luf under the Dirichlet boundary condition, and show that Bayesian methods with priors placed either on uf or Luf for this problem 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 numerical experiments on synthetic data sets.