Normal equation for a linear inverse problem ($A^{\ast }Au=A^{\ast }f$)

The normal equation for $Au=f$ is $A^{\ast }Au=A^{\ast }f$. Its solutions minimize $\Vert Au-f\Vert$; when they exist, the unique solution orthogonal to $\ker A$ is the Moore--Penrose solution $A^{†}f$.

A singular system $({\sigma }_j,u_j,v_j)$ for a compact operator $A:X\to Y$ has positive singular values ${\sigma }_j\to 0$ and orthonormal vectors satisfying $A{v}_j={\sigma }_j{u}_j$ and $A^{\ast }{u}_j={\sigma }_j{v}_j$. It diagonalizes spectral regularization methods.

For a compact operator with singular system $({\sigma }_j,u_j,v_j)$, the datum $f$ lies in the domain of $A^{†}$ exactly when its component in $\overline{\mathrm{ran}A}$ satisfies

Landweber iteration applies $u_{n+1}=u_n+\tau A^{\ast }(f-A{u}_n)$. Starting at zero,and early stopping regularizes the inverse problem.