Nonlinear Tikhonov regularization in Hilbert scales with balancing principle tuning parameter in statistical inverse problems

Figures & data

Figure 1. Upper bounds for the nonlinear data noise error (dotted line), Lepskii rule $\mathrm{\Phi }$ (dashed line), linear data noise error (full line) for error level ${\delta }^2=0.5$ (left, diamond) and 0.01 (right, diamond), and the corresponding minimum points for $\mathrm{\Psi }$ (circle).

Figure 2. Exact parameter $a^{†}$ (full line - $q=2.5$, dashed line - $q=3.5$, dotted line - $q=4.5$), and its corresponding empirical and theoretical (dashed - dotted) rate of convergence on a log-log scale

Table 1. Overview of the constants.

No.	Parameter	Definition	Value
1	q	Sobolev smoothness of $a^{†}$	2.5, 3.5, 4.5
2	$\gamma$	Upper bound on $a^{†}$	1
3	s	Smoothness of the penalization term in Tikhonov functional	2
4	p	Degree of ill-posedness	2
5	$q+2$	Sobolev smoothness of the exact data $u^{†}$	4.5, 5.5, 6.5
6	$q+1.5$	Hölder smoothness of the exact data $u^{†}$	4,5,6
7	$\beta$	Theoretical order optimal rate for local pol. est.	$n^{-\frac{4}{9}},n^{-\frac{5}{11}},n^{-\frac{6}{13}}$
8	$c_h$	Factor for bandwidth	[0.1, 1]
9	$l_2$	Factor in the tail behaviour of the loc.pol.est.	$\frac{1}{e}$
10	$\eta$	Exponent of the tail behaviour of the loc.pol.est.	1
11	$k_{\ast }$	Factor in the Lepskii rule	[0.05, 0.1]
12	n	Sample size	1000,2512,6310,15849,39811,10000
13	D	Length of the interval	1
14	g	Boundary conditions	1
15	$\lambda$	Lipschitz constant	1
16	$\mathrm{\Lambda }$	Lipschitz constant	4