Analysis of regularized inversion of data corrupted by white Gaussian noise

Abstract

Tikhonov regularization is studied in the case of linear pseudodifferential operator as the forward map and additive white Gaussian noise as the measurement error. The measurement model for an unknown function $u(x)$ is \begin{eqnarray*} m(x) = Au(x) + δ\hspace{.2mm}\varepsilon(x), \end{eqnarray*} where $δ>0$ is the noise magnitude. If $\varepsilon$ was an $L^2$-function, Tikhonov regularization gives an estimate \begin{eqnarray*} T_α(m) = \text{argmin}_{u\in H^r}\big\{\|A u-m\|_{L^2}^2+ α\|u\|_{H^r}^2 \big\}\end{eqnarray*} for $u$ where $α=α(δ)$ is the regularization parameter. Here penalization of the Sobolev norm $ \|u\|_{H^r}$ covers the cases of standard Tikhonov regularization ($r=0$) and first derivative penalty ($r=1$). Realizations of white Gaussian noise are almost never in $L^2$, but do belong to $H^s$ with probability one if $s<0$ is small enough. A modification of Tikhonov regularization theory is presented, covering the case of white Gaussian measurement noise. Furthermore, the convergence of regularized reconstructions to the correct solution as $δ\rightarrow 0$ is proven in appropriate function spaces using microlocal analysis. The convergence of the related finite-dimensional problems to the infinite-dimensional problem is also analysed.