偏微分方程约束反问题的超微分灵敏度分析
Hyper-Differential Sensitivity Analysis for Inverse Problems Constrained by Partial Differential Equations
摘要：
在许多科学和工程应用中使用的高保真模型耦合多个物理状态和参数。当模型参数不能直接确定，而是使用状态的测量（通常是稀疏和有噪声的）来估计时，就会出现逆问题。数据通常不足以同时通知所有参数。因此，控制模型通常包含不确定的参数，但必须指定完整的模型表征，以反演相关参数。我们将附加的模型参数（未被反演的那些参数）和测量的数据状态的组合称为“互补参数”。我们试图量化这些互补参数的反问题的解决方案的相对重要性。为了解决这个问题，我们提出了一个框架的基础上超差分灵敏度分析（HDSA）。HDSA计算逆问题的解关于互补参数的导数。我们提出了一个数学框架，HDSA在大规模的偏微分方程约束的反问题，并显示如何HDSA可以解释给洞察力的反问题。我们证明了该方法的有效性，通过估计渗透率场的反问题，使用压力和浓度测量，在多孔介质流动的应用程序中的边界条件，源注入，和扩散系数的不确定性。
英文摘要：
High fidelity models used in many science and engineering applications couple multiple physical states and parameters. Inverse problems arise when a model parameter cannot be determined directly, but rather is estimated using (typically sparse and noisy) measurements of the states. The data is usually not sufficient to simultaneously inform all of the parameters. Consequently, the governing model typically contains parameters which are uncertain but must be specified for a complete model characterization necessary to invert for the parameters of interest. We refer to the combination of the additional model parameters (those which are not inverted for) and the measured data states as the "complementary parameters". We seek to quantify the relative importance of these complementary parameters to the solution of the inverse problem. To address this, we present a framework based on hyper-differential sensitivity analysis (HDSA). HDSA computes the derivative of the solution of an inverse problem with respect to complementary parameters. We present a mathematical framework for HDSA in large-scale PDE-constrained inverse problems and show how HDSA can be interpreted to give insight about the inverse problem. We demonstrate the effectiveness of the method on an inverse problem by estimating a permeability field, using pressure and concentration measurements, in a porous medium flow application with uncertainty in the boundary conditions, source injection, and diffusion coefficient.
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