Posterior consistency and convergence rates for Bayesian inversion with hypoelliptic operators

TL;DR

This paper investigates the Bayesian approach to inverse problems involving hypoelliptic operators with Gaussian noise, establishing posterior consistency and convergence rates using microlocal analysis.

Contribution

It extends Bayesian inversion theory to hypoelliptic operators with white Gaussian noise, providing new convergence and consistency results.

Findings

Proves posterior consistency as noise level decreases.

Establishes convergence rates for Bayesian estimates.

Analyzes contraction of confidence regions.

Abstract

Bayesian approach to inverse problems is studied in the case where the forward map is a linear hypoelliptic pseudodifferential operator and measurement error is additive white Gaussian noise. The measurement model for an unknown Gaussian random variable $U(x,\omega)$ is \begin{eqnarray*} M(y,\omega) = A(U(x,\omega) )+ \delta\hspace{.2mm}\mathcal{E}(y,\omega), \end{eqnarray*} where $A$ is a finitely many times smoothing linear hypoelliptic operator and $\delta>0$ is the noise magnitude. The covariance operator $C_U$ of $U$ is $2r$ times smoothing, self-adjoint, injective and elliptic pseudodifferential operator. If $\mathcal{E}$ was taking values in $L^2$ then in Gaussian case solving the conditional mean (and maximum a posteriori) estimate is linked to solving the minimisation problem \begin{eqnarray*} T_\delta(M) = \text{argmin}_{u\in H^r} \big\{\|A u-m\|_{L^2}^2+ \delta^2\|C_U^{-1/2}u\|_{L^2}^2 \big\}. \end{eqnarray*} However, Gaussian white noise does not take values in $L^2$ but in $H^{-s}$ where $s>0$ is big enough. A modification of the above approach to solve the inverse problem is presented, covering the case of white Gaussian measurement noise. Furthermore, the convergence of conditional mean estimate to the correct solution as $\delta\rightarrow 0$ is proven in appropriate function spaces using microlocal analysis. Also the contraction of the confidence regions is studied.