Numerical spectral analysis of Cauchy-type inverse problems: A probabilistic approach

We investigate the inverse Cauchy and data completion problems for elliptic partial differential equations in a bounded domain $D \subset \mathbb{R}^d$, $d \ge 2$, with a special emphasis on the steady-state heat conduction in anisotropic media. More precisely, boundary conditions are prescribed on an accessible part of the boundary $\varnothing \neq \Gamma_0 \subsetneqq \partial{D}$ and/or internal conditions are available inside the domain $D$ and the aim is to reconstruct the solution to these inverse problems in the domain and on the inaccessible remaining boundary $\Gamma_1 := \partial{D} \setminus \Gamma_0$. Although such severely ill-posed problems have been studied intensively in the past decades, deriving efficient methods for approximating their solution still remains challenging in the general setting, e.g., in high dimensions, for solutions and/or domains with singularities, in complex geometries, etc. Herein, we derive a fundamental probabilistic framework for the stable reconstruction of the solution to the Cauchy and data completion problems in steady-state anisotropic heat conduction, as well as enhancing the knowledge on the impact of the geometry of the domain $D$ and the structure of the conductivity tensor $\mathbf{K}$ on the stability of these inverse problems. This is achieved in three steps: ({\it i}) the spectrum of the direct problem is simulated using stochastic estimators; ({\it ii}) the singular value decomposition of the corresponding direct operator is performed; and ({\it iii}) for the prescribed measurements, a natural subspace of approximate solutions is constructed. This approach is based on elliptic measures, in conjunction with probabilistic representations and parallel Monte Carlo simulations. Thorough numerical simulations performed on GPU, for various two- and three-dimensional geometries, are also provided.