Bayesian non-linear statistical inverse problems (Q6116625)

The book gives a broad introduction to this very class of problems and discusses (nonparametric) statistical and computational aspects related to them. The work achieves its goal to present them on a level suitable for graduate students in mathematics and mathematical statistics. The first chapter introduces inverse problems conceptually -- along with a few relevant model problems: the Calderón problem, (non-abelian) X-ray reconstruction, the elliptic inverse problems (and how to use parabolic data), and the problem of estimating the initial state in a nonlinear Navier-Stokes equation. The author then proceeds to explain the Bayesian approach to inverse problems together with appropriate Markov chain Monte Carlo algorithms for computational approximation and the frequentist view point on Bayesian inference. Chapter 2 investigates the posterior consistency for some Bayesian inverse problems, e.g., whether in the large sample limit the posterior converges to a Dirac mass concentrated in the true model. Chapter 3 studies local properties of the posterior when the forward operator is linearised in a neighbourhood of the true model. Chapter 4, on the one hand, considers the asymptotic normality for a class of Bayesian inverse problems through nonparametric Bernstein-von Mises theorems. Chapter 5, on the other hand, shows that where asymptotic normality is not given, posteriors may still be approximately log-concave and studies related implications in the theory of Markov chain Monte Carlo algorithms. Readers with a background in variational inverse problems or uncertainty quantification will be able to use this book to study the statistical (and, especially, asymptotic) aspects behind variational and Bayesian inverse problems. (Bayesian) nonparametric statisticians are introduced to new inference problems (such appearing in image reconstruction, geotechnical engineering, weather forecasting,...) and their analysis. They will also profit from the presented results on non-asymptotic approximate log-concavity of posteriors under non-linear models.