  We develop a fast and scalable computational framework to solve large-scale
and high-dimensional Bayesian optimal experimental design problems. In
particular, we consider the problem of optimal observation sensor placement for
Bayesian inference of high-dimensional parameters governed by partial
differential equations (PDEs), which is formulated as an optimization problem
that seeks to maximize an expected information gain (EIG). Such optimization
problems are particularly challenging due to the curse of dimensionality for
high-dimensional parameters and the expensive solution of large-scale PDEs. To
address these challenges, we exploit two essential properties of such problems:
the low-rank structure of the Jacobian of the parameter-to-observable map to
extract the intrinsically low-dimensional data-informed subspace, and the high
correlation of the approximate EIGs by a series of approximations to reduce the
number of PDE solves. We propose an efficient offline-online decomposition for
the optimization problem: an offline stage of computing all the quantities that
require a limited number of PDE solves independent of parameter and data
dimensions, and an online stage of optimizing sensor placement that does not
require any PDE solve. For the online optimization, we propose a swapping
greedy algorithm that first construct an initial set of sensors using leverage
scores and then swap the chosen sensors with other candidates until certain
convergence criteria are met. We demonstrate the efficiency and scalability of
the proposed computational framework by a linear inverse problem of inferring
the initial condition for an advection-diffusion equation, and a nonlinear
inverse problem of inferring the diffusion coefficient of a log-normal
diffusion equation, with both the parameter and data dimensions ranging from a
few tens to a few thousands.
