SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1278-1312, June 2024.
Abstract. 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.

中文翻译：

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求解不完整信息的偏微分方程


《SIAM 数值分析杂志》，第 62 卷，第 3 期，第 1278-1312 页，2024 年 6 月。

抽象的。当没有足够的信息来确定唯一解时，我们考虑对偏微分方程 (PDE) 的解进行数值近似的问题。我们的主要例子是泊松边值问题，当边界数据未知时，人们会观察到解的有限多个线性测量。我们将此设置视为最佳恢复问题，并为其解决方案开发理论和数值算法。所采用的主要工具是这些泛函的 Riesz 表示相对于调和函数的相关希尔伯特空间的推导和近似。