This work is devoted to inverse problems for elliptic partial differential equations in an Euclidean domain, in which the boundary and/or interior conditions are given merely on some accessible portion of the boundary and/or inside the domain, the goal being the efficient construction of an approximation for the unknown solution in the remaining part of the domain; such inverse problems are usually called data-completion problems or inverse Cauchy problems. They have been intensively studied in the past decades, but due to their severe instability it has remained an up-to-date challenge to derive both theoretical and numerical methods that can efficiently treat such inverse problems in general settings, especially in high dimensions or in which the solution or the domain exhibit singularities or complex geometries. In this paper we establish a fundamental probabilistic framework in which such inverse problems can be analyzed both theoretically and numerically in terms of the geometry of the domain and the structure of the coefficients. The methods we develop are different from what has been previously proposed in the literature, and are designed to accurately quantify the instability of the inverse problem, as well as to construct a natural subspace of approximate solutions given the available measurements, by simulating the spectrum of the direct problem and performing a singular value decomposition.The approach is based on elliptic measures in conjunction with probabilistic representations and parallel Monte Carlo simulations. The proposed methods are accompanied by a full probabilistic error analysis, showing the convergence of the approximations and providing explicit error bounds. The complexity of the methods is also taken into discussion.We provide thorough numerical simulations performed on graphical processing units, in dimensions two and three, and for various types of domains.

翻译：本研究致力于欧几里得域中椭圆型偏微分方程的反问题，其中边界和/或内部条件仅给定于边界或域内的部分可访问区域，目标在于高效构建未知解在域内剩余部分的近似；此类反问题通常称为数据完备问题或反柯西问题。过去数十年间，这些问题得到了深入研究，但由于其严重的不适定性，在一般设定下——特别是在高维情形或解及区域存在奇异性或复杂几何结构时——推导能有效处理此类反问题的理论与数值方法，至今仍是前沿挑战。本文建立了一个基础的概率框架，使得此类反问题能够依据区域的几何结构与系数特性进行理论与数值分析。我们所发展的方法不同于以往文献中提出的方案，旨在通过模拟正问题的谱并执行奇异值分解，精确量化反问题的不稳定性，同时基于现有测量数据构建自然的近似解子空间。该方法基于椭圆测度结合概率表示与并行蒙特卡洛模拟。所提出的方法辅以完整的概率误差分析，证明了近似解的收敛性并给出了显式误差界。方法的计算复杂度亦在讨论之列。我们在图形处理器上进行了详尽的数值模拟，涵盖二维与三维空间及多种区域类型。