  We study the efficient numerical solution of linear inverse problems with
operator valued data which arise, e.g., in seismic exploration, inverse
scattering, or tomographic imaging. The high-dimensionality of the data space
implies extremely high computational cost already for the evaluation of the
forward operator which makes a numerical solution of the inverse problem, e.g.,
by iterative regularization methods, practically infeasible. To overcome this
obstacle, we take advantage of the underlying tensor product structure of the
problem and propose a strategy for constructing low-dimensional certified
reduced order models of quasi-optimal rank for the forward operator which can
be computed much more efficiently than the truncated singular value
decomposition. A complete analysis of the proposed model reduction approach is
given in a functional analytic setting and the efficient numerical construction
of the reduced order models as well as of their application for the numerical
solution of the inverse problem is discussed. In summary, the setup of a
low-rank approximation can be achieved in an offline stage at essentially the
same cost as a single evaluation of the forward operator, while the actual
solution of the inverse problem in the online phase can be done with extremely
high efficiency. The theoretical results are illustrated by application to a
typical model problem in fluorescence optical tomography.

Numerical Analysis;Numerical Analysis