English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Tikhonov regularization is studied in the case of linear pseudodifferentialoperator as the forward map and additive white Gaussian noise as themeasurement error. The measurement model for an unknown function $u(x)$ is\begin{eqnarray*} m(x) = Au(x) + \delta\hspace{.2mm}\varepsilon(x),\end{eqnarray*} where $\delta>0$ is the noise magnitude. If $\varepsilon$ wasan $L^2$-function, Tikhonov regularization gives an estimate \begin{eqnarray*}T_\alpha(m) = \text{argmin}_{u\in H^r}\big\{\|A u-m\|_{L^2}^2+\alpha\|u\|_{H^r}^2 \big\}\end{eqnarray*} for $u$ where $\alpha=\alpha(\delta)$is the regularization parameter. Here penalization of the Sobolev norm $\|u\|_{H^r}$ covers the cases of standard Tikhonov regularization ($r=0$) andfirst derivative penalty ($r=1$).  Realizations of white Gaussian noise are almost never in $L^2$, but do belongto $H^s$ with probability one if $s<0$ is small enough. A modification ofTikhonov regularization theory is presented, covering the case of whiteGaussian measurement noise. Furthermore, the convergence of regularizedreconstructions to the correct solution as $\delta\rightarrow 0$ is proven inappropriate function spaces using microlocal analysis. The convergence of therelated finite-dimensional problems to the infinite-dimensional problem is alsoanalysed.

Analysis of regularized inversion of data corrupted by white Gaussian noise