Inverse problems constrained by partial differential equations (PDEs) play a critical role in model development and calibration. In many applications, there are multiple uncertain parameters in a model which must be estimated. Although the Bayesian formulation is attractive for such problems, computational cost and high dimensionality frequently prohibit a thorough exploration of the parametric uncertainty. A common approach is to reduce the dimension by fixing some parameters (which we will call auxiliary parameters) to a best estimate and use techniques from PDE-constrained optimization to approximate properties of the Bayesian posterior distribution. For instance, the maximum a posteriori probability (MAP) and the Laplace approximation of the posterior covariance can be computed. In this article, we propose using hyper-differential sensitivity analysis (HDSA) to assess the sensitivity of the MAP point to changes in the auxiliary parameters. We establish an interpretation of HDSA as correlations in the posterior distribution. Our proposed framework is demonstrated on the inversion of bedrock topography for the Greenland ice sheet with uncertainties arising from the basal friction coefficient and climate forcing (ice accumulation rate)
翻译:受部分差异方程限制的反面问题(PDEs)在模型开发和校准中起着关键作用。在许多应用中,模型中有许多必须估算的不确定参数。虽然Bayesian的配方对此类问题具有吸引力,但计算成本和高维性经常禁止彻底探索参数不确定性。一个共同的方法是通过确定某些参数(我们将称之为辅助参数),将HDSA解释为后表分布的关联关系,从而减少其尺寸。我们提议的框架显示格陵兰冰层基本地形与因堡面摩擦系数和气候压力(冰累积率)产生的不确定性的转化。