Regularization of an inverse problem

A regularization is a family of bounded approximate inverses $R_{\alpha }$ together with a parameter rule $\alpha (\delta )$ such that $R_{\alpha (\delta )}{f}^{\delta }\to A^{†}f$ whenever $\Vert f^{\delta }-f\Vert \le \delta$ and $f$ lies in the domain of the Moore--Penrose inverse.

Tikhonov regularization minimizesgiving $x_{\alpha }^{\delta }=(A^{\ast }A+\alpha I{)}^{-1}{A}^{\ast }{y}^{\delta }$. The spectral filter $\sigma /({\sigma }^2+\alpha )$ suppresses unstable division by small singular values.

Iterated Tikhonov regularization repeatedly applies a filtered correction based on $A^{\ast }(y-A{x}_n)$. Its singular-system representation exposes the iteration as a family of scalar spectral filters.

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 $\alpha$.