Multiscale analysis for ill-posed problems with semi-discrete Tikhonov regularization
\textit{H. Wendland}'s results [Numer. Math. 116, No. 3, 493--517 (2010; Zbl 1208.65067)] on multiscale analysis in Sobolev spaces on bounded domains can be applied to the interpretation of indirect measurements when the available data are discrete and noisy. In this context, compactly supported radial basis functions of varying radii are exploited. The present paper deals with the application of the theory from Wendland's paper to the stable approximate solution of ill-posed linear operator equations \(Af=g\) in a Hilbert space setting by a semi-discrete version of Tikhonov's regularization method. In particular, an a posteriori choice of the regularization parameter based on Morozov's discrepancy principle with corresponding error estimates for noisy data is included in the discussion. The authors have in mind linear inverse problems, where the (mostly compact) forward operator \(A\) has the structure of an integral operator NEWLINE\[NEWLINE[Af](x)=\int \limits _\Omega k(x,t) f(t) dt NEWLINE\]NEWLINE over a bounded domain \(\Omega \subset \mathbb{R}^k,\;k=1,2,\dots\). The developed theory and algorithm suppose that the problem is moderately ill-posed, which means here that there are constants \(0<\underline c \leq \overline c<\infty\) such that NEWLINE\[NEWLINE\underline c\,\|f\|_{H^\theta(\Omega)} \leq \|Af\|_{H^{\theta+a}(\Omega)} \leq \overline c\,\|f\|_{H^\theta(\Omega)} NEWLINE\]NEWLINE is valid for all \(f \in H^\theta(\Omega)\) and some \(\theta \in \mathbb{R}\). The constant \(a>0\) represents the degree of ill-posedness of the problem and is finite. Some numerical examples with the simple integration operator as \(A\) on \(\Omega=[0,2] \subset \mathbb{R}\) and two different exact solutions \(f\) illustrate the approach and associated ingredients like reproducing kernel Hilbert spaces, discretization schemes and error behavior.
scientific article; zbMATH DE number 6054457