Dual source certificates for TV regularization

Assume Countable Choice. Let $\mathrm{\Omega }={\mathbb{R}}^2$, or let $\mathrm{\Omega }\subseteq {\mathbb{R}}^2$ be a bounded Lipschitz domain. Set $X=L^2(\mathrm{\Omega };\mathbb{R})$ and let $Y=L^2(\mathrm{\Sigma },\mu ;\mathbb{R})$ for a measure space $(\mathrm{\Sigma },\mu )$, using the real integral inner products; under Countable Choice these are real Hilbert spaces. Let $A:X\to Y$ be bounded and real-linear, and let $A^{\ast }:Y\to X$ be its Hilbert adjoint, characterized by $⟨Au,p{⟩}_Y=⟨u,A^{\ast }p{⟩}_X$. Define $R:X\to [0,+\mathrm{\infty }]$ as follows. If $\mathrm{\Omega }={\mathbb{R}}^2$, take $R(u)=T{V}_2(u;{\mathbb{R}}^2)$, the supremum of $\int u\mathrm{div}\phi$ over smooth compactly supported vector fields of Euclidean norm at most one. For bounded $\mathrm{\Omega }$, either take Neumann TV, $R(u)=TV(u;\mathrm{\Omega })$ when $u\in BV(\mathrm{\Omega })$ and $+\mathrm{\infty }$ otherwise, or Dirichlet TV, $R(u)=TV(\tilde{u};{\mathbb{R}}^2)$ when the zero extension $\tilde{u}$ belongs to $BV({\mathbb{R}}^2)$ and $+\mathrm{\infty }$ otherwise. Thus the bounded-domain choices count, respectively, only the interior derivative or also the jump of the zero extension across the boundary. In the full-space case finite $R$ does not require global $L^1$ integrability.

Identify each $q\in X$ with the continuous linear functional $h\mapsto ⟨q,h{⟩}_X$ by the real Hilbert-space Riesz representation. For exact data $f=A{u}^{†}$, a minimum-TV exact solution is a point $u^{†}\in X$ satisfying $A{u}^{†}=f$ and $R(u^{†})\le R(u)$ for every $u\in X$ with $Au=f$. A vector $p^{†}\in Y$ is a dual source certificate for $u^{†}$ if $A^{\ast }{p}^{†}\in \mathrm{\partial }R(u^{†})$, where $\mathrm{\partial }R$ is the convex subdifferential on the real Banach space $X$. The TV source condition at $u^{†}$ is that such a certificate exists; it does not assert uniqueness of the truth or of the certificate. For a chosen certificate, the associated Bregman distance is

$D_R^{A^{\ast }{p}^{†}}(u,u^{†}):=R(u)-R(u^{†})-⟨A^{\ast }{p}^{†},u-u^{†}{⟩}_X,$

for $u\in \mathrm{dom}R$.

Spatial TV is the mass of the distributional derivative.

Full-space TV on L² uses the local distributional test-field supremum.

This extended functional is proper, convex, lower semicontinuous, and agrees with BV TV when globally integrable.

Full-space, Dirichlet, and Neumann TV have distinct boundary conventions.

The subdifferential uses the supporting-functional inequality on the real dual.

The Bregman distance fixes the comparison/reference-point orientation.

The real L² pairing is linear in its first argument and induces the L² norm.

Real L² spaces with the integral pairing are Hilbert spaces under Countable Choice.

Riesz representation defines the unique bounded Hilbert adjoint.

Countable Choice is assumed by the real L² Hilbert-space and Riesz-representation results.

Iglesias, Mercier, and Scherzer, A note on convergence of solutions of total variation regularized linear inverse problems, Definition 1

Chambolle, Duval, Peyré, and Poon, Geometric properties of solutions to the total variation denoising problem, §§1.3 and 3.1

Kaltenbacher, Convergence rates under a range invariance condition with application to electrical impedance tomography, Definition (1.25)