Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 326 / 1

A problem is well posed in the sense of Hadamard when a solution exists for every admissible datum, is unique, and depends continuously on the datum. It is ill posed if any one of these three properties fails.

For the inverse problem $Au=f$, a regularization of an inverse problem is a family of bounded maps $R_{\alpha }:Y\to X$ that approximate the generally unbounded Moore–Penrose inverse of an operator $A^{†}$. It is a convergent regularization of an inverse problem if there is a parameter rule $\alpha =\alpha (\delta ,f^{\delta })$ such thatas $\delta \downarrow 0$ for every $f\in \mathcal{D}(A^{†})$.

Let $({\sigma }_i,x_i,y_i)$ be the singular system of a compact operator. Since $P_n{x}_i=x_i$ for $i\le n$ and $P_n{x}_i=0$ for $i>n$, the truncated operator $T_n=A{P}_n$ has the finite singular value decompositionConsequentlyso $\boxed{T_n^{†}=A^{†}{Q}_n}$.

For completeness, $T_n^{†}{T}_n=P_n$ on $(\ker A{)}^{\perp }$ and $T_n{T}_n^{†}=Q_n$. These are self-adjoint orthogonal projectors, and henceThus all four Penrose equations hold, which verifies the formula independently of the singular expansion.

Repeated substitution in Landweber iteration from $u_0=0$ givesOn the $i$th singular vector, $A^{\ast }A$ has eigenvalue ${\sigma }_i^2$ and $A^{\ast }{y}_i={\sigma }_i{x}_i$. The finite geometric series therefore givesThis is a spectral regularization method with filter

The regularization parameter is the stopping index $k$: increasing $k$ reduces the approximation bias but amplifies noise. Equivalently one may use the parameter $\alpha =1/k$, which tends to zero as $k\to \infty$.

It is sufficient thatand that the stopping rule obeyFor example, $k(\delta )=\lfloor {\delta }^{-1}\rfloor$ works.

For exact data $f\in \mathcal{D}(A^{†})$, each factor $(1-\tau {\sigma }_i^2{)}^k$ tends to zero. The Picard criterion makes $⟨f,y_i⟩/{\sigma }_i$ square summable, while $|1-(1-\tau {\sigma }_i^2{)}^k|$ is uniformly bounded. The dominated convergence theorem on the resulting series yields $u_k(f)\to A^{†}f$.

For noisy data with $\Vert f^{\delta }-f\Vert \le \delta$, the filter representation givesIndeed, when $\tau {\sigma }^2\le 1$, Bernoulli's inequality gives $1-(1-\tau {\sigma }^2{)}^k\le \min (k\tau {\sigma }^2,1)$; when $\tau {\sigma }^2>1$, the quotient is uniformly bounded because $\tau \Vert A{\Vert }^2<2$. Hence the triangle inequality giveswhich proves that early-stopped Landweber iteration is a convergent regularization of an inverse problem.