When estimating quantities and fields that are difficult to measure directly, such as the fluidity of ice, from point data sources, such as satellite altimetry, it is important to solve a numerical inverse problem that is formulated with Bayesian consistency. Otherwise, the resultant probability density function for the difficult to measure quantity or field will not be appropriately clustered around the truth. In particular, the inverse problem should be formulated by evaluating the numerical solution at the true point locations for direct comparison with the point data source. If the data are first fitted to a gridded or meshed field on the computational grid or mesh, and the inverse problem formulated by comparing the numerical solution to the fitted field, the benefits of additional point data values below the grid density will be lost. We demonstrate, with examples in the fields of groundwater hydrology and glaciology, that a consistent formulation can increase the accuracy of results and aid discourse between modellers and observationalists. To do this, we bring point data into the finite element method ecosystem as discontinuous fields on meshes of disconnected vertices. Point evaluation can then be formulated as a finite element interpolation operation (dual-evaluation). This new abstraction is well-suited to automation, including automatic differentiation. We demonstrate this through implementation in Firedrake, which generates highly optimised code for solving PDEs with the finite element method. Our solution integrates with dolfin-adjoint/pyadjoint, allowing PDE-constrained optimisation problems, such as data assimilation, to be solved through forward and adjoint mode automatic differentiation.

翻译：在从卫星测高等点数据源估计难以直接测量的量（如冰流动性）时，必须以贝叶斯一致性方式构建数值反问题。否则，所得到的关于这些难以测量量或场的概率密度函数将无法恰当集中在真实值附近。具体而言，反问题应通过直接在真实点位置处评估数值解以与点数据源进行对比来构建。如果先将数据拟合到计算网格上的插值场，再通过比较数值解与拟合场来构建反问题，则将丢失网格密度以下额外点数据值带来的优势。我们通过地下水文学和冰川学领域的实例证明，一致性的公式化表述能提高结果精度并促进建模者与观测者之间的交流。为此，我们将点数据作为由孤立顶点构成的网格上的不连续场引入有限元方法生态系统中。点评估可被表述为有限元插值操作（对偶评估）。这种新的抽象方法非常适合自动化，包括自动微分。我们通过Firedrake中的实现展示了这一点，该实现能生成用于求解偏微分方程的有限元方法的高优化代码。我们的解决方案集成了dolfin-adjoint/pyadjoint，使得数据同化等偏微分方程约束优化问题可通过正向和伴随模式自动微分求解。