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：当从点数据源（如卫星测高）估计难以直接测量的量或场（如冰的流动性）时，以贝叶斯一致性求解数值反问题至关重要。否则，所得的概率密度函数将无法使难以测量的量或场围绕真值合理聚集。具体而言，反问题的构建应通过在真实点位置评估数值解，以便与点数据源直接比较。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问题，则网格密度以下额外点数据值的优势将丧失。我们通过地下水文学和冰川学领域的实例证明，一致性的表述能够提高结果精度，并促进建模者与观测者之间的对话。为此，我们将点数据以不连续场的形式引入有限元方法体系，其网格由孤立顶点组成。点评估可被表述为有限元插值运算（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中的实现证明了这一点——该工具能生成用于求解偏微分方程的高度优化有限元代码。我们的方案集成了dolfin-adjoint/pyadjoint，使得数据同化等偏微分方程约束优化问题可通过前向和伴随模式自动微分求解。