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

Abstract

Parameter identification tasks for partial differential equations are non-linear illposed problems where the parameters are typically assumed to be in $$L^\infty $$ L ∞ . This Banach space is non-smooth, non-reflexive and non-separable and requires therefore a more sophisticated regularization treatment than the more regular $$L^p$$ L p -spaces with $$1<p<\infty $$ 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 $$L^\infty $$ L ∞ 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.