# Linear methods for non-linear inverse problems

Botond Tibor Szabo, Bocconi University, Italy 2.9.2.2214:00 - 16:00

We consider recovering an unknown function *f* from a noisy observation of the solution* u _{f }*to a partial differential equation of the type

*Lu*=

_{f}*c(f, u*for a differential operator

_{f})*L*, and invertible function

*c*, i.e.

*f = e(Lu*

_{f}*)*. Examples include amongst others the time-independent Schrödinger equation 1/2 ∆

*u*=

_{f}*u*

_{f}*f*and the heat equation with absorption term

*du*/

_{f}*dt*− 1/2 ∆

*u*=

_{f}*f*. We transform this problem into the linear inverse problem of recovering

*Lu*under Dirichlet boundary condition, and show that Bayesian methods (with priors placed either on

_{f}*u*or

_{f }*Lu*) for this problem may yield optimal recovery rates not only for

_{f}*u*, but also for

_{f}*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).