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A nonlinear Tikhonov functional can have two global minimizers

Example

Let $X = Y = R$ with the usual inner product, let $U = R$ , and define $F : U \to Y$ by $F (t) = t^{2}$ . Take $x_{0} = 0$ , $y = 1$ , and $α = 1$ . The domain $U$ is weakly closed and, whenever $t_{n}$ converges weakly to $t$ and $F (t_{n})$ converges weakly to $z$ , one has $z = F (t)$ ; weak convergence means convergence under every bounded real-linear functional. The Tikhonov functional is

J (t) = \frac{1}{2} (t^{2} - 1)^{2} + \frac{1}{2} t^{2} (t \in R) .

It has exactly two distinct global minimizers, $t = 1 / \sqrt{2}$ and $t = - 1 / \sqrt{2}$ .

Example references

Nonlinear forward maps on an open real Hilbert domain

Weakly closed forward maps

The Tikhonov functional

Real Hilbert spaces

Weak sequence convergence

Christian Clason, Regularization of Inverse Problems, Chapter 10

Benedikt Wirth, Inverse Problems, Remark 84

Facts & Assumptions

Given: $X = Y = R$ with the standard real inner product and norm, $U = R$ , $F (t) = t^{2}$ , $x_{0} = 0$ , $y = 1$ , and $α = 1$ .

[F1]

The standard real inner product induces the usual norm, and $R$ is complete for it; hence it is a real Hilbert space (Real and complex inner-product spaces and their induced length, The reals are complete, Hilbert space).

[F2]

Weak convergence is convergence under every bounded real-linear functional; in particular the identity functional shows that weak convergence in $R$ implies ordinary convergence (Weak convergence of nets and sequences).

[F3]

A weakly closed forward map has weakly closed domain and sequentially weak-to-weak closed graph (Weakly closed nonlinear forward maps on Hilbert spaces).

[F4]

The nonlinear Tikhonov functional is $\frac{1}{2} ∥ F (t) - y ∥^{2} + \frac{α}{2} ∥ t - x_{0} ∥^{2}$ (The nonlinear Tikhonov functional).

[F5]

A nonlinear forward map is a map from a nonempty norm-open subset of a real Hilbert space to a real Hilbert space; no linearity is assumed (Nonlinear forward maps between Hilbert spaces).

Verification

1.1givenF1F2F3F5algebra

The domain $U = R$ is open and weakly closed. The map is nonlinear because $F (1) + F (- 1) = 2 \neq 0 = F (0)$ . If $t_{n} ⇀ t$ , [F2] gives $t_{n} \to t$ , hence $t_{n}^{2} \to t^{2}$ ; if also $F (t_{n}) ⇀ z$ , applying [F2] to the output gives $t_{n}^{2} \to z$ , so uniqueness of real limits gives $z = t^{2}$ . Thus the graph has the required sequential weak-to-weak closure, and [F1, F5] identify this as an open-domain nonlinear forward map between real Hilbert spaces.

1.2givenF4algebra

By [F4] and the given parameters, $J (t) = \frac{1}{2} (t^{2} - 1)^{2} + \frac{1}{2} t^{2}$ . Set $u = t^{2} \geq 0$ . Then $J (t) = \frac{1}{2} (u - 1)^{2} + \frac{1}{2} u$ .

2.1step 1.2algebra∎

Completing the square gives $J (t) = \frac{1}{2} (u - \frac{1}{2})^{2} + \frac{3}{8}$ . This is at least $3 / 8$ , with equality exactly when $u = 1 / 2$ , equivalently $t = 1 / \sqrt{2}$ or $t = - 1 / \sqrt{2}$ . These two distinct points are therefore exactly the global minimizers.

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Sources

Christian Clason, Regularization of Inverse Problems, Chapter 10, nonuniqueness discussion (standard reference, not scraped)
Benedikt Wirth, Inverse Problems, Remark 84 (standard reference, not scraped)