Abstract:We present a review of methods for optimal experimental design (OED) for Bayesian inverse problems governed by partial differential equations with infinite-dimensional parameters. The focus is on problems where one seeks to optimize the placement of measurement points, at which data are collected, such that the uncertainty in the estimated parameters is minimized. We present the mathematical foundations of OED in this context and survey the computational methods for the class of OED problems under study. We al… Show more
“…It is worth noting that this approach of determining successive measurement locations is a naive approach to maximize mutual information. The purpose of this exercise is to demonstrate the impact of using different data locations on the inferred posterior distributions rather than optimizing the locations of measurements for this example of a (non-linear) Bayesian inverse problem for which work exists (e.g., Huan and Marzouk, 2013; Alexanderian et al, 2016; Alexanderian, 2021). Figure 7. Dendrogram indicating spatial proximity of measurement points.…”
Understanding the subsurface is crucial in building a sustainable future, particularly for urban centers. Importantly, the thermal effects that anthropogenic infrastructure, such as buildings, tunnels, and ground heat exchangers, can have on this shared resource need to be well understood to avoid issues, such as overheating the ground, and to identify opportunities, such as extracting and utilizing excess heat. However, obtaining data for the subsurface can be costly, typically requiring the drilling of boreholes. Bayesian statistical methodologies can be used towards overcoming this, by inferring information about the ground by combining field data and numerical modeling, while quantifying associated uncertainties. This work utilizes data obtained in the city of Cardiff, UK, to evaluate the applicability of a Bayesian calibration (using GP surrogates) approach to measured data and associated challenges (previously not tested) and to obtain insights on the subsurface of the area. The importance of the data set size is analyzed, showing that more data are required in realistic (field data), compared to controlled conditions (numerically-generated data), highlighting the importance of identifying data points that contain the most information. Heterogeneity of the ground (i.e., input parameters), which can be particularly prominent in large-scale subsurface domains, is also investigated, showing that the calibration methodology can still yield reasonably accurate results under heterogeneous conditions. Finally, the impact of considering uncertainty in subsurface properties is demonstrated in an existing shallow geothermal system in the area, showing a higher than utilized ground capacity, and the potential for a larger scale system given sufficient demand.
“…It is worth noting that this approach of determining successive measurement locations is a naive approach to maximize mutual information. The purpose of this exercise is to demonstrate the impact of using different data locations on the inferred posterior distributions rather than optimizing the locations of measurements for this example of a (non-linear) Bayesian inverse problem for which work exists (e.g., Huan and Marzouk, 2013; Alexanderian et al, 2016; Alexanderian, 2021). Figure 7. Dendrogram indicating spatial proximity of measurement points.…”
Understanding the subsurface is crucial in building a sustainable future, particularly for urban centers. Importantly, the thermal effects that anthropogenic infrastructure, such as buildings, tunnels, and ground heat exchangers, can have on this shared resource need to be well understood to avoid issues, such as overheating the ground, and to identify opportunities, such as extracting and utilizing excess heat. However, obtaining data for the subsurface can be costly, typically requiring the drilling of boreholes. Bayesian statistical methodologies can be used towards overcoming this, by inferring information about the ground by combining field data and numerical modeling, while quantifying associated uncertainties. This work utilizes data obtained in the city of Cardiff, UK, to evaluate the applicability of a Bayesian calibration (using GP surrogates) approach to measured data and associated challenges (previously not tested) and to obtain insights on the subsurface of the area. The importance of the data set size is analyzed, showing that more data are required in realistic (field data), compared to controlled conditions (numerically-generated data), highlighting the importance of identifying data points that contain the most information. Heterogeneity of the ground (i.e., input parameters), which can be particularly prominent in large-scale subsurface domains, is also investigated, showing that the calibration methodology can still yield reasonably accurate results under heterogeneous conditions. Finally, the impact of considering uncertainty in subsurface properties is demonstrated in an existing shallow geothermal system in the area, showing a higher than utilized ground capacity, and the potential for a larger scale system given sufficient demand.
“…Here, the method can be used to generate fast surrogate models as a replacement for the computationally expensive highdimensional problem. Uncertainty quantification (Iglesias and Stuart, 2014) the optimal position of new measurement locations by employing optimal experimental design methods (Alexanderian, 2021).…”
Abstract. An accurate assessment of the physical states of the Earth system is an essential component of many scientific, societal and economical considerations. These assessments are becoming an increasingly challenging computational task since we aim to resolve models with high resolutions in space and time, to consider complex coupled partial differential equations, and to estimate uncertainties, which often requires many realizations. Machine learning methods are becoming a very popular method for the construction of surrogate 5 models to address these computational issues. However, they also face major challenges in producing explainable, scalable, interpretable and robust models. In this manuscript, we evaluate the perspectives of geoscience applications of physics-based machine learning, which combines physics-based and data-driven methods to overcome the limitations of each approach taken alone. Through three designated examples (from the fields of geothermal energy, geodynamics, and hydrology), we show that the non-intrusive reduced basis method as a physics-based machine learning approach is able to 10 produce highly precise surrogate models that are explainable, scalable, interpretable, and robust.
“…In the present work, we let the prior distribution law be a Gaussian µ pr = N (m pr , C pr ), with mean m pr and covariance operator C pr . We let C pr = A −2 where A is a Laplace-like differential operator; see, e.g., [18,2,23]. The Gaussian prior measure is meaningful since A −2 is a trace class operator which guarantees bounded variance and almost surely pointwise well-defined samples.…”
Section: Bayesian Inverse Problems and Complementary Parametersmentioning
confidence: 99%
“…Of greater difficulty is obtaining sensitivities of a measure of the posterior uncertainty. A natural setting for defining such measures is provided by the theory of optimal experimental design (OED) [13,14,15,16,17,18]. Recall that in OED, one seeks experiments that minimize posterior uncertainty or, more generally, optimize the statistical quality of the estimated parameters.…”
We consider hyper-differential sensitivity analysis (HDSA) of nonlinear Bayesian inverse problems governed by PDEs with infinite-dimensional parameters. In previous works, HDSA has been used to assess the sensitivity of the solution of deterministic inverse problems to additional model uncertainties and also different types of measurement data. In the present work, we extend HDSA to the class of Bayesian inverse problems governed by PDEs. The focus is on assessing the sensitivity of certain key quantities derived from the posterior distribution. Specifically, we focus on analyzing the sensitivity of the MAP point and the Bayes risk and make full use of the information embedded in the Bayesian inverse problem. After establishing our mathematical framework for HDSA of Bayesian inverse problems, we present a detailed computational approach for computing the proposed HDSA indices. We examine the effectiveness of the proposed approach on a model inverse problem governed by a PDE for heat conduction.
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