= Inverse problem
{title2=$A(x)=y$}
{wiki}
An inverse problem reconstructs an unknown object $x$ from observed data $y$ and a specified model $A(x)=y$. The associated <forward problem> predicts the observations from $x$. In <image deblurring>, $x$ contains sharp pixel values and a <matrix> $A$ mixes neighbouring pixels; in <tomography>, the observations are integrals through the object. The appeal is that indirect measurements can reveal otherwise inaccessible structure. The difficulty is that the <forward problem> may erase precisely the distinctions we want to recover.
Consider this completely computable two-pixel <image deblurring> model:
$$
A=\begin{pmatrix}0.55&0.45\\0.45&0.55\end{pmatrix},\qquad
x_{\mathrm{true}}=\begin{pmatrix}1\\0\end{pmatrix},\qquad
y=Ax_{\mathrm{true}}=\begin{pmatrix}0.55\\0.45\end{pmatrix}.
$$
Each measured pixel retains $55\%$ of its own value and receives $45\%$ of its neighbour's. Adding <measurement noise> $e=(0.01,-0.01)^T$ gives $y^\delta=(0.56,0.44)^T$, whose direct reconstruction is $A^{-1}y^\delta=(1.1,-0.1)^T$. The <Euclidean norm> of the reconstruction error is $0.1\sqrt2$, ten times the data error $0.01\sqrt2$. A negative second pixel appears even though both observed pixels are positive.
Despite this wrong reconstruction, its <residual of an inverse problem> is exactly zero: the predicted data equal the noisy observations. Thus matching the data perfectly does not certify reconstruction accuracy. The relevant question is how large a change in the unknown could hide within the measurement uncertainty. A <condition number> quantifies that sensitivity for an invertible <matrix>; the <null space> identifies changes that no improvement in precision can reveal.
The exact <singular value decomposition> explains this amplification. With the <orthonormal basis>
$$
q_+=\frac{(1,1)^T}{\sqrt2},\qquad q_-=\frac{(1,-1)^T}{\sqrt2},\qquad
A=Q\begin{pmatrix}1&0\\0&0.1\end{pmatrix}Q^T,\quad Q=(q_+,q_-),
$$
the mean mode survives unchanged while the difference mode shrinks tenfold. Direct inversion multiplies difference-mode <measurement noise> by ten. The <condition number> is $\kappa_2(A)=\sigma_{\max}/\sigma_{\min}=10$. The beauty of the <singular value decomposition> is this separation into independent channels: it identifies which information survives and which needs protection by <regularization of an inverse problem>.
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