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On the iterative regularization of non-linear illposed problems in $L^{\infty }$

Pieronek, Lukas 1; Rieder, Andreas ORCID iD icon 1
1 Fakultät für Mathematik (MATH), Karlsruher Institut für Technologie (KIT)


Parameter identification tasks for partial differential equations are non-linear illposed problems where the parameters are typically assumed to be in L∞. This Banach space is non-smooth, non-reflexive and non-separable and requires therefore a more sophisticated regularization treatment than the more regular Lp-spaces with 1<p<∞. We propose a novel inexact Newton-like iterative solver where the Newton update is an approximate minimizer of a smoothed Tikhonov functional over a finite-dimensional space whose dimension increases as the iteration progresses. In this way, all iterates stay bounded in L∞ and the regularizer, delivered by a discrepancy principle, converges weakly-⋆ 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.

Verlagsausgabe §
DOI: 10.5445/IR/1000160566
Veröffentlicht am 27.07.2023
DOI: 10.1007/s00211-023-01359-7
Zitationen: 1
Web of Science
Zitationen: 1
Zitationen: 1
Cover der Publikation
Zugehörige Institution(en) am KIT Fakultät für Mathematik (MATH)
Publikationstyp Zeitschriftenaufsatz
Publikationsmonat/-jahr 06.2023
Sprache Englisch
Identifikator ISSN: 0029-599X, 0945-3245
KITopen-ID: 1000160566
Erschienen in Numerische Mathematik
Verlag Springer
Band 154
Heft 1-2
Seiten 209–247
Vorab online veröffentlicht am 22.06.2023
Nachgewiesen in Scopus
Web of Science
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