Spotlight Inversion by Orthogonal Projections

Abstract. Many computational problems involve solving a linear system of equations, although only a subset of the entries of the solution are needed. In inverse problems, where the goal is to estimate unknown parameters from indirect noisy observations, it is not uncommon that the forward model linking the observed variables to the unknowns depends on variables that are not of primary interest, often referred to as nuisance parameters. In this article, we consider linear problems and propose a novel projection technique to eliminate, or at least mitigate, the contribution of the nuisance parameters in the model. We refer to this approach as spotlight inversion, as it allows to focus on only the portion of primary interest of the unknown parameter vector, leaving the uninteresting part in the shadow. The viability of the approach is illustrated with two computed examples, one where it works as model reduction for a finite element approximation of an elliptic PDE, the other amounting to local fanbeam X-ray tomography, spotlighting the region of interest that is part of the full target.