The goal of optimal experimental design (OED) is to find the optimal design of a data acquisition system (e.g., location of sensors, what quantities are measured and how often, what sources are used in each experiment), so that the uncertainty in the inferred parameters—or some predicted quantity derived from them—is minimized with respect to a statistical criterion. OED for Bayesian inverse problems governed by partial differential equations (PDEs) is an extremely challenging problem. First, the parameter to be inferred is often a spatially correlated field, leading to a high dimensional parameter space upon discretization. Second, the forward PDE model is often complex and computationally expensive to solve. Third, the design space for the data acquisition system may be high dimensional and constrained. And fourth, the Bayesian inverse problem—a difficult problem in itself—is a part of the OED formulation and needs to be repeated many times. This minisymposium brings together leading experts to present recent advances in numerical methods for Bayesian OED that address these difficulties.
14:00
A sample-driven transport approach to Bayesian optimal experimental design
Fengyi Li | Massachusetts Institute of Technology | United States
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Authors:
Fengyi Li | Massachusetts Institute of Technology | United States
Ricardo Baptista | Massachusetts Institute of Technology | United States
Youssef Marzouk | Massachusetts Institute of Technology | United States
The optimal design of experiments is essential in many fields of science and engineering, especially when each experiment is expensive and resources are limited. Given a prior and a design-dependent likelihood function, we would like to choose the design that maximizes the expected information gain (EIG) in the posterior. Efficient and accurate estimation of EIG therefore becomes crucial. We introduce a flexible transport-map based framework that enables fast estimation of EIG by solving only convex optimization problems. This framework is also compatible with implicit models, where one can simulate from the likelihood but the conditional probability density function of the data is unknown. Several estimators naturally appear within our framework---in particular, positively and negatively biased estimators that provide bounds for the true EIG. We explore the bias and variance of our estimators and assess their dependence on the transport map parameterization and learning process. We then demonstrate the performance of our approach by comparing it to previous methods on both synthetic and real data.
14:30
Multilevel Double-loop Monte Carlo and its counterpart in stochastic collocation
Luis Espath | RWTH Aachen University | Germany
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Authors:
Joakim Beck | King Abdullah University of Science and Technology | Saudi Arabia
Dia Ben Mansour | King Fahd University of Petroleum & Minerals | Saudi Arabia
Luis Espath | RWTH Aachen University | Germany
Raul F. Tempone | King Abdullah University of Science and Technology | Saudi Arabia
Here, we propose two multilevel methods, which efficiently compute the expected information gain using a Kullback-Leibler divergence measure in simulation-based Bayesian optimal experimental design. Firstly, we introduce the multilevel double-loop Monte Carlo (MLDLMC) with importance sampling that reduces the computational work of the inner loop. Secondly, we propose the multilevel double loop stochastic collocation (MLDLSC) with importance sampling, which performs a high-dimensional integration by deterministic quadrature on sparse grids. In both methods, we employ the Laplace approximation as the importance sampling measure, where the optimal values of the method parameters are determined by minimizing the average computational work, subject to the desired tolerance. The computational efficiency of the methods is demonstrated by computing the expected information gain from an electrical impedance tomography experiment where the fiber orientation in composite laminate materials is inferred through Bayesian inversion. This talk is based on: 'Multilevel Double Loop Monte Carlo and Stochastic Collocation Methods with Importance Sampling for Bayesian Optimal Experimental Design.' arXiv preprint arXiv:1811.11469.
15:00
Power posterior implicit sampling for the expected information gain in Bayesian experimental design
Dia Ben Mansour | King Fahd University of Petroleum and Minerals | Saudi Arabia
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Dia Ben Mansour | King Fahd University of Petroleum and Minerals | Saudi Arabia
We address the computation of the information metric in simulation-based Bayesian optimal experimental design (OED). We consider the Shannon expected information gain (EIG) as the utility function to assess the data goodness in statistical inference. For data formed of observed quantities of a forward model outputs polluted with Gaussian noise, the EIG comes as the difference of the data entropy and the expected entropy of the Gaussian likelihood function. The computational complexity is reduced by using the thermodynamic integration to get ride of the normalizing Bayes factor. The estimation of the EIG is replaced by the approximation of the expected deviance. Implicit sampling methods, with linear mapping (Laplace approximation) and with random mapping, are therefore utilized to approximate the expected deviance.
We derive the optimal setting of the method’s parameters for the average computational cost and interpret the sensitivity of the expected deviance to the tempering variable to formulate a high precision tempering quadrature with an extra degree of freedom for controlling the bias. We assess the computational efficiency by addressing a chemical EOR core flooding experiment in a petroleum reservoir, where the objective is the efficient learning of the reservoir’s relative permeability.
15:30
Optimal Experimental Design Problem as Mixed-integer Optimal Control Problem
Ekaterina Kostina | Heidelberg University | Germany
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Ekaterina Kostina | Heidelberg University | Germany
The optimization of one or more dynamic experiments in order to maximize the accuracy of the parameter estimates subject to cost and other inequality constraints leads to very complex non-standard optimal control problems. One of the difficulties is that in case of sampling design, we deal with an optimal control problems with mixed-integer controls. We are interested in the application of the Pontryagin's maximum principle in order to understand the structure of the sampling design.