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

Pieronek, Lukas; Rieder, Andreas

## Abstract:

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.

 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
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