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 smoothed Tikhonov functional over a finite-dimensional space whose dimension increases as the iteration progresses. In this way, all iterates stay bounded in \(L^\infty \) 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.

偏微分方程的参数识别任务是非线性病态问题，其中参数通常假设在\(L^\infty \)中。这个 Banach 空间是非光滑、非自反和不可分的，因此需要比更规则的\(L^p\)空间更复杂的正则化处理，其中\(1<p<\infty \)。我们提出了一种新颖的不精确类牛顿迭代求解器，其中牛顿更新是有限维空间上平滑吉洪诺夫函数的近似最小化，该空间的维数随着迭代的进行而增加。通过这种方式，所有迭代都保持在\(L^\infty \)范围内，并且由差异原理提供的正则化器弱收敛 - \(\star \)当噪声水平降低到零时得到解决方案。我们的理论结果通过基于一空间维声波方程的数值实验得到了证明。该模型问题满足我们理论分析的所有假设。