We investigate the ill-posed inverse problem of recovering unknown spatially dependent parameters in nonlinear evolution PDEs. We propose a bi-level Landweber scheme, where the upper-level parameter reconstruction embeds a lower-level state approximation. This can be seen as combining the classical reduced setting and the newer all-at-once setting, allowing us to, respectively, utilize well-posedness of the parameter-to-state map, and to bypass having to solve nonlinear PDEs exactly. Using this, we derive stopping rules for lower- and upper-level iterations and convergence of the bi-level method. We discuss application to parameter identification for the Landau-Lifshitz-Gilbert equation in magnetic particle imaging.

翻译：本文研究非线性演化偏微分方程中未知空间依赖参数恢复的不适定反问题。我们提出一种双层Landweber格式，其中上层参数重建嵌入了下层状态近似。该方法可视为将经典简化设定与新型"一次性"设定相结合，从而既能利用参数-状态映射的适定性，又能避免精确求解非线性PDE。基于此框架，我们推导了上下层迭代的停止准则及双层方法的收敛性。最后讨论了该方法在磁粒子成像中Landau-Lifshitz-Gilbert方程参数识别中的应用。