Joint regularization rates under differential range invariance

Apply [F2] to [F1] with $c_1=1$, $c_2=1-b$, $C_3=C_{\ast }$, $r=S_{\delta }$, $e=e_{\delta }$, and $d=(2+C_{\eta }){\delta }^p$. The first bound gives $e_{\delta }\le \frac{1}{1-b}((2+C_{\eta }){\delta }^p\mathrm{/}\alpha +{\rho }_{\delta })$. By [A1], ${\delta }^p\mathrm{/}\alpha \le {\rho }_{\delta }\mathrm{/}{c}_{-}$, hence $e_{\delta }=O({\rho }_{\delta })$.

The second bound from [F2] gives $S_{\delta }\le 2((2+C_{\eta }){\delta }^p+\alpha H(1\mathrm{/}(2C_{\ast }\alpha )))$. By [A1]–[A2], $\alpha H(1\mathrm{/}(2C_{\ast }\alpha ))\le C_H{c}_{+}{\delta }^p$, so $S_{\delta }=O({\delta }^p)$.

For the EIT clauses use [F4]. Every admitted projected coefficient is physical, the ambient indicator leaves the quadratic objective unchanged, and its Bregman error is $e_{\delta }=\frac{1}{2}\mathrm{\parallel }{\widehat{\sigma }}_{\delta }-{\sigma }^{†}{\mathrm{\parallel }}_{H^{s_1}}^2$, where ${\widehat{\sigma }}_{\delta }$ is the first range coordinate translated by ${\sigma }_0$. The first-coordinate identity of the range map gives $\mathrm{\parallel }{\sigma }_{\delta }-{\widehat{\sigma }}_{\delta }{\mathrm{\parallel }}_{H^{s_1}}\le Q(r(x_{\delta }),{\widehat{r}}_{\delta })$. Thus a Bregman rate and the coupling rate control the actual physical conductivity, without confusing the relaxed coefficient with the admitted output coefficient.

Each nonnegative summand defining $S_{\delta }$ is bounded by $S_{\delta }$. Since $\beta$ is fixed and positive, the definitions of $S_{\delta }$ and $p\ge 1$ imply the three displayed $O(\delta )$ bounds. Moreover, $K{\widehat{r}}_{\delta }+F(x_0)-y^{\delta }=K({\widehat{r}}_{\delta }-r(x^{†}))+(F(x^{†})-y^{\delta })$ by [F3]; the triangle inequality and the noise bound then give the $O(\delta )$ data-residual estimate.

For $\alpha \asymp {\delta }^k$ with $0<k<2$, apply the strict-power scalar conclusion of [F5] directly to the joint comparison [F1], with $d=(2+C_{\eta }){\delta }^2$. This yields $e_{\delta }=O(L_{\delta }^{-\nu })$ and $S_{\delta }=O({\delta }^k{L}_{\delta }^{-\nu })$. The fixed positive coupling coefficient and nonnegative summands of $S_{\delta }$ give the claimed artificial-voltage and coupling bounds. Step 1.3 then gives the same physical conductivity norm rate because ${\delta }^{k\mathrm{/}2}{L}_{\delta }^{-\nu \mathrm{/}2}=O(L_{\delta }^{-\nu \mathrm{/}2})$.

By [F5], for $\alpha \asymp {\delta }^2{L}_{\delta }^{\nu }$ the values $H(1\mathrm{/}(C_{\ast }\alpha ))$ and $H(1\mathrm{/}(2C_{\ast }\alpha ))$ are both of order $L_{\delta }^{-\nu }$, their ratio is bounded, and $\alpha H(1\mathrm{/}(C_{\ast }\alpha ))\asymp {\delta }^2$. Hence the generic theorem assumptions apply and steps 1.1–2.1 give $e_{\delta }=O(L_{\delta }^{-\nu })$, $S_{\delta }=O({\delta }^2)$, and the stated residual, coupling and projection bounds. Taking square roots in step 1.3 and using the coupling bound proves the physical conductivity norm rate.

At the endpoint use the actual noise ${\epsilon }_{\delta }=\mathrm{\parallel }{y}^{\delta }-y\mathrm{\parallel }$ in the approximate-minimum comparison, so its scalar right-side constant is $d=2{\epsilon }_{\delta }^2+{\eta }_{\delta }$. The explicit endpoint assumption and [F6] give $e_{\delta }=O(L_{\delta }^{-\nu })$ and $S_{\delta }=O({\delta }^2{L}_{\delta }^{-\nu })$. Nonnegativity of the projection and coupling terms and step 1.3 give all displayed norm bounds. In each regime the relaxed data residual is at most $\mathrm{\parallel }K({\widehat{r}}_{\delta }-r(x^{†}))\mathrm{\parallel }+{\epsilon }_{\delta }=O(\sqrt{S_{\delta }}+{\epsilon }_{\delta })$, exactly as in step 2.1. This completes every stated EIT regime with its explicit noise assumptions.