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 Partial Differential Equations (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：当从卫星测高等点数据源估计难以直接测量的量（如冰的流动性）时，以贝叶斯一致性方式求解数值反演问题至关重要。否则，难以测量量或场的概率密度函数将无法合理聚集于真实值附近。具体而言，反演问题应通过在真实点位评估数值解并与点数据源直接比较来构建。若先将数据拟合到计算网格上的网格化场，再通过比较数值解与拟合场构建反演问题，则低于网格密度的额外点数据值带来的优势将丧失。我们通过地下水水文学和冰川学领域的实例证明，一致性表述可提高结果精度，并促进建模者与观测者之间的交流。为此，我们将点数据以非连续场的形式引入有限元方法生态体系，在由孤立顶点组成的网格上表达。点评估可被形式化为有限元插值操作（对偶评估）。这种新的抽象方法适用于自动化，包括自动微分。我们通过在Firedrake中的实现验证了这一点——该框架可生成高度优化的代码，用于求解偏微分方程（PDE）的有限元方法。我们的解决方案与dolfin-adjoint/pyadjoint集成，使得数据同化等偏微分方程约束的优化问题可通过正向和伴随模式的自动微分求解。