KIT | KIT-Bibliothek | Impressum | Datenschutz

On the iterative regularization of non-linear illposed problems in L∞

Pieronek, Lukas; Rieder, Andreas ORCID iD icon


Parameter identification tasks for partial differential equations are non-linear illposed problems where the parameters are typically assumed to be in $L^{\infty}$ . This Banach space is non-smooth, non-reflexive and non-separable and requires therefore a more sophisticated regularization treatment than the more regular $L^p$-spaces with $1 < p < \infty$. We propose a novel inexact Newton-like iterative solver where the Newton update is an approximate minimizer of a smooth Tikhonov functional over a finite-dimensional space whose dimension increases as the iteration progresses. In this way, all iterates stay bounded and the regularizer, delivered by a discrepancy principle, converges weakly-$\star$? to a solution when the noise level decreases to zero. Our theoretical results are demonstrated by numerical experiments based on the acoustic wave equation in one spatial dimension. This model problem satisfies all assumptions from our theoretical analysis.

Volltext §
DOI: 10.5445/IR/1000140578
Veröffentlicht am 03.12.2021
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Angewandte und Numerische Mathematik (IANM)
Sonderforschungsbereich 1173 (SFB 1173)
Publikationstyp Forschungsbericht/Preprint
Publikationsmonat/-jahr 12.2021
Sprache Englisch
Identifikator ISSN: 2365-662X
KITopen-ID: 1000140578
Verlag Karlsruher Institut für Technologie (KIT)
Umfang 26 S.
Serie CRC 1173 Preprint ; 2021/46
Projektinformation SFB 1173/2 (DFG, DFG KOORD, SFB 1173/2 2019)
Externe Relationen Siehe auch
Schlagwörter non-linear inverse and illposed problem, inexact Newton regularization, non-reflexive and non-smooth Banach space
KIT – Die Forschungsuniversität in der Helmholtz-Gemeinschaft
KITopen Landing Page