Nonlinear Landweber iteration with discrepancy stopping

Let $X,Y$ be real Hilbert spaces, let $U\subseteq X$ be open, and let $F:U\to Y$ be Fréchet differentiable at every point with a derivative map $F^{\mathrm{\prime }}:U\to \mathcal{B}(X,Y)$ that is continuous in operator norm. This is the real-Hilbert forward-map convention. Assume Countable Choice, so every bounded derivative $F^{\mathrm{\prime }}(x)$ has the Hilbert adjoint $F^{\mathrm{\prime }}(x{)}^{\ast }:Y\to X$ characterized by $⟨F^{\mathrm{\prime }}(x)u,v{⟩}_Y=⟨u,F^{\mathrm{\prime }}(x{)}^{\ast }v{⟩}_X$.

An admissible local instance consists of an initial point $x_0\in U$, a radius $r>0$, an exact solution $x^{†}\in {\bar{B}}_r(x_0):=\{x\in X:\mathrm{\parallel }x-x_0{\mathrm{\parallel }}_X\le r\}$ with exact datum $y=F(x^{†})$, a noise level $\delta \ge 0$, noisy datum $y^{\delta }\in Y$ with $\mathrm{\parallel }{y}^{\delta }-y{\mathrm{\parallel }}_Y\le \delta$, a constant $0\le \eta <1\mathrm{/}2$, a discrepancy parameter $\tau >2(1+\eta )\mathrm{/}(1-2\eta )$, and a finite update budget $M\in {\mathbb{N}}_0$. The closed outer ball ${\bar{B}}_{2r}(x_0)$ is contained in $U$; $\mathrm{\parallel }{F}^{\mathrm{\prime }}(x)\mathrm{\parallel }\le 1$ for every $x\in {\bar{B}}_{2r}(x_0)$; and the uniform two-point tangential-cone inequality

$$\mathrm{\parallel }F(x)-F(\tilde{x})-F^{\mathrm{\prime }}(x)(x-\tilde{x}){\mathrm{\parallel }}_Y\le \eta \mathrm{\parallel }F(x)-F(\tilde{x}){\mathrm{\parallel }}_Y$$

holds for every $x,\tilde{x}\in {\bar{B}}_{2r}(x_0)$. This is the uniform family of the one-basepoint local tangential-cone inequality; the shared constant and pair domain are part of the admissibility hypothesis. The exact solution and the bounds are analysis data and need not be available to the computational loop.

Inputs: $F$, $x_0$, $y^{\delta }$, $\delta$, $\tau$, and $M$ from an admissible local instance.

Output: Either stopped with an index $k$, iterate $x_k^{\delta }$, and residual $\mathrm{\parallel }{y}^{\delta }-F(x_k^{\delta }){\mathrm{\parallel }}_Y$, or budget_exhausted with the last iterate and its residual.

Procedure: Set $x_0^{\delta }:=x_0$. For $k=0,1,\dots ,M$, compute $r_k^{\delta }:=\mathrm{\parallel }{y}^{\delta }-F(x_k^{\delta }){\mathrm{\parallel }}_Y$. If $r_k^{\delta }\le \tau \delta$, return stopped with $(k,x_k^{\delta },r_k^{\delta })$. If $k=M$, return budget_exhausted with $(k,x_k^{\delta },r_k^{\delta })$. Otherwise set

$$x_{k+1}^{\delta }:=x_k^{\delta }+F^{\mathrm{\prime }}(x_k^{\delta }{)}^{\ast }(y^{\delta }-F(x_k^{\delta }))$$

and continue at the next index. Each update is evaluated at a current point in $U$, where the derivative and its Hilbert adjoint are defined.

Real-Hilbert forward-map and exact-data interface

Fréchet derivatives are bounded linear operators, with the derivative map defined pointwise

Completeness and real-Hilbert-space convention

Inner-product convention, linear in the first argument

One-basepoint local tangential-cone inequality

Operator norm of each derivative

Hilbert adjoints under Countable Choice

Adjoint identity and norm equality

Axiom of Countable Choice

Christian Clason, Regularization of Inverse Problems, Chapter 11

S. Hubmer, E. Sherina, S. Kindermann, and K. Raik, A numerical comparison of some heuristic stopping rules for nonlinear Landweber iteration, §2.1