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Multilevel Double Loop Monte Carlo and Stochastic Collocation Methods with Importance Sampling for Bayesian Optimal Experimental Design

An optimal experimental set-up maximizes the value of data for statistical inferences and predictions. The efficiency of strategies for finding optimal experimental set-ups is particularly important for experiments that are time-consuming or expensive to perform. For instance, in the situation when the experiments are modeled by Partial Differential Equations (PDEs), multilevel methods have been proven to dramatically reduce the computational complexity of their single-level counterparts when estimating expected values. For a setting where PDEs can model experiments, we propose two multilevel methods for estimating a popular design criterion known as the expected information gain in simulation-based Bayesian optimal experimental design. The expected information gain criterion is of a nested expectation form, and only a handful of multilevel methods have been proposed for problems of such form. We propose a Multilevel Double Loop Monte Carlo (MLDLMC), which is a multilevel strategy with Double Loop Monte Carlo (DLMC), and a Multilevel Double Loop Stochastic Collocation (MLDLSC), which performs a high-dimensional integration by deterministic quadrature on sparse grids. For both methods, the Laplace approximation is used for importance sampling that significantly reduces the computational work of estimating inner expectations. The optimal values of the method parameters are determined by minimizing the average computational work, subject to satisfying the desired error tolerance. The computational efficiencies of the methods are demonstrated by estimating the expected information gain for Bayesian inference of the fiber orientation in composite laminate materials from an electrical impedance tomography experiment. MLDLSC performs better than MLDLMC when the regularity of the quantity of interest, with respect to the additive noise and the unknown parameters, can be exploited.