Variational source conditions on a regularizer sublevel

Let $X,Y$ be real Banach spaces, let $F:D(F)\subseteq X\to Y$, let $x^{†}\in D(F)$ with exact datum $y=F(x^{†})$, and let $R:X\to (-\mathrm{\infty },+\mathrm{\infty }]$ be proper and convex with $x^{†}\in \mathrm{dom}R$. Fix ${\xi }^{†}\in \mathrm{\partial }R(x^{†})$, an exponent $1\le p<\mathrm{\infty }$, a continuous nondecreasing concave function $\psi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ with $\psi (0)=0$ and $\psi (t)>0$ for $t>0$, a coefficient $0\le \beta <1$, and a level $R_0\ge R(x^{†})$. The variational source condition with these data holds on the named regularizer sublevel $L_{R_0}:=\{x\in D(F):R(x)\le R_0\}$ when

$-⟨{\xi }^{†},x-x^{†}{⟩}_{X^{\ast },X}\le \beta D_R^{{\xi }^{†}}(x,x^{†})+\psi \text{\hspace{0.17em}⁣}(\mathrm{\parallel }F(x)-F(x^{†}){\mathrm{\parallel }}_Y^p)(x\in L_{R_0})\mathrm{.}$

The condition is an assumption on the selected subgradient and the stated sublevel; it makes no assertion that a subgradient exists at every point or that a regularized minimizer exists.

Banach subgradients and dual pairings

Bregman distance orientation

Christian Clason, Regularization of Inverse Problems, equation (10.17)

Frederic Weidling, Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs, equations (2.22) and (2.26)

Barbara Kaltenbacher, Convergence rates under a range invariance condition with application to electrical impedance tomography, equations (1.28)–(1.31)