Linear methods for non-linear inverse problems

17.10.2024, 14:00 - 16:00  –  2.9.2.22
SFB-Seminar

Botond Tibor Szabo, Bocconi University, Italy

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_f ) for a differential operator 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_ff 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_f  under Dirichlet boundary condition, and show that Bayesian methods (with priors placed either on u_f  or Lu_f ) for this problem may 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 several numerical analysis on synthetic data sets.

This is a joint work with Aad van der Vaart (Delft) and Geerten Koers (Delft).