Hyper-differential sensitivity analysis for inverse problems constrained by partial differential equations
Hyper-differential sensitivity analysis for inverse problems constrained by partial differential equations
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DOI：
10.1088/1361-6420/abaf63
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发表时间：
2020-12-01
期刊：
影响因子：
2.1
通讯作者：
Alexanderian, Alen
中科院分区：
文献类型：
--
作者：
Sunseri, Isaac;Hart, Joseph;Alexanderian, Alen
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。当不能直接确定模型参数，而是使用(通常稀疏和有噪声的)状态测量来估计模型参数时，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，但必须指定用于对感兴趣的参数进行倒置所需的完整模型表征。我们把附加的模型参数(那些没有倒置的参数)和测量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对反问题解的相对重要性。为了解决这一问题，我们提出了一个基于超差分灵敏度分析(HDSA)的框架。HDSA计算反问题的解关于互补参数的导数。我们给出了大规模PDE约束反问题中HDSA的数学框架，并展示了如何解释HDSA来洞察反问题。我们通过在边界条件、源注入和扩散系数不确定的多孔介质流动应用中使用压力和浓度测量来估计渗透场，从而证明了该方法在反问题上的有效性。
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.