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Let $({\sigma }_j,u_j,v_j)$ be a singular system of a compact operator, withThe Moore–Penrose inverse of an operator is the generally unbounded mapdefined when the Picard criterion holds, with the component in $\ker A^{\ast }$ sent to zero. It is the minimum-norm least-squares solution of $Af=g$.

A regularization of an inverse problem consists of bounded maps $R_{\alpha }:Y\to X$ and a parameter rule $\alpha =\alpha (\delta ,g^{(\delta )})$ such thatwhenever $\Vert g^{(\delta )}-g\Vert \le \delta$ and $g$ is in the domain of $A^{†}$.

For Tikhonov regularization, minimizinggivesThe scalar spectral filter satisfiesand consequentlyFor exact data, each filter factor ${\sigma }_j^2/({\sigma }_j^2+\alpha )$ tends to one, so $R_{\alpha }g\to A^{†}g$. Choosingtherefore makes both the approximation error and propagated data error vanish. For example, $\alpha (\delta )=\delta$ is an admissible a priori regularization parameter choice.

A regularization of an inverse problem consists of bounded operators $R_{\alpha }:Y\to X$ and a parameter rule $\alpha =\alpha (\delta ,y^{\delta })$ such that, whenever $\Vert y^{\delta }-y\Vert \le \delta$ and $y$ lies in the domain of $A^{†}$,as $\delta \to 0$. It is needed because a compact operator on an infinite-dimensional space has singular values tending to zero, so direct inversion divides noisy data by arbitrarily small numbers and is generally discontinuous.

Tikhonov regularization definesand its normal equation givesFor every fixed $\alpha >0$, $A^{\ast }A+\alpha I$ is bounded below by $\alpha I$, and the data-to-solution operator is bounded. Small changes in $y^{\delta }$ therefore produce small changes in $x_{\alpha }^{\delta }$.

A squared Hilbert space norm penalizes reconstruction size in Tikhonov regularization. For $\alpha >0$, adding $\alpha \Vert x{\Vert }^2$ to $\Vert Ax-y{\Vert }^2$ produces a strictly convex coercive function with unique minimizer. Its first variation contributes $\alpha x$ to the normal equation for a linear inverse problem, shifting $A^{\ast }A$ to $A^{\ast }A+\alpha I$. It selects a minimum-norm representative and controls amplification of noisy small-singular-value components.

Regularization replaces sensitive inversion by a controlled approximation. For a bounded linear operator between Hilbert spaces, a linear regularization is a family of bounded linear operators $R_{\alpha }$ satisfying $R_{\alpha }y\to A^{†}y$ as $\alpha \downarrow 0$ for each datum in the domain of the Moore–Penrose inverse of an operator. Noisy-data convergence additionally requires a suitable regularization parameter choice.^[ref] Nonlinear variational regularization can instead use nonquadratic penalties.

The noise-bias decomposition for linear regularization separates two costs: amplifying measurement noise and changing the exact-data reconstruction. Truncated singular value decomposition discards weak modes; Tikhonov regularization attenuates them smoothly. A quadratic norm penalty expresses a preference for smaller reconstructions. Neither method manufactures a missing null space component: with complete averaging, the minimum-norm least-squares solution for $(0.5,0.5{)}^T$ is $(0.5,0.5{)}^T$, even if the original image was $(1,0{)}^T$. The preference selects a representative; extra observations or justified constraints are needed to identify the original.

For Tikhonov regularization, the Tikhonov stability bound limits transmitted measurement noise of norm at most $\delta$ to $\delta /(2\sqrt{\alpha })$. Taking $\alpha \to 0$ while $\delta /\sqrt{\alpha }\to 0$ gives a sufficient rule for convergent regularization of an inverse problem. Holding $\alpha$ fixed leaves approximation bias; letting it vanish too quickly can admit overwhelming measurement noise.

For a singular system of a compact operator with $A{v}_i={\sigma }_i{u}_i$, Tikhonov regularization gives $x_{\alpha }={\sum }_i{g}_{\alpha }({\sigma }_i)(y,u_i)v_i$, where $g_{\alpha }(\sigma )=\sigma /({\sigma }^2+\alpha )$ and the inner product is linear in the first argument. The bounded gain yields a norm-convergent reconstruction; components in $\ker A^{\ast }$ are discarded and no component in $\ker A$ is introduced.

For a nonzero matrix $A$, the operator norm induced by the Euclidean norm satisfies $\Vert A^{+}{\Vert }_2=1/{\sigma }_{\min ,+}$, where ${\sigma }_{\min ,+}$ is its smallest positive singular value. The singular value decomposition makes the danger visible: a small nonzero stretch becomes a large amplification of errors when inverted.

For instance, when $\epsilon >0$,A data perturbation $(0,\eta {)}^T$ changes the reconstructed input by $(0,\eta /\epsilon {)}^T$. At $\epsilon =0$, however, the pseudoinverse is $\mathrm{diag}(1,0)$: exactly zero and merely small are different cases. Thus the pseudoinverse need not be continuous across a change of matrix rank. With uncertain data, truncated singular value decomposition or Tikhonov regularization can trade exact inversion of weak directions for stability; the cutoff or penalty adds a modelling choice.

The Tikhonov regularization of the Volterra integration operator replaces the unbounded differentiation gain $a_n$ by $a_n/(1+\alpha a_n^2)$. The series converges in $L^2$ for every datum and $\alpha >0$. On the range of the Volterra integration operator, it converges to the weak derivative of the datum as $\alpha \downarrow 0$. No pointwise differentiation of arbitrary noisy L2 space data is required.

For a bounded linear operator between Hilbert spaces, the spectral regularization method for quadratic Tikhonov regularization has scalar coefficient $s/(s^2+\alpha )$. Since $(s-\sqrt{\alpha }{)}^2\ge 0$, this coefficient is at most $1/(2\sqrt{\alpha })$. The spectral theorem for normal operators transfers this scalar bound to the operator norm, providing the noise term in the noise-bias decomposition for linear regularization.

The two Born approximation measurement operators $A_{\pm }$ map a supported scattering potential to fields on planes above and below it. The displayed Tikhonov regularization functional balances their residuals against a positive quadratic penalty. With an $L^2$ penalty, the unconstrained minimizer is $(A^{\ast }A+\alpha I{)}^{-1}{A}^{\ast }d$, where $A=(A_{+},A_{-})$. Regularization stabilizes estimation but cannot remove the nonuniqueness of Born non-scattering contrasts at one incident direction without extra information.

Let $d_n=k{i}^{n+1}{h}_n^{(1)}(kR)$ and map the near-field trace to the far field by $Ag=(g_n^m/d_n{)}_{n,m}$. Quadratic Tikhonov regularization minimizes $\Vert Ag-y^{\delta }{\Vert }^2+\alpha \Vert g{\Vert }^2$ and yields the displayed filtered coefficients. Its data-error gain is at most $1/(2\sqrt{\alpha })$. Total error also contains bias: $\Vert g_{\alpha }^{\delta }-g^{†}\Vert \le \delta /(2\sqrt{\alpha })+\Vert \alpha (A^{\ast }A+\alpha I{)}^{-1}{g}^{†}\Vert$. A convergent parameter rule has $\alpha \to 0$ and $\delta /\sqrt{\alpha }\to 0$. Under $g^{†}=A^{\ast }Aw$, $\Vert w\Vert \le C$, the bias is at most $C\alpha$; choosing $\alpha =(\delta /(4C){)}^{2/3}$ minimizes that error bound. Making $\alpha$ indefinitely large removes noise by shrinking the reconstruction to zero, not by recovering the true field.

The reconstruction operator in Tikhonov regularization has scalar gain $\sigma /({\sigma }^2+\alpha )$. Its maximum over $\sigma \ge 0$ is $1/(2\sqrt{\alpha })$, so a data perturbation of norm at most $\delta$ changes the solution by at most $\delta /(2\sqrt{\alpha })$. A convergent noisy-data parameter choice satisfies $\alpha \to 0$ and $\delta /\sqrt{\alpha }\to 0$. Fixed-parameter stability is distinct from stability of the unregularized inverse.