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For bounded linear operators $K:U\to V$ and $B:U\to W$ between real Hilbert spaces, $\alpha >0$ and $\Vert Bu\Vert \ge \beta \Vert u\Vert$ with $\beta >0$, the unique minimizer of $\Vert Ku-f{\Vert }^2/2+\alpha \Vert Bu{\Vert }^2/2$ satisfies $(K^{\ast }K+\alpha B^{\ast }B)u_{\alpha }=K^{\ast }f$. Differentiate the objective in every direction and use the adjoint operator to obtain this equation. The coercive quadratic penalty makes its left-hand operator boundedly invertible. Setting $B=I$ gives the Tikhonov normal equation; omitting the penalty instead gives the normal equation for a linear inverse problem.

The nullspace in the Fourier diffraction theorem already prevents uniqueness, and the rapid decay of evanescent high-frequency information prevents stable recovery. Both make this an ill-posed inverse problem.

Let $A_{\pm }$ be the linear Born approximation operators mapping a real contrast supported in a known region $D$ to the measured fields on the two observation planes. For data $d_{\pm }$, a suitable Tikhonov reconstruction from two-plane scattering data minimizesA Sobolev norm penalty can instead express a smoothness preference. Reality, known support and $n=1-V/k>0$ can be imposed as constraints. For the unconstrained complex Hilbert-space problem with the displayed quadratic penalty, write $A=(A_{+},A_{-})$ and $d=(d_{+},d_{-})$. Its Tikhonov normal equation isTikhonov regularization selects a stable preferred estimate; it does not establish that the original two-plane data uniquely determine the true contrast.

For $\alpha >0$, Tikhonov regularization minimizes the strictly convex quadraticDifferentiating in every direction gives the Tikhonov normal equation $(A^{T}A+\alpha I)v=A^T{y}_{\delta }$. Symmetry of $A$ therefore givesThe spectral filter suppresses unstable division by small eigenvalues. Put $e_i=(y_{\delta }-y,u_i)$ and $x_i=(x,u_i)$. The exact noise-bias decomposition for linear regularization isBy orthonormality and the triangle inequality,The scalar gain satisfies $\lambda /({\lambda }^2+\alpha )\le 1/(2\sqrt{\alpha })$, because $(\lambda -\sqrt{\alpha }{)}^2\ge 0$. Hence the Tikhonov stability bound givesThis explicitly separates data noise from regularization bias. For a known bound $X\ge \Vert x\Vert >0$, minimizing this last bound gives an a priori regularization parameter choiceOne data-based bound is $X=(\Vert y_{\delta }\Vert +\delta )/{\lambda }_1$. More generally, $\alpha (\delta )\to 0$ and $\delta /\sqrt{\alpha (\delta )}\to 0$ make the robust estimate converge to zero.

For this fixed finite matrix, the sharper gain bound ${\max }_i{\lambda }_i/({\lambda }_i^2+\alpha )\le 1/{\lambda }_1$ is also available. Thus the ${\delta }^{2/3}$ rate is a consequence of the robust bound, not an optimality claim: choosing $\alpha X/{\lambda }_1^2\le C\delta /{\lambda }_1$ gives an $O(\delta /{\lambda }_1)$ estimate. No unique statistically optimal parameter is specified without further assumptions on the signal or noise.

A linear regularization is a family of bounded operators $R_{\alpha }:Y\to X$ such that $R_{\alpha }y\to A^{†}y$ for every $y$ in the domain of the Moore–Penrose inverse of an operator as $\alpha \downarrow 0$. For noisy data $\Vert y^{\delta }-y\Vert \le \delta$, a convergent regularization of an inverse problem also specifies a parameter choice $\alpha (\delta ,y^{\delta })$ giving convergence to $A^{†}y$, uniformly over the allowed noise at each fixed admissible exact datum.

Expand $x_{\alpha }$ in the input singular vectors. The Tikhonov normal equation impliesThe component in $\ker A$ is zero because $\alpha >0$. Comparing with the given filter convention givesThe singular-system Tikhonov filter is often described by its gain $g_{\alpha }(\sigma )=\sigma /(\alpha +{\sigma }^2)$; the requested $f_{\alpha }$ includes one extra factor of $\sigma$. For admissible data, $f_{\alpha }\to 1$ and $0\le f_{\alpha }\le 1$, so dominated convergence theorem applied to the squared Picard criterion coefficients proves exact-data convergence. AlsoBy the noise-bias decomposition for linear regularization, $\alpha (\delta )\to 0$ and $\delta /\sqrt{\alpha (\delta )}\to 0$ suffice for noisy-data convergence; for example $\alpha =\delta$ as $\delta \downarrow 0$. This establishes both consistency and stability of Tikhonov regularization.

Fix a singular system of a compact operator, with $K{v}_j={\sigma }_j{w}_j$ and $K^{\ast }{w}_j={\sigma }_j{v}_j$, ${\sigma }_j>0$. A spectral regularization method has the formIt annihilates $\mathcal{R}(K{)}^{\perp }$. Standard sufficient spectral filter conditions are boundedness of ${\sup }_{0<\sigma \le \Vert K\Vert }|\sigma g_{\alpha }({\sigma }^2)|$ for each $\alpha$, pointwise $\lambda g_{\alpha }(\lambda )\to 1$ as $\alpha \downarrow 0$ for $\lambda >0$, and a uniform bound on $|\lambda g_{\alpha }(\lambda )|$. These give bounded linear operators and convergence on the domain of the Moore–Penrose inverse of an operator by the Picard criterion and dominated convergence theorem.

For Tikhonov regularization,This can be computed by solving the Tikhonov normal equation, without knowing the singular value decomposition.

For spectral cutoff regularization, takewith value zero at $\lambda =0$. Only singular values ${\sigma }_j\ge \sqrt{\alpha }$ are inverted. Here $g_{\alpha }$ multiplies $\sigma$; conventions calling the full coefficient $\sigma g_{\alpha }({\sigma }^2)$ the spectral filter are equivalent.

The equation $(A^{\ast }A+{\alpha }_{\ast }I)u_{\ast }=A^{\ast }f$ is the Tikhonov normal equation. For $u=u_{\ast }+h$, expand the Tikhonov regularization functional:The normal equation makes the linear term zero. The final two terms are nonnegative and are strictly positive for $h\ne 0$ because ${\alpha }_{\ast }>0$. Thus $u_{\ast }$ is the unique global minimizer.