ABSTRACT The parameters of many physical processes are unknown and have to be inferred from experimental data. The corresponding parameter estimation problem is often solved using iterative methods such as steepest descent methods combined with trust regions. For a few problem classes also continuous analogues of iterative methods are available. In this work, we expand the application of continuous analogues to function spaces and consider PDE (partial differential equation)-constrained optimization problems. We derive a class of continuous analogues, here coupled ODE (ordinary differential equation)–PDE models, and prove their convergence to the optimum under mild assumptions. We establish sufficient bounds for local stability and convergence for the tuning parameter of this class of continuous analogues, the retraction parameter. To evaluate the continuous analogues, we study the parameter estimation for a model of gradient formation in biological tissues. We observe good convergence properties, indicating that the continuous analogues are an interesting alternative to state-of-the-art iterative optimization methods.

中文翻译：

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对 PDE 约束逆问题的迭代优化的连续模拟

摘要 许多物理过程的参数是未知的，必须从实验数据中推断出来。相应的参数估计问题通常使用迭代方法解决，例如结合信任区域的最速下降方法。对于一些问题类别，迭代方法的连续类似物也是可用的。在这项工作中，我们将连续类似物的应用扩展到函数空间并考虑 PDE（偏微分方程）约束优化问题。我们推导出一类连续类似物，这里是耦合 ODE（常微分方程）-PDE 模型，并证明它们在温和假设下收敛到最优。我们为此类连续类似物的调谐参数（回缩参数）建立了足够的局部稳定性和收敛界限。为了评估连续类似物，我们研究了生物组织中梯度形成模型的参数估计。我们观察到良好的收敛特性，表明连续类似物是最先进的迭代优化方法的有趣替代方案。