We develop a computational framework for D-optimal experimental design forPDE-based Bayesian linear inverse problems with infinite-dimensionalparameters. We follow a formulation of the experimental design problem thatremains valid in the infinite-dimensional limit. The optimal design is obtainedby solving an optimization problem that involves repeated evaluation of thelog-determinant of high-dimensional operators along with their derivatives.Forming and manipulating these operators is computationally prohibitive forlarge-scale problems. Our methods exploit the low-rank structure in the inverseproblem in three different ways, yielding efficient algorithms. Our mainapproach is to use randomized estimators for computing the D-optimal criterion,its derivative, as well as the Kullback--Leibler divergence from posterior toprior. Two other alternatives are proposed based on a low-rank approximation ofthe prior-preconditioned data misfit Hessian, and a fixed low-rankapproximation of the prior-preconditioned forward operator. Detailed erroranalysis is provided for each of the methods, and their effectiveness isdemonstrated on a model sensor placement problem for initial statereconstruction in a time-dependent advection-diffusion equation in two spacedimensions.
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