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Compact-operator Tikhonov residuals are continuous and increase from range distance to data norm

Statement

Assume Countable Choice. Let $A : X \to Y$ be compact between real or complex Hilbert spaces. For data $g \in Y$ and $λ > 0$ , let $x_{λ} (g) = (A^{*} A + λ I)^{- 1} A^{*} g$ and $r_{g} (λ) = ∥ A x_{λ} (g) - g ∥$ . Then $r_{g}$ is continuous and nondecreasing on $(0, \infty)$ , and $g - A x_{λ} (g) = λ (A A^{*} + λ I_{Y})^{- 1} g, ∥ λ (A A^{*} + λ I_{Y})^{- 1} ∥ \leq 1,$ and $\lim_{λ ↓ 0} r_{g} (λ) = dist (g, \overline{Ran A}), \lim_{λ \to \infty} r_{g} (λ) = ∥ g ∥ .$

Facts & Assumptions

Given: Countable Choice, compact $A : X \to Y$ between Hilbert spaces, data $g \in Y$ , and $λ > 0$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is the hypothesis of the compact spectral theorem; no full Choice is used.

[F1]

Compact spectral filters, source conditions, and qualification: The Tikhonov solution is $x_{λ} (g) = (A^{*} A + λ I)^{- 1} A^{*} g$ .

[F2]

A compact Hilbert-space Gram operator is positive and has the forward kernel: $A A^{*}$ is compact, positive and self-adjoint, with $\ker (A A^{*}) = \ker A^{*}$ .

[F3]

Spectral theorem for compact self adjoint operators: The nonzero eigenspaces of $A A^{*}$ are mutually orthogonal, and together with its kernel they give an orthogonal decomposition of $Y$ .

[F4]

Bounded multipliers on compact self-adjoint eigenspaces define bounded operators: A bounded scalar multiplier $q$ on the spectrum of a compact self-adjoint operator defines a bounded operator with norm at most $\sup ∣ q ∣$ .

[F5]

The shifted Hilbert-space normal operator is invertible with inverse norm at most $1 / λ$ : For bounded $T$ and $λ > 0$ , $T^{*} T + λ I$ is invertible.

[F6]

Orthogonality and the orthogonal complement: $z ⊥ Ran A$ exactly when $z$ is orthogonal to every $A x$ .

[F7]

Orthogonal decomposition by a closed subspace: A closed subspace of a Hilbert space has an orthogonal projection, and the distance to it is the norm of the orthogonal-complement component.

[F8]

Hilbert-adjoint identities: The adjoint identity relates $⟨ A x, z ⟩$ to $⟨ x, A^{*} z ⟩$ .

Proof

1.1F1F2F4F5algebraA1

Put $T = A A^{*}$ . The normal equation $(A^{*} A + λ I) x_{λ} (g) = A^{*} g$ follows from [F1] and [F5]. Applying $A$ gives $(T + λ I) A x_{λ} (g) = T g$ , whence $g - A x_{λ} (g) = λ (T + λ I)^{- 1} g .$ The inverse exists by [F5] applied to $A^{*} : Y \to X$ . The multiplier $q (s) = λ / (s + λ)$ , including $q (0) = 1$ , has supremum at most one; [F4] gives the operator norm bound.

2.1F3step 1.1algebraA1

Let $P_{s}$ be the eigenspace projections of $T$ for its nonzero eigenvalues $s$ , and $P_{0}$ the projection onto $\ker T$ . By [F3] and step 1.1, $r_{g} (λ)^{2} = ∥ P_{0} g ∥^{2} + \sum_{s > 0} {(\frac{λ}{s + λ})}^{2} ∥ P_{s} g ∥^{2} .$ Each coefficient $λ / (s + λ)$ is nondecreasing in $λ$ ; therefore the sum and its nonnegative square root are nondecreasing.

2.2F2F3step 1.1algebraA1

On any compact interval $[a, b] \subset (0, \infty)$ , $∣ \frac{λ}{s + λ} - \frac{μ}{s + μ} ∣ = \frac{s ∣ λ - μ ∣}{(s + λ) (s + μ)} \leq \frac{∥ T ∥}{a^{2}} ∣ λ - μ ∣$ for $0 \leq s \leq ∥ T ∥$ and $λ, μ \in [a, b]$ . Hence the residual multipliers vary continuously in operator norm, which makes $λ \mapsto λ (T + λ I)^{- 1} g$ and its norm continuous.

2.3F3step 1.1algebraA1

As $λ ↓ 0$ , the coefficient on $P_{0} g$ stays $1$ and each coefficient on $P_{s} g$ for $s > 0$ tends to zero. For a given $ε > 0$ , choose a finite spectral set whose orthogonal tail of $g$ has norm below $ε / 2$ ; the multipliers have norm at most one, and the finite positive-eigenvalue sum tends to zero. Thus the residual tends to $P_{0} g$ . As $λ \to \infty$ , the bound $\sup_{0 \leq s \leq ∥ T ∥} ∣ 1 - λ / (s + λ) ∣ \leq ∥ T ∥ / λ$ shows that the residual tends to $g$ .

3.1F2F6F7F8step 2.3A1∎

The adjoint identity [F8] gives $(Ran A)^{⊥} = \ker A^{*}$ : orthogonality to every $A x$ is equivalent to $⟨ x, A^{*} z ⟩ = 0$ for every $x$ , which is equivalent to $A^{*} z = 0$ . Taking closures leaves the orthogonal complement unchanged. By [F2], $\ker T = \ker A^{*}$ , and [F7] identifies $dist (g, \overline{Ran A}) = ∥ P_{0} g ∥$ . Step 2.3 gives the claimed lower endpoint, while its upper endpoint is $∥ g ∥$ .

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Sources

Guido Clason, Regularization of Inverse Problems, full lecture notes (standard reference, not scraped)
Per Christian Hansen, Introduction to Inverse Problems, Chapter 5: Regularization Parameter Choice (standard reference, not scraped)