  We propose optimal dimensionality reduction techniques for the solution of
goal-oriented linear-Gaussian inverse problems, where the quantity of interest
(QoI) is a function of the inversion parameters. These approximations are
suitable for large-scale applications. In particular, we study the
approximation of the posterior covariance of the QoI as a low-rank negative
update of its prior covariance, and prove optimality of this update with
respect to the natural geodesic distance on the manifold of symmetric positive
definite matrices. Assuming exact knowledge of the posterior mean of the QoI,
the optimality results extend to optimality in distribution with respect to the
Kullback-Leibler divergence and the Hellinger distance between the associated
distributions. We also propose approximation of the posterior mean of the QoI
as a low-rank linear function of the data, and prove optimality of this
approximation with respect to a weighted Bayes risk. Both of these optimal
approximations avoid the explicit computation of the full posterior
distribution of the parameters and instead focus on directions that are well
informed by the data and relevant to the QoI. These directions stem from a
balance among all the components of the goal-oriented inverse problem: prior
information, forward model, measurement noise, and ultimate goals. We
illustrate the theory using a high-dimensional inverse problem in heat
transfer.
