  We develop a computational framework for D-optimal experimental design for
PDE-based Bayesian linear inverse problems with infinite-dimensional
parameters. We follow a formulation of the experimental design problem that
remains valid in the infinite-dimensional limit. The optimal design is obtained
by solving an optimization problem that involves repeated evaluation of the
log-determinant of high-dimensional operators along with their derivatives.
Forming and manipulating these operators is computationally prohibitive for
large-scale problems. Our methods exploit the low-rank structure in the inverse
problem in three different ways, yielding efficient algorithms. Our main
approach is to use randomized estimators for computing the D-optimal criterion,
its derivative, as well as the Kullback--Leibler divergence from posterior to
prior. Two other alternatives are proposed based on a low-rank approximation of
the prior-preconditioned data misfit Hessian, and a fixed low-rank
approximation of the prior-preconditioned forward operator. Detailed error
analysis is provided for each of the methods, and their effectiveness is
demonstrated on a model sensor placement problem for initial state
reconstruction in a time-dependent advection-diffusion equation in two space
dimensions.
