Source condition in variational regularization

A $J$-minimizing exact solution $u^{†}$ satisfies the source condition when some $w^{†}\in Y^{\ast }$ obeysIt connects a penalty subgradient to the range of the adjoint forward operator.

This stronger source condition in variational regularization requires a subgradient $w=K^{\ast }Kv$ at $u^{†}$. For each $\alpha >0$ it is equivalent to making $u^{†}$ minimize $\Vert Ku-K\bar{u}{\Vert }^2/2+\alpha J(u)$ for artificial input $\bar{u}=u^{†}+\alpha v$. The subgradient optimality condition proves both implications. The source vector may be zero: $K=I$ and $J\equiv 0$ satisfy the condition but admit no nonzero source vector.

Let $u^{†}$ be a least-squares solution and $w=K^{\ast }Kv\in \partial J(u^{†})$. For a minimizer $u_{\alpha }$ of $\Vert Ku-f{\Vert }^2/2+\alpha J(u)$, expand the objective about $u^{†}$. The normal equation for a linear inverse problem removes the data-residual cross term, while the Bregman distance definition givesCompare the minimizer with $u^{†}-\alpha v$, where the square vanishes, and divide by $\alpha$. The bound is informative when that comparator has finite penalty.

If $Au=f$ and $A^{\ast }\mu \in \partial J(u)$, then $u$ is a J-minimizing solution. For every other exact solution $v$, the subgradient inequality gives $J(v)\ge J(u)+⟨\mu ,Av-Au⟩=J(u)$. This is a certificate of global optimality supplied by the range of the adjoint operator.

The range condition asks whether a prescribed exact solution is also a minimizer of a penalized problem with suitably chosen artificial data. The subgradient optimality condition makes it equivalent to a source subgradient lying in the range of the adjoint forward operator.