Variational regularization

Variational regularization balances data fidelity against a lower-semicontinuous penalty, for example by minimizing $\frac{1}{2}\Vert Au-f{\Vert }^2+\alpha J(u)$.

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.

For a convex functional $J$ and $p\in \partial J(v)$, the Bregman divergence isIt is nonnegative but need not be symmetric or satisfy the triangle inequality.

An exact penalty method uses a nonsquared residual such as $\Vert Au-f\Vert +\alpha J(u)$. Under a source condition it can recover an exact constrained minimizer for every sufficiently small fixed positive $\alpha$.