Error budget for computed Tikhonov inversion

Let $X,Y$ be real or complex Hilbert spaces over the same scalar field, let $A:X\to Y$ be a compact linear forward map (the closure of $A(B)$ is compact in $Y$ for every bounded set $B\subseteq X$), and let $A_h:X\to Y$ be a bounded linear discretized map with finite-dimensional range. For a datum $b\in Y$, let $\lambda >0$ be the coefficient of $\mathrm{\parallel }x{\mathrm{\parallel }}_X^2$ in the Tikhonov objective $\mathrm{\parallel }Ax-b{\mathrm{\parallel }}_Y^2+\lambda \mathrm{\parallel }x{\mathrm{\parallel }}_X^2$. Supply adjoints $A^{\ast }:Y\to X$ and $A_h^{\ast }:Y\to X$ satisfying $⟨Ax,y{⟩}_Y=⟨x,A^{\ast }y{⟩}_X$ and $⟨A_{h}x,y{⟩}_Y=⟨x,A_h^{\ast }y{⟩}_X$, where the inner products are linear in their first arguments. Assume the shifted normal operators $A^{\ast }A+\lambda I_X$ and $A_h^{\ast }{A}_h+\lambda I_X$ are bijective, and define $R_{\lambda }=(A^{\ast }A+\lambda I_X{)}^{-1}{A}^{\ast },R_{\lambda ,h}=(A_h^{\ast }{A}_h+\lambda I_X{)}^{-1}{A}_h^{\ast }\mathrm{.}$

For a truth $x^{†}\in X$, data perturbation $e\in Y$, and $\delta \ge 0$ with $\mathrm{\parallel }e\mathrm{\parallel }\le \delta$, put $y=A{x}^{†},y_{\delta }=y+e,y_{h,\delta }=A_h{x}^{†}+e,d_h=(A-A_h)x^{†}=y_{\delta }-y_{h,\delta }\mathrm{.}$ For any returned output ${\tilde{x}}_h\in X$, define its exact discrete normal-equation residual by $r_{h,h}=(A_h^{\ast }{A}_h+\lambda I_X){\tilde{x}}_h-A_h^{\ast }{y}_{h,\delta }\mathrm{.}$ The five-channel error budget is the ordered tuple $(b_{\mathrm{b}\mathrm{i}\mathrm{a}\mathrm{s}},b_{\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{a}},b_{\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}},b_{\mathrm{p}\mathrm{r}\mathrm{e}},b_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{v}\mathrm{e}})$, where $\begin{array}{rl}{\displaystyle b_{\mathrm{b}\mathrm{i}\mathrm{a}\mathrm{s}}}& {\displaystyle =x^{†}-R_{\lambda }y,b_{\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{a}}=R_{\lambda }y-R_{\lambda }{y}_{\delta },}\\ {\displaystyle b_{\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}}}& {\displaystyle =(R_{\lambda }-R_{\lambda ,h})y_{h,\delta },b_{\mathrm{p}\mathrm{r}\mathrm{e}}=R_{\lambda }{d}_h,}\\ {\displaystyle b_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{v}\mathrm{e}}}& {\displaystyle =R_{\lambda ,h}{y}_{h,\delta }-{\tilde{x}}_h\mathrm{.}}\end{array}$ The five entries are, in order, the truth-bias, data-perturbation, regularized-operator, pre-inversion, and algebraic-solve error vectors. The scalar total budget is the sum of the five channel norms. The raw pre-inversion forward residual is $d_h$; its reconstruction contribution is $b_{\mathrm{p}\mathrm{r}\mathrm{e}}$. The operator channel measures the difference between the regularized exact and discretized operators. The solve channel measures the returned output's difference from the exact discrete Tikhonov output. The definition supplies no rate or arithmetic residual certificate.

Compact Hilbert-space inverse model

Clason, Regularization of Inverse Problems, Chapter 8

Hansen, Introduction to Inverse Problems, Chapter 5

Van Leeuwen and Brune, 10 Lectures on Inverse Problems and Imaging, Chapter 3