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:我们认为,当没有足够的信息来决定一个独特的解决方案时,在数字上接近部分差异方程式(PDE)的解决方案(PDE)就存在问题。我们的主要例子是Poisson边界值问题,当时的边界数据未知,相反,我们观察了该解决方案的有限数量的线性测量。我们认为这一设置是一个最佳的回收问题,并为其解决方案制定了理论和数字算法。使用的主要工具是这些功能的Riesz代表的推算和近似,与相关的Hilbert调力空间有关。