DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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The nonlinear Tikhonov functional

Definition

Let $X$ and $Y$ be real Hilbert spaces, let $D \subseteq X$ be a nonempty domain, and let $F : D \to Y$ be a forward map. For a regularization parameter $α > 0$ , reference element $x_{0} \in X$ , and datum $y \in Y$ , the nonlinear Tikhonov functional is the map $J_{α, y, x_{0}} : D \to [0, \infty)$ defined by

J_{α, y, x_{0}} (x) : = \frac{1}{2} ∥ F (x) - y ∥_{Y}^{2} + \frac{α}{2} ∥ x - x_{0} ∥_{X}^{2} (x \in D) .

A global minimizer is a point $x_{α} \in D$ such that $J_{α, y, x_{0}} (x_{α}) \leq J_{α, y, x_{0}} (x)$ for every $x \in D$ . This definition asserts neither existence nor uniqueness.

Definition references

Forward maps and their domains

Real Hilbert spaces and their norms

Christian Clason, Regularization of Inverse Problems, equation (10.1)

Benedikt Wirth, Inverse Problems, Definition 83

Depends on

Used by

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Christian Clason, Regularization of Inverse Problems, equation (10.1) (standard reference, not scraped)
Benedikt Wirth, Inverse Problems, Definition 83 (standard reference, not scraped)