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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., Comment: 73 pages