= Normal equation for coercive Tikhonov regularization
{title2=$(K^*K+\alpha B^*B)u_\alpha=K^*f$}
For <bounded linear operators> $K:U\to V$ and $B:U\to W$ between real <Hilbert spaces>, $\alpha>0$ and $\|Bu\|\geq\beta\|u\|$ with $\beta>0$, the unique minimizer of $\|Ku-f\|^2/2+\alpha\|Bu\|^2/2$ satisfies $(K^*K+\alpha B^*B)u_\alpha=K^*f$. 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>.
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