Case Studies and a Pitfall for Nonlinear Variational Regularization Under Conditional Stability

Abstract

Conditional stability   estimates are a popular tool for the regularization of ill-posed problems. A drawback in particular under nonlinear operators is that additional regularization is needed for obtaining stable approximate solutions if the validity area of such estimates is not completely known. In this paper we consider Tikhonov regularization under conditional stability estimates for nonlinear ill-posed operator equations in Hilbert scales. We summarize assertions on convergence and convergence rate in three cases describing the relative smoothness of the penalty in the Tikhonov functional and of the exact solution. For oversmoothing penalties, for which the true solution no longer attains a finite value, we present a result with modified assumptions for a priori choices of the regularization parameter yielding convergence rates of optimal order for noisy data. We strongly highlight the local character of the conditional stability estimate and demonstrate that pitfalls may occur through incorrect stability estimates. Then convergence can completely fail and the stabilizing effect of conditional stability may be lost. Comprehensive numerical case studies for some nonlinear examples illustrate such effects.

Appendix: Proof of Proposition 9.4

In this proof we set $E:=\Vert x^\dagger \Vert _p$. To prove the convergence rate result (9.20) under the a priori parameter choice (9.21) it is sufficient to show that for sufficiently small $\delta >0$ there are two constants $K>0$ and $\tilde{E}>0$ such that the inequalities

$$\begin{aligned} \Vert x_{\alpha _{*}}^\delta - x^\dag \Vert _{-a} \le K \delta \end{aligned}$$

(9.42)

and

$$\begin{aligned} \Vert x_{\alpha _{*}}^\delta - x^\dag \Vert _{p} \le \tilde{E} \end{aligned}$$

(9.43)

hold. Namely, the convergence rate (9.20) follows directly from inequality chain

$$\begin{aligned} \Vert x_{\alpha _{*}}^\delta - x^\dag \Vert _{X} \le \Vert x_{\alpha _{*}}^\delta - x^\dag \Vert _{-a}^{\frac{p}{a + p}} \Vert x_{\alpha _{*}}^\delta - x^\dag \Vert _{p} ^{\frac{a}{a + p}} \le K^{\frac{p}{a +p}} \tilde{E}^{\frac{a}{a + p}}\, \delta ^{\frac{p}{a + p}}, \end{aligned}$$

which is valid for sufficiently small $\delta >0$ as a consequence of (9.42), (9.43) and of the interpolation inequality for the Hilbert scale $\{X_{\tau }\}_{\tau \in \mathbb {R}}$.

As an essential tool for the proof we use auxiliary elements $x_{\alpha }$, which are for all $\alpha >0$ the uniquely determined minimizers over all $x \in X$ of the artificial Tikhonov functional

$$\begin{aligned} T_{-a,\alpha }(x) := \Vert x - x^\dag \Vert _{-a}^{2} + \alpha \Vert B x\Vert _{X}^{2}. \end{aligned}$$

(9.44)

Note that the elements $x_{\alpha }$ are independent of the noise level $\delta >0$ and belong by definition to $X_1$, which is in strong contrast to $x^\dagger \notin X_1$.

The following lemma is an immediate consequence of [13, Prop. 2], see also [14, Prop. 3].

Lemma 9.2

Let $\Vert x^\dag \Vert _p=E$ and $x_{\alpha }$ be the minimizer of the functional $T_{-a,\alpha }$ from (9.44) over all $x \in X$. Given $\alpha _{*}=\alpha _{*}(\delta )>0$ as defined by formula (9.21) the resulting element $x_{\alpha _{*}}$ obeys the bounds

$$\begin{aligned} \Vert x_{\alpha _{*}}- x^\dag \Vert _{X}&\le E \delta ^{p/(a + p)}, \end{aligned}$$

(9.45)

$$\begin{aligned} \Vert B^{-a}(x_{\alpha _{*}}- x^\dag )\Vert _{X}&\le E \delta ,\end{aligned}$$

(9.46)

$$\begin{aligned} \Vert Bx_{\alpha _{*}}\Vert _{X}&\le E \delta ^{(p-1)/(a + p)} = E \frac{\delta }{\sqrt{\alpha _{*}}} \end{aligned}$$

(9.47)

and

$$\begin{aligned} \Vert x_{\alpha _{*}}- x^\dag \Vert _{p} \le E. \end{aligned}$$

Due to (9.45) we have $\Vert x_{\alpha _{*}}-x^\dag \Vert _X \rightarrow 0$ as $\delta \rightarrow 0$. Hence by Assumption 9.4, in particular because $x^\dagger$ is an interior point of $\mathcal {D}(F)$, for sufficiently small $\delta >0$ the element $x_{\alpha _{*}}$ belongs to $\mathcal {B}^X_r(x^\dagger )\subset \mathcal {D}(F)$ and moreover with $x_{\alpha _{*}}\in X_1$ the inequality (9.19) applies for $x=x_{\alpha _{*}}$ and such small $\delta$.

Instead of the inequality (9.8), which is missing in case of oversmoothing penalties, we can use here the inequality

$$\begin{aligned} T^{\delta }_{\alpha _{*}}(x_{\alpha _*}^\delta )\le T^{\delta }_{\alpha _{*}}(x_{\alpha _{*}}). \end{aligned}$$

(9.48)

as minimizing property for the Tikhonov functional. Using (9.48) it is enough to bound $T^{\delta }_{\alpha _{*}}(x_{\alpha _{*}})$ by $\overline{C}^2 \delta ^2$ with

$$\begin{aligned} \overline{C}:= \left( (\overline{K} E +1)^{2}+ E^{2}\right) ^{1/2} \end{aligned}$$

(9.49)

in order to obtain the estimates

$$\begin{aligned} \Vert F(x_{\alpha _{*}}^\delta ) - y^\delta \Vert _{Y} \le \overline{C} \delta \end{aligned}$$

(9.50)

and

$$\begin{aligned} \Vert Bx_{\alpha _{*}}^\delta \Vert _{X} \le \overline{C} \frac{\delta }{\sqrt{\alpha _{*}}}. \end{aligned}$$

(9.51)

Since the inequality (9.19) applies for $x=x_{\alpha _{*}}$ and sufficiently small $\delta >0$, we can estimate for such $\delta$ as follows:

$$\begin{aligned} T^{\delta }_{\alpha _{*}}(x_{\alpha _{*}})&\le \left( \Vert F(x_{\alpha _{*}}) - F(x^\dag )\Vert _Y + \Vert F(x^\dag ) - y^\delta \Vert _{Y}\right) ^2 + \alpha _{*}\Vert Bx_{\alpha _{*}}\Vert _{X}^{2}\\&\le \left( \overline{K} \Vert x_{\alpha _{*}}- x^\dag \Vert _{-a} + \delta \right) ^{2} + E^{2} \alpha _{*}\delta ^{2(p-1)/(a + p)}\\&\le \left( \overline{K} E\delta + \delta \right) ^{2} + E^{2}\delta ^{2} \\&= \left( (\overline{K} E +1)^{2} + E^{2}\right) \delta ^{2}. \end{aligned}$$

This ensures the estimates (9.50) and (9.51). Based on this we are going now to show that an inequality (9.42) is valid for some $K>0$. Here, we use the inequality (9.18) of Assumption 9.4, which applies for $x=x_{\alpha _{*}}^\delta$, and we find

$$\begin{aligned} \Vert x_{\alpha _{*}}^\delta - x^\dag \Vert _{-a}&\le \frac{1}{\underline{K}} \Vert F(x_{\alpha _{*}}^\delta ) - F(x^\dag )\Vert _{Y}\\&\le \frac{1}{\underline{K}} \left( \Vert F(x_{\alpha _{*}}^\delta ) -y^\delta \Vert _{Y} + \Vert F(x^\dag ) - y^\delta \Vert _{Y}\right) \\&\le \frac{1}{\underline{K}} \left( \overline{C} \delta + \delta \right) = \frac{1}{\underline{K}} \left( \overline{C} +1 \right) \delta =K \delta , \end{aligned}$$

where $\overline{C}$ is the constant from (9.49) and we derive $K:=\frac{1}{\underline{K}} \left( \overline{C} +1 \right)$.

Secondly, we still have to show the existence of a constant $\tilde{E}>0$ such that the inequality (9.43) holds. By using the triangle inequality in combination with (9.51) and (9.47) we find that

$$ \Vert B(x_{\alpha _{*}}^\delta - x_{\alpha _{*}})\Vert _{X} \le \Vert Bx_{\alpha _{*}}^\delta \Vert _{X} +\Vert Bx_{\alpha _{*}}\Vert _{X} \le (\overline{C}+E) \frac{\delta }{\sqrt{\alpha _{*}}}. $$

Again, we use the interpolation inequality and can estimate further as

$$ \Vert x_{\alpha _{*}}^\delta - x_{\alpha _{*}}\Vert _{p} \le \Vert x_{\alpha _{*}}^\delta - x_{\alpha _{*}}\Vert _{1}^{\frac{a + p}{a + 1}} \Vert x_{\alpha _{*}}^\delta - x_{\alpha _{*}}\Vert _{-a} ^{\frac{1 - p}{a + 1}} $$

$$ \le \left( (\overline{C}+E) \frac{\delta }{\sqrt{\alpha _{*}}}\right) ^{\frac{a + p}{a + 1}}\left( \Vert x_{\alpha _{*}}^\delta - x^\dag \Vert _{-a}+\Vert x^\dag - x_{\alpha _{*}}\Vert _{-a} \right) ^{\frac{1 - p}{a + 1}} $$

$$ \le \left( (\overline{C}+E) \frac{\delta }{\sqrt{\alpha _{*}}}\right) ^{\frac{a + p}{a + 1}}\left( (K+E)\delta \right) ^{\frac{1 - p}{a + 1}} $$

$$ \left( (\overline{C}+E)\delta ^{(p-1)/(a + p)} \right) ^{\frac{a + p}{a + 1}}\left( (K+E)\delta \right) ^{\frac{1 - p}{a + 1}}=:\bar{E}. $$

Finally, we have now

$$ \Vert x_{\alpha _{*}}^\delta - x^\dag \Vert _{p} \le \Vert x_{\alpha _{*}}^\delta - x_{\alpha _{*}}\Vert _{p} +\Vert x_{\alpha _{*}}- x^\dag \Vert _{p} \le \bar{E}+E =:\tilde{E}. $$

This shows (9.43) and thus completes the proof of Proposition 9.4.    $\square$