Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 326 / 2 / 1

Fix a singular system of a compact operator, with $K{v}_j={\sigma }_j{w}_j$ and $K^{\ast }{w}_j={\sigma }_j{v}_j$, ${\sigma }_j>0$. A spectral regularization method has the formIt annihilates $\mathcal{R}(K{)}^{\perp }$. Standard sufficient spectral filter conditions are boundedness of ${\sup }_{0<\sigma \le \Vert K\Vert }|\sigma g_{\alpha }({\sigma }^2)|$ for each $\alpha$, pointwise $\lambda g_{\alpha }(\lambda )\to 1$ as $\alpha \downarrow 0$ for $\lambda >0$, and a uniform bound on $|\lambda g_{\alpha }(\lambda )|$. These give bounded linear operators and convergence on the domain of the Moore–Penrose inverse of an operator by the Picard criterion and dominated convergence theorem.

For Tikhonov regularization,This can be computed by solving the Tikhonov normal equation, without knowing the singular value decomposition.

For spectral cutoff regularization, takewith value zero at $\lambda =0$. Only singular values ${\sigma }_j\ge \sqrt{\alpha }$ are inverted. Here $g_{\alpha }$ multiplies $\sigma$; conventions calling the full coefficient $\sigma g_{\alpha }({\sigma }^2)$ the spectral filter are equivalent.

Suppose each $R_{\alpha }:\mathcal{V}\to \mathcal{U}$ is a bounded linear operator and $R_{\alpha }f\to K^{†}f$ for every $f\in \mathcal{D}(K^{†})$ as $\alpha \downarrow 0$. A standard sufficient a priori regularization parameter choice satisfiesFor every datum with $\Vert f^{\delta }-f\Vert \le \delta$, the noise-bias decomposition for linear regularization givesThus the parameter rule gives a convergent regularization of an inverse problem, uniformly over data in the prescribed noise ball for each fixed admissible exact datum. For Tikhonov regularization, $\Vert R_{\alpha }\Vert \le 1/(2\sqrt{\alpha })$, so $\alpha \to 0$ and $\delta /\sqrt{\alpha }\to 0$ suffice. The same sufficient noise scaling holds for spectral cutoff regularization.

Starting from zero, the Landweber iteration isFor $K\ne 0$, take the step size $0<\tau <2/\Vert K{\Vert }^2$. Iterating the linear update givesand its Landweber spectral filter in a singular system of a compact operator isThe strict upper step size bound makes $|1-\tau {\sigma }_j^2|<1$ for every positive singular value. A common more restrictive choice is $0<\tau \le \Vert K{\Vert }^{-2}$, giving nonnegative damping factors. Each finite iterate is bounded and linear; early stopping of Landweber iteration controls the amplification of noise as smaller singular values are progressively inverted. If $K=0$, every iterate is zero for any positive step size.