Quartic-norm variational regularization

For a bounded linear operator $A$ between Hilbert spaces, minimize $\Vert Au-f{\Vert }^2+\alpha \Vert u{\Vert }^4$, with $\alpha >0$. The objective is coercive and weakly lower semicontinuous, and its strictly convex norm penalty gives a unique minimizer. Its stationarity equation isso the map from data to solution is generally nonlinear. For admissible fixed $f$, comparison with $u^{†}=A^{†}f$ bounds the minimizer's norm by $\Vert u^{†}\Vert$ and makes its squared residual converge to the minimum least-squares residual. Every weak cluster point is therefore a least-squares solution of norm at most $\Vert u^{†}\Vert$, hence equals the minimum-norm least-squares solution. Weak lower semicontinuity then forces convergence of norms, and the Radon-Riesz theorem gives strong convergence. This supplies a nonlinear regularization of an inverse problem.

For completeness, the fixed-$\alpha$ data-to-solution map is continuous. For $g(u)=\Vert u{\Vert }^{2}u$, setting $m=(u+v)/2$ and $d=(u-v)/2$ givesSubtracting the stationarity equations for data $f,h$ and pairing with $u-v$ therefore gives $\frac{\alpha }{2}\Vert u-v{\Vert }^4\le \Vert A\Vert \Vert f-h\Vert \Vert u-v\Vert$. Thuswhich proves continuity directly.