Efficient Nonparametric Bayesian Inference for some inverse problems arising in Partial Differential Equation Models
schedule le mardi 03 octobre 2017 de 10h45 à 11h45
Organisé par : I. Castillo, A. Fischer, E. Roquain, M. Sangnier
Intervenant : Richard Nickl (University of Cambridge)
Lieu : UPMC, salle 15-16.201
Sujet : Efficient Nonparametric Bayesian Inference for some inverse problems arising in Partial Differential Equation Models
We consider the problem of making inference on the coefficients of partial differential operators based on observing the solutions of the partial differential equations (PDEs) associated with these operators, corrupted by statistical noise. Such problems are often non-linear and non-parametric, and have a wide range of applications in mathematical inverse problems, including scattering problems, image tomography, diffusion dynamics etc. Bayesian inference methods have recently become popular in such settings, thanks to the effective use of MCMC techniques, but it remains largely unclear whether such Bayesian algorithms provide solutions to these inverse problems that can be objectively trusted. We will discuss recent results that establish frequentist validity and optimality of nonparametric Bayes procedures in some model examples from elliptic, parabolic and transport PDEs. Applications include the problem of inference on the unknown potential in the Schroedinger equation, inference on images based on observing their X-ray transforms, as well as inference on diffusion parameters from discrete data.