TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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Variational source conditions yield nonlinear regularization rates

Statement

Let $X, Y$ be real Banach spaces, $D (F) \subseteq X$ , and $F : D (F) \to Y$ . Let $R : X \to (- \infty, + \infty]$ be proper and convex, let $x^{†} \in D (F) \cap dom R$ , set $y = F (x^{†})$ , and choose $ξ^{†} \in \partial R (x^{†})$ . Fix $1 \leq p < \infty$ , $0 \leq β < 1$ , and a level $R_{0} > R (x^{†})$ . Let $ψ : [0, \infty) \to [0, \infty)$ be continuous, nondecreasing and concave, with $ψ (0) = 0$ and $ψ (t) > 0$ for $t > 0$ , and assume the variational source inequality

$- ⟨ ξ^{†}, x - x^{†} ⟩_{X^{*}, X} \leq β D_{R}^{ξ^{†}} (x, x^{†}) + ψ ⁣ (∥ F (x) - F (x^{†}) ∥_{Y}^{p}) (x \in L_{R_{0}} : = {x \in D (F) : R (x) \leq R_{0}}),$

where $D_{R}^{ξ^{†}} (x, x^{†}) : = R (x) - R (x^{†}) - ⟨ ξ^{†}, x - x^{†} ⟩_{X^{*}, X}$ . Extend $- ψ$ by $+ \infty$ on negative arguments and write $H (q) : = (- ψ)^{*} (- q) = \sup_{s \geq 0} (ψ (s) - q s)$ ; assume $H (q)$ is finite for sufficiently large $q$ and $H (q) \to 0$ as $q \to \infty$ . For sufficiently small noise levels $δ > 0$ and data $y^{δ} \in Y$ with $∥ y^{δ} - y ∥_{Y} \leq δ$ , define $J_{α}^{δ} (x) : = p^{- 1} ∥ F (x) - y^{δ} ∥_{Y}^{p} + α R (x)$ . Suppose $α = α (δ) > 0$ tends to zero, $0 \leq ε_{δ} \leq C_{ε} δ^{p}$ , and $x_{α}^{δ, ε} \in D (F)$ is an $ε_{δ}$ -minimizer, meaning $J_{α}^{δ} (x_{α}^{δ, ε}) \leq \inf_{z \in D (F)} J_{α}^{δ} (z) + ε_{δ}$ . Set $c_{p} : = 2^{1 - p} / p$ and $ρ_{δ} : = H (c_{p} / α (δ))$ . Assume that for constants $c_{-}, c_{+}, C_{H} > 0$ independent of $δ$ ,

$c_{-} δ^{p} \leq α (δ) ρ_{δ} \leq c_{+} δ^{p}, H (c_{p} / (2 α (δ))) \leq C_{H} ρ_{δ}$

for all sufficiently small $δ$ . Then for all sufficiently small $δ$ , $x_{α}^{δ, ε} \in L_{R_{0}}$ and

$D_{R}^{ξ^{†}} (x_{α}^{δ, ε}, x^{†}) = O (ρ_{δ}), ∥ F (x_{α}^{δ, ε}) - y^{δ} ∥_{Y} = O (δ) .$

If, in addition, $ψ (C_{0} δ^{p}) = O (ρ_{δ})$ for a fixed $C_{0}$ such that $T_{δ} : = ∥ F (x_{α}^{δ, ε}) - F (x^{†}) ∥_{Y}^{p} \leq C_{0} δ^{p}$ for all sufficiently small $δ$ , then $∣ R (x_{α}^{δ, ε}) - R (x^{†}) ∣ = O (ρ_{δ})$ . If also $D_{R}^{ξ^{†}} (x, x^{†}) \geq c_{R} ∥ x - x^{†} ∥_{X}^{s}$ on $L_{R_{0}}$ for constants $c_{R}, s > 0$ , then $∥ x_{α}^{δ, ε} - x^{†} ∥_{X}^{s} = O (ρ_{δ})$ .

In the Hilbert special case, set $p = 2$ and suppose $X, Y$ are real Hilbert spaces, $F : U \to Y$ has the real open-domain interface of a nonlinear forward map, $x^{†}$ is an $x_{0}$ -minimum-norm exact solution, $R (x) = \frac{1}{2} ∥ x - x_{0} ∥_{X}^{2}$ , and the source inequality on a regularizer sublevel contained in $U$ is

$⟨ x^{†} - x_{0}, x^{†} - x ⟩_{X} \leq β_{1} \frac{1}{2} ∥ x - x^{†} ∥_{X}^{2} + β_{2} ∥ F (x) - F (x^{†}) ∥_{Y}$

with $0 \leq β_{1} < 1$ and $β_{2} \geq 0$ . For exact global minimizers of the nonlinear Tikhonov functional with $α (δ) ≍ δ$ , the theorem gives $∥ x_{α (δ)}^{δ} - x^{†} ∥_{X} = O (\sqrt{δ})$ and $∥ F (x_{α (δ)}^{δ}) - y^{δ} ∥_{Y} = O (δ)$ . The assertion is conditional on such a global minimizer being supplied; it does not assert existence or cover local numerical minima.

Statement references

Christian Clason, Regularization of Inverse Problems, equation (10.17) and Theorem 10.7

Frederic Weidling, Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs, Theorem 2.20

Barbara Kaltenbacher, Convergence rates under a range invariance condition with application to electrical impedance tomography, Lemma 2 and Theorem 3

Facts & Assumptions

Given: All hypotheses in the Statement hold. Write $x_{δ} : = x_{α}^{δ, ε}$ , $T_{δ} : = ∥ F (x_{δ}) - F (x^{†}) ∥_{Y}^{p}$ , and $r_{δ} : = ∥ F (x_{δ}) - y^{δ} ∥_{Y}$ .

[A1]

$δ > 0$ , $x_{δ}$ is an $ε_{δ}$ -minimizer of $J_{α}^{δ}$ , and $∥ y^{δ} - y ∥_{Y} \leq δ$ , with $0 \leq ε_{δ} \leq C_{ε} δ^{p}$ .

[A2]

The VSC holds on $L_{R_{0}}$ with the chosen $ξ^{†}$ , $β$ , and $ψ$ .

[A3]

$α ρ_{δ} ≍ δ^{p}$ , $H (c_{p} / (2 α)) \leq C_{H} ρ_{δ}$ , $α \to 0$ , and $ρ_{δ} \to 0$ .

[A4]

For the optional penalty conclusion, $ψ (C_{0} δ^{p}) = O (ρ_{δ})$ for a fixed $C_{0}$ that bounds $T_{δ} / δ^{p}$ for small $δ$ .

[A5]

For the optional norm conclusion, $D_{R}^{ξ^{†}} (x, x^{†}) \geq c_{R} ∥ x - x^{†} ∥_{X}^{s}$ on $L_{R_{0}}$ for constants $c_{R}, s > 0$ .

[A6]

For the Hilbert conclusion, $R (x) = \frac{1}{2} ∥ x - x_{0} ∥_{X}^{2}$ , $ξ^{†} (h) = ⟨ x^{†} - x_{0}, h ⟩_{X}$ , the displayed $x_{0}$ -minimum-norm VSC holds with $0 \leq β_{1} < 1$ and $β_{2} \geq 0$ , and $α ≍ δ$ .

[F1]

Banach-space convex subgradients and dual pairings: for $x \in dom R$ , $\partial R (x) : = {ξ \in X^{*} : R (z) \geq R (x) + ⟨ ξ, z - x ⟩_{X^{*}, X} for every z \in X}$ .

[F2]

Bregman distances for convex regularizers: $D_{R}^{ξ^{†}} (x, x^{†}) = R (x) - R (x^{†}) - ⟨ ξ^{†}, x - x^{†} ⟩_{X^{*}, X}$ and is nonnegative.

[F3]

Variational source conditions on a regularizer sublevel: $- ⟨ ξ^{†}, x - x^{†} ⟩_{X^{*}, X} \leq β D_{R}^{ξ^{†}} (x, x^{†}) + ψ (∥ F (x) - F (x^{†}) ∥_{Y}^{p})$ on $L_{R_{0}}$ .

[F4]

A conjugate bound for residual and Bregman errors: from $c_{1} r + c_{2} α e \leq d + α ψ (C_{3} r)$ , the stated conjugate bounds hold for $e$ and $r$ .

[F5]

Nonlinear forward maps between Hilbert spaces: in the Hilbert specialization $F : U \to Y$ is a real-Hilbert forward map on an open domain.

[F6]

x₀-minimum-norm solutions of nonlinear equations: the Hilbert specialization assumes the exact solution globally minimizes distance to $x_{0}$ over its exact solution set.

[F7]

The nonlinear Tikhonov functional: for a datum $y \in Y$ , $J_{α, y, x_{0}} (x) = \frac{1}{2} ∥ F (x) - y ∥_{Y}^{2} + \frac{α}{2} ∥ x - x_{0} ∥_{X}^{2}$ .

[F8]

A norm on a real vector space, the induced metric, and the dictionary with the metric axioms: each Banach norm satisfies the triangle inequality.

[F9]

Real and complex inner-product spaces and their induced length: a real Hilbert norm satisfies $∥ v ∥_{X}^{2} = ⟨ v, v ⟩_{X}$ .

[F10]

Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs: $∣ ⟨ u, v ⟩_{X} ∣ \leq ∥ u ∥_{X} ∥ v ∥_{X}$ .

Proof

1.1givenA1A3algebra

Comparing the approximate-minimum inequality with the exact solution gives $J_{α}^{δ} (x_{δ}) \leq δ^{p} / p + α R (x^{†}) + ε_{δ}$ . Since the data term is nonnegative, $R (x_{δ}) - R (x^{†}) \leq δ^{p} / (p α) + ε_{δ} / α = O (ρ_{δ})$ by [A1] and [A3]. As $ρ_{δ} \to 0$ and $R_{0} > R (x^{†})$ , $x_{δ} \in L_{R_{0}}$ for sufficiently small $δ$ .

1.2A6F1F2F3F5F6F7F9F10algebra

By [F5]–[F7] the special case uses the open-domain Hilbert map, the stipulated global $x_{0}$ -minimum-norm exact solution, and the existing quadratic nonlinear Tikhonov functional. Under [A6], define $ξ^{†} (h) : = ⟨ x^{†} - x_{0}, h ⟩_{X}$ . By [F10] this is a bounded real-linear functional; expansion using [F9] gives $R (x) - R (x^{†}) - ⟨ ξ^{†}, x - x^{†} ⟩_{X^{*}, X} = \frac{1}{2} ∥ x - x^{†} ∥_{X}^{2} \geq 0$ , so [F1] gives $ξ^{†} \in \partial R (x^{†})$ and [F2] gives $D_{R}^{ξ^{†}} (x, x^{†}) = \frac{1}{2} ∥ x - x^{†} ∥_{X}^{2}$ . Taking $ψ (s) = (β_{2} + 1) \sqrt{s}$ , the stated Hilbert source inequality implies the VSC with $p = 2$ .

2.1A1F8step 1.1algebra

By [F8], $T_{δ}^{1 / p} \leq r_{δ} + δ$ . For $p = 1$ , $(u + v)^{p} = u^{p} + v^{p}$ ; for $p > 1$ , $t \mapsto t^{p}$ is continuous at zero and has nonnegative second derivative $p (p - 1) t^{p - 2}$ on $(0, \infty)$ , hence is convex on $[0, \infty)$ , and its defining convexity inequality at the midpoint gives $(u + v)^{p} \leq 2^{p - 1} (u^{p} + v^{p})$ . Taking $u = r_{δ}$ , $v = δ$ yields $T_{δ} \leq 2^{p - 1} (r_{δ}^{p} + δ^{p})$ and thus $r_{δ}^{p} / p \geq c_{p} T_{δ} - δ^{p} / p$ . Substituting into the comparison inequality from 1.1 gives $c_{p} T_{δ} + α (R (x_{δ}) - R (x^{†})) \leq 2 δ^{p} / p + ε_{δ}$ .

2.2A3A6step 1.2algebra

For $B : = β_{2} + 1 > 0$ , the conjugate is $H (q) = \sup_{u \geq 0} (B u - q u^{2}) = B^{2} / (4 q)$ , by completing the square. With $p = 2$ , $c_{p} = 1 / 4$ , so $α H (c_{p} / α) = B^{2} α^{2} ≍ δ^{2}$ and $ρ_{δ} = H (c_{p} / α) = B^{2} α ≍ δ$ when $α ≍ δ$ . The scale comparability in [A3] also holds because $H (c_{p} / (2 α)) = 2 H (c_{p} / α)$ .

3.1F2F3step 2.1algebra

By [F2] and [F3], $R (x_{δ}) - R (x^{†}) = D_{R}^{ξ^{†}} (x_{δ}, x^{†}) + ⟨ ξ^{†}, x_{δ} - x^{†} ⟩_{X^{*}, X} \geq (1 - β) D_{R}^{ξ^{†}} (x_{δ}, x^{†}) - ψ (T_{δ})$ . Combining this with the objective comparison in step 2.1 gives $c_{p} T_{δ} + α (1 - β) D_{R}^{ξ^{†}} (x_{δ}, x^{†}) \leq d_{δ} + α ψ (T_{δ})$ , where $d_{δ} : = 2 δ^{p} / p + ε_{δ}$ .

4.1F4step 3.1

Apply [F4] with $c_{1} = c_{p}$ , $c_{2} = 1 - β$ , $C_{3} = 1$ , $r = T_{δ}$ , $e = D_{R}^{ξ^{†}} (x_{δ}, x^{†})$ , and $d = d_{δ}$ . It follows that $D_{R}^{ξ^{†}} (x_{δ}, x^{†}) \leq \frac{1}{1 - β} (\frac{d_{δ}}{α} + ρ_{δ})$ and $T_{δ} \leq \frac{2}{c_{p}} (d_{δ} + α H (c_{p} / (2 α)))$ .

5.1A1A3F8step 4.1algebra

From [A1] and [A3], $d_{δ} = O (δ^{p})$ , $d_{δ} / α = O (ρ_{δ})$ , and $d_{δ} + α H (c_{p} / (2 α)) = O (δ^{p})$ . Thus $D_{R}^{ξ^{†}} (x_{δ}, x^{†}) = O (ρ_{δ})$ and $T_{δ} = O (δ^{p})$ . The triangle inequality gives $r_{δ} \leq T_{δ}^{1 / p} + δ$ , so the noisy-data residual is $O (δ)$ .

6.1A4A5F2F3step 1.1step 5.1

For the penalty rate, [F2]–[F3] give $R (x_{δ}) - R (x^{†}) \geq - ψ (T_{δ})$ , while step 1.1 supplies its upper bound. If [A4] holds for a fixed $C_{0}$ bounding $T_{δ} / δ^{p}$ , then $∣ R (x_{δ}) - R (x^{†}) ∣ = O (ρ_{δ})$ . If [A5] also holds, the Bregman bound in step 5.1 gives $∥ x_{δ} - x^{†} ∥_{X}^{s} = O (ρ_{δ})$ .

7.1A6step 5.1step 1.2step 2.2algebra∎

Applying the Bregman bound from step 5.1 and the exact Bregman identity from step 1.2 gives $∥ x_{α (δ)}^{δ} - x^{†} ∥_{X}^{2} = O (δ)$ and $∥ F (x_{α (δ)}^{δ}) - y^{δ} ∥_{Y} = O (δ)$ , hence the asserted $O (\sqrt{δ})$ solution error and $O (δ)$ residual.

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Sources

Christian Clason, Regularization of Inverse Problems, Theorem 10.7 (standard reference, not scraped)
Frederic Weidling, Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs, Theorem 2.20 (standard reference, not scraped)
Barbara Kaltenbacher, Convergence rates under a range invariance condition with application to electrical impedance tomography, Lemma 2 and Theorem 3 (standard reference, not scraped)