We investigate the ill-posed inverse problem of recovering unknown spatially dependent parameters in nonlinear evolution partial differential equations (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.

中文翻译：

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非线性偏微分方程反问题的双层迭代正则化

我们研究了在非线性演化偏微分方程（PDE）中恢复未知空间相关参数的不适定反问题。我们提出了一种双层 Landweber 方案，其中上层参数重建嵌入了下层状态近似。这可以看作是结合了经典的简化设置和较新的一次性设置，使我们能够分别利用参数到状态映射的适定性，并绕过必须精确求解非线性偏微分方程。利用这一点，我们推导出上下层迭代的停止规则以及双层方法的收敛。我们讨论了磁粒子成像中 Landau-Lifshitz-Gilbert 方程参数识别的应用。