2022/12/23 by William D. Reese, Joseph Hart, Reese, William +9 · 1 citation
Decision Sciences · Earth and Planetary Sciences · Physics and Astronomy · #Cryospheric studies and observations #FOS: Computer and information sciences #FOS: Mathematics #Methodology (stat.ME) #Model Reduction and Neural Networks #Optimization and Control (math.OC) #Probabilistic and Robust Engineering Design
paper · pdf · doi:10.48550/arxiv.2212.12386
openalex publication_date 2022/12/23 · openalex created_date 2023/01/04 · openalex updated_date 2026/07/28
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)