Optimal Bayesian experimental design for subsurface flow problems

Abstract

Optimal Bayesian design techniques provide an estimate for the best parameters of an experiment in order to maximize the value of measurements prior to the actual collection of data. In other words, these techniques explore the space of possible observations and determine an experimental setup that produces maximum information about the system parameters on average. Generally, optimal Bayesian design formulations result in multiple high-dimensional integrals that are difficult to evaluate without incurring significant computational costs as each integration point corresponds to solving a coupled system of partial differential equations. In the present work, we propose a novel approach for development of polynomial chaos expansion (PCE) surrogate model for the design utility function. In particular, we demonstrate how the orthogonality of PCE basis polynomials can be utilized in order to replace the expensive integration over the space of possible observations by direct construction of PCE approximation for the expected information gain. This novel technique enables the derivation of a reasonable quality response surface for the targeted objective function with a computational budget comparable to several singlepoint evaluations. Therefore, the proposed technique reduces dramatically the overall cost of optimal Bayesian experimental design. We evaluate this alternative formulation utilizing PCE on few numerical test cases with various levels of complexity to illustrate the computational advantages of the proposed approach. * The current manuscipt is a preprint of the paper published in Computer Methods in Applied Mechanics and Engineering.

Machine Learning for Bayesian Experimental Design in the Subsurface

“…High-fidelity response surface models replicate the original model's numerical results. This approach employs techniques like kriging (Baú and Mayer, 2006;Garcet et al, 2006), artificial neural networks (Kourakos and Mantoglou, 2009;Yan and Minsker, 2006), radial basis functions (Regis and Shoemaker, 2005), and polynomial chaos expansion (Laloy et al, 2013;Tarakanov and Elsheikh, 2020) to determine a relationship between model parameters and one or more model response variables using statistical models or empirical data-driven models.…”

“…Once a suitable surrogate model is identified, stochastic inversion and experimental design issues can be effectively resolved at a minimal computational cost. The approximation, however, may result in a sizable bias in the prediction, leading to an inaccurate estimation of the data utility function (Asher et al, 2015;Babaei et al, 2015;Laloy et al, 2013;Razavi et al, 2012;Tarakanov and Elsheikh, 2020;Zhang et al, 2015Zhang et al, , 2020.…”

“…However, the computational burden is even more important because experimental design assumes that the observed data is not yet known, requiring the search for the stochastic solution to the inverse problem for any possible outcome of the unknown data (e.g., Leube et al 2012;Neuman et al 2012a). Two main simplifications have been proposed to make practical applications tractable: (1) Bayesian Model Averaging (BMA) combined with preposterior estimation (Kikuchi et al, 2015;Neuman et al, 2012b;Pham and Tsai, 2016;Raftery et al, 2005;Samadi et al, 2020;Tsai and Li, 2008;Vrugt and Robinson, 2007;Wöhling and Vrugt, 2008), and (2) surrogate modelling (Asher et al, 2015;Babaei et al, 2015;Laloy et al, 2013;Razavi et al, 2012;Tarakanov and Elsheikh, 2020;Zhang et al, 2015Zhang et al, , 2020.…”

Multimodal Information Gain in Bayesian Design of Experiments

“…The Laplace approximation has been used together with the gradient descent method to tackle the continuous optimization problems in optimal Bayesian experimental design in [10] [12]. Some authors used surrogate to approximate the expected information gain against the design space [25] [33]. Additionally, it is note-worthy that consistent formulas have been derived for Bayesian optimal experimental design based on infinite dimensional models, for example, the partial differential equations (pdes) [1] [2].…”

Variational Bayesian Approximation of Inverse Problems using Sparse Precision Matrices

“…In particular, the mean of the posterior probability distribution corresponds to the point estimate of the solution while the credible intervals capture the range of the parameters consistent with the observed measurements and prior assumptions. For these reasons, Bayesian methods have gained popularity in computational mechanics for experimental design and inverse problems with uncertainty quantification; see, e.g., the recent papers by Abdulle and Garegnani (2021); Pandita et al (2021); Pyrialakos et al (2021); Ni et al (2021); Sabater et al (2021); Huang et al (2021); Ibrahimbegovic et al (2020); Tarakanov and Elsheikh (2020); Michelén Ströfer et al (2020); Carlon et al (2020); Wu et al (2020); Uribe et al (2020); Rizzi et al (2019); Arnst and Soize (2019); Beck et al (2018); Betz et al (2018); Chen et al (2017); Asaadi and Heyns (2017); Huang et al (2017); Karathanasopoulos et al (2017); Babuška et al (2016); Girolami et al (2021).…”