  We develop a framework for goal-oriented optimal design of experiments
(GOODE) for large-scale Bayesian linear inverse problems governed by PDEs. This
framework differs from classical Bayesian optimal design of experiments (ODE)
in the following sense: we seek experimental designs that minimize the
posterior uncertainty in the experiment end-goal, e.g., a quantity of interest
(QoI), rather than the estimated parameter itself. This is suitable for
scenarios in which the solution of an inverse problem is an intermediate step
and the estimated parameter is then used to compute a QoI. In such problems, a
GOODE approach has two benefits: the designs can avoid wastage of experimental
resources by a targeted collection of data, and the resulting design criteria
are computationally easier to evaluate due to the often low-dimensionality of
the QoIs. We present two modified design criteria, A-GOODE and D-GOODE, which
are natural analogues of classical Bayesian A- and D-optimal criteria. We
analyze the connections to other ODE criteria, and provide interpretations for
the GOODE criteria by using tools from information theory. Then, we develop an
efficient gradient-based optimization framework for solving the GOODE
optimization problems. Additionally, we present comprehensive numerical
experiments testing the various aspects of the presented approach. The driving
application is the optimal placement of sensors to identify the source of
contaminants in a diffusion and transport problem. We enforce sparsity of the
sensor placements using an $\ell_1$-norm penalty approach, and propose a
practical strategy for specifying the associated penalty parameter.
