Multilevel double loop Monte Carlo and stochastic collocation methods with importance sampling for Bayesian optimal experimental design

• Published in: International journal for numerical methods in engineering Vol. 121; no. 15; pp. 3482 - 3503
• Main Authors: Beck, Joakim, Mansour Dia, Ben, Espath, Luis, Tempone, Raúl
• Format: Journal Article
• Language: English
• Published: Hoboken, USA John Wiley & Sons, Inc 15.08.2020; Wiley Subscription Services, Inc
• Subjects: Approximation; Bayesian analysis; Collocation methods; Computer simulation; Design of experiments; Electrical impedance; electrical impedance tomography; Estimation; expected information gain; Experiments; Fiber orientation; Importance sampling; Monte Carlo simulation; Multilevel; Partial differential equations; stochastic collocation
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.6367

Summary An optimal experimental set‐up maximizes the value of data for statistical inferences. The efficiency of strategies for finding optimal experimental set‐ups is particularly important for experiments that are time‐consuming or expensive to perform. In the situation when the experiments are modeled by partial differential equations (PDEs), multilevel methods have been proven to 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 criterion known as the expected information gain (EIG) in Bayesian optimal experimental design. We propose a multilevel double loop Monte Carlo, which is a multilevel strategy with double loop Monte Carlo, and a multilevel double loop stochastic collocation, which performs a high‐dimensional integration 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 values of the method parameters are determined by minimizing the computational work, subject to satisfying the desired error tolerance. The efficiencies of the methods are demonstrated by estimating EIG for inference of the fiber orientation in composite laminate materials from an electrical impedance tomography experiment.