随机正交化中的参数识别问题
A parameter identification problem in stochastic homogenization
AI总结：
本文通过最小二乘法解决随机正交化中的参数识别问题，探讨了如何选择宏观可观测量以唯一确定参数，并在单维情况下展示牛顿算法在存在统计噪声时的有效性。
摘要：
在多孔介质物理中，通过实验校准模型参数是一个挑战。这一过程受到建模、测量和通过随机正交化（正向问题）计算宏观可观测量的误差影响，以及来自参数拟合过程（反向问题）的误差。本文通过考虑最小二乘法来识别微观模型的参数，基于宏观可观测量。特别地，我们讨论了需要选择哪些宏观可观测量以唯一确定这些参数。为了获得更好的直观并探索问题而不产生过高的计算负载，我们主要关注一维情况。我们展示即使系统中存在一些小的统计噪声，牛顿算法也可以高效地用于稳健地确定最优参数。
英文摘要：
In porous media physics, calibrating model parameters through experiments is a challenge. This process is plagued with errors that come from modelling, measurement and computation of the macroscopic observables through random homogenization -- the forward problem -- as well as errors coming from the parameters fitting procedure -- the inverse problem. In this work, we address these issues by considering a least-square formulation to identify parameters of the microscopic model on the basis on macroscopic observables. In particular, we discuss the selection of the macroscopic observables which we need to know in order to uniquely determine these parameters. To gain a better intuition and explore the problem without a too high computational load, we mostly focus on the one-dimensional case. We show that the Newton algorithm can be efficiently used to robustly determine optimal parameters, even if some small statistical noise is present in the system.
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