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

International Journal for Numerical Methods in Engineering, vol. 121, pp. 3482–3503

Abstract

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.

Authors 4

1. Joakim Beck

   King Abdullah University of Science and Technology

   Affiliation as printed

   Computer, Electrical and Mathematical Science and Engineering Division King Abdullah University of Science and Technology Thuwal 23955‐6900 Saudi Arabia

   KING ABDULLAH UNIVERSITY OF SCIENCE AND TECHNOLOGY

2. Ben Mansour Dia

   King Fahd University of Petroleum and Minerals

   Affiliation as printed

   CIPR, College of Petroleum Engineering and Geosciences King Fahd University of Petroleum and Minerals Dhahran Saudi Arabia

   CIPR, College of Petroleum Engineering and Geosciences King Fahd University of Petroleum and Minerals Dhahran Saudi Arabia

3. Luis Espath corresponding Aachen

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen University Department of Mathematics Aachen Germany

   RWTH Aachen University Department of Mathematics Aachen Germany

4. Raul F. Tempone Aachen

   King Abdullah University of Science and Technology · RWTH Aachen University

   Affiliation as printed

   Computer, Electrical and Mathematical Science and Engineering Division King Abdullah University of Science and Technology Thuwal 23955‐6900 Saudi Arabia

   RWTH Aachen University Alexander von Humboldt Professor in Mathematics of Uncertainty Quantification Aachen Germany

   KING ABDULLAH UNIVERSITY OF SCIENCE AND TECHNOLOGY