= Ill-posed inverse problem
An <inverse problem> is an <ill-posed problem> if it fails any requirement of a <well-posed problem>: existence for every admissible datum, uniqueness, or <continuity> of the reconstructed object as the data change. These requirements depend on the chosen spaces and <norms>, not just the equation. Clason's regularization notes formulate this distinction explicitly.https://arxiv.org/pdf/2001.00617[Christian Clason, Regularization of Inverse Problems, §3]{ref}
Complete averaging illustrates lost information:
$$
B=\begin{pmatrix}0.5&0.5\\0.5&0.5\end{pmatrix},\qquad
B(t,1-t)^T=(0.5,0.5)^T\quad\hbox{for every }t.
$$
The difference mode $q_-=(1,-1)^T/\sqrt2$ belongs to the <null space> because $Bq_-=0$. Even restricting both pixels to $[0,1]$ leaves infinitely many indistinguishable inputs. This is failure of uniqueness. Data such as $(0.51,0.49)^T$ also have no exact preimage under $B$, illustrating failure of existence when arbitrary two-component observations are admissible.
The two-pixel <matrix> $A$ in the <inverse problem> example has a weakly observed difference mode, not a <null space> mode. Its inverse is continuous: this finite-dimensional problem is a <well-posed problem> with noise-sensitive conditioning. Small positive <singular values> alone do not imply mathematical ill-posedness. In infinite-dimensional problems, positive <singular values> approaching zero can make inversion unbounded: unit solution modes then produce data tending to zero. This destroys <continuity> even without a <null space>. <Regularization of an inverse problem> must distinguish missing information from information recoverable only at the cost of amplifying <measurement noise>.
\Image[/inverse-problem-nullspace.png]
{title=Two-pixel averaging compresses a square to a thin diamond or collapses it to a line}
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