We consider the problem of numerically approximating the solutions to a partial differential equation (PDE) when there is insufficient information to determine a unique solution. Our main example is the Poisson boundary value problem, when the boundary data is unknown and instead one observes finitely many linear measurements of the solution. We view this setting as an optimal recovery problem and develop theory and numerical algorithms for its solution. The main vehicle employed is the derivation and approximation of the Riesz representers of these functionals with respect to relevant Hilbert spaces of harmonic functions.

翻译：本文研究了在信息不足以确定唯一解的情况下，数值逼近偏微分方程（PDE）解的问题。我们的主要示例是泊松边值问题，其中边界数据未知，而仅能观测到解的有限个线性测量值。我们将此设定视为最优恢复问题，并为其求解发展了理论框架与数值算法。所采用的核心手段是推导并近似这些泛函在相关调和函数希尔伯特空间中的里斯表示元。