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We develop a fast and scalable computational framework to solve large-scaleand high-dimensional Bayesian optimal experimental design problems. Inparticular, we consider the problem of optimal observation sensor placement forBayesian inference of high-dimensional parameters governed by partialdifferential equations (PDEs), which is formulated as an optimization problemthat seeks to maximize an expected information gain (EIG). Such optimizationproblems are particularly challenging due to the curse of dimensionality forhigh-dimensional parameters and the expensive solution of large-scale PDEs. Toaddress these challenges, we exploit two essential properties of such problems:the low-rank structure of the Jacobian of the parameter-to-observable map toextract the intrinsically low-dimensional data-informed subspace, and the highcorrelation of the approximate EIGs by a series of approximations to reduce thenumber of PDE solves. We propose an efficient offline-online decomposition forthe optimization problem: an offline stage of computing all the quantities thatrequire a limited number of PDE solves independent of parameter and datadimensions, and an online stage of optimizing sensor placement that does notrequire any PDE solve. For the online optimization, we propose a swappinggreedy algorithm that first construct an initial set of sensors using leveragescores and then swap the chosen sensors with other candidates until certainconvergence criteria are met. We demonstrate the efficiency and scalability ofthe proposed computational framework by a linear inverse problem of inferringthe initial condition for an advection-diffusion equation, and a nonlinearinverse problem of inferring the diffusion coefficient of a log-normaldiffusion equation, with both the parameter and data dimensions ranging from afew tens to a few thousands.

A fast and scalable computational framework for large-scale and high-dimensional Bayesian optimal experimental design