Linear methods for non-linear inverse problems

We consider the recovery of an unknown function $f$ from a noisy observation of the solution $u_f$ to a partial differential equation that can be written in the form $\mathcal{L}{u}_f=c(f,u_f)$, for a differential operator $\mathcal{L}$ that is rich enough to recover $f$ from $\mathcal{L}{u}_f$. Examples include the time-independent Schrödinger equation $\mathrm{\Delta }{u}_f=2u_{f}f$, the heat equation with absorption term $({\mathrm{\partial }}_t-{\mathrm{\Delta }}_x\mathrm{/}2)u_f=f{u}_f$, and the Darcy problem $\mathrm{\nabla }\cdot (f\mathrm{\nabla }{u}_f)=h$. We transform this problem into the linear inverse problem of recovering $\mathcal{L}{u}_f$ under the Dirichlet boundary condition, and show that Bayesian methods with priors placed either on $u_f$ or $\mathcal{L}{u}_f$ for this problem yield optimal recovery rates not only for $u_f$, 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.