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：我们研究了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。提出了一种双层Landweber方案，其中上层参数重构嵌入了下层状态近似。该方案可视为经典简化设定与新型全同步设定的结合，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偏微分方程。基于此，我们推导出下层与上层迭代的停止准则以及双层方法的收敛性。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。