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We study the efficient numerical solution of linear inverse problems withoperator valued data which arise, e.g., in seismic exploration, inversescattering, or tomographic imaging. The high-dimensionality of the data spaceimplies extremely high computational cost already for the evaluation of theforward operator which makes a numerical solution of the inverse problem, e.g.,by iterative regularization methods, practically infeasible. To overcome thisobstacle, we take advantage of the underlying tensor product structure of theproblem and propose a strategy for constructing low-dimensional certifiedreduced order models of quasi-optimal rank for the forward operator which canbe computed much more efficiently than the truncated singular valuedecomposition. A complete analysis of the proposed model reduction approach isgiven in a functional analytic setting and the efficient numerical constructionof the reduced order models as well as of their application for the numericalsolution of the inverse problem is discussed. In summary, the setup of alow-rank approximation can be achieved in an offline stage at essentially thesame cost as a single evaluation of the forward operator, while the actualsolution of the inverse problem in the online phase can be done with extremelyhigh efficiency. The theoretical results are illustrated by application to atypical model problem in fluorescence optical tomography.

A model reduction approach for inverse problems with operator valued data