Large Scale Inverse Problems And Quantification Of Uncertainty

Author : Siddhant Wahal
Publisher :
ISBN 13 :
Total Pages : 354 pages
Book Rating : 4.:/5 (119 download)

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Download or read book Novel Algorithms for Uncertainty Quantification in Large Scale Systems written by Siddhant Wahal and published by . This book was released on 2020 with total page 354 pages. Available in PDF, EPUB and Kindle. Book excerpt: Uncertainty Quantification (UQ) algorithms are of increasing significance in science and engineering. The process of modeling physical reality on computers is rife with uncertainties. These uncertainties get propagated through the computer model, leading to uncertain outputs. As decision-makers from every facet of society come to increasingly rely on computer predictions, the need to characterize this uncertainty has never been greater. However, doing so efficiently remains challenging. This is primarily because computer models are often time consuming to run and because their inputs live in high-dimensional spaces that are difficult to explore. In this thesis, we seek to address this challenge in the context of two UQ problems. In the first UQ problem, we study rare-event simulation: given a smooth non-linear map with uncertain inputs, what is the probability that the output evaluates inside a specified interval? Standard statistical approaches for computing this probability, such as the Monte Carlo method, become computationally inefficient as the event under consideration becomes rare. To address this inefficiency, we present two Importance Sampling (IS) algorithms. Our first algorithm, called the Bayesian Inverse Monte Carlo (BIMC) method, relies on solving a fictitious Bayesian inverse problem. The solution of the inverse problem yields a posterior PDF, a local Gaussian approximation to which serves as the importance sampling density. We subject BIMC to rigorous theoretical and experimental analysis, which establishes that BIMC can lead to speedups of several orders-of-magnitude (over the Monte Carlo method) when the forward map is nearly affine, or weakly non-linear. When these conditions are violated, that is, when the forward map is significantly nonlinear, BIMC leads to a poor-quality IS distribution. Motivated by these limitations, we propose modifications to BIMC. The modified algorithm, which we term Adaptive-BIMC (A-BIMC), proceeds in two stages. The first stage roughly identifies those regions in input space that trigger a rare event. The second stage then refines the approximation from the first stage of the algorithm. We study A-BIMC’s performance on synthetic problems and demonstrate that its performance doesn’t depend on how small the target probability is. Rather it depends on the nonlinearity of the input-output map. Through these experiments, we also find that A-BIMC’s performance deteriorates with increasing ambient dimensionality of the problem. To address this issue, we lay the foundation for a general dimension reduction strategy for rare-event probability estimation. The second UQ problem concerns the statistical calibration of model inputs from observed data, with the ultimate aim of issuing uncertainty-equipped predictions of a Quantity-of- Interest (QoI). The physical system that we study here is a hydrocarbon reservoir containing geological faults. Operational decisions concerning the reservoir rely on predictions of financial summaries of the reservoir, such as its Net Present Value. These summaries depend on the nature of fluid flow within the reservoir, which is itself controlled by the extent to which an individual fault inhibits or facilitates flow. This fault property, known as the fault transmissibility, isn’t directly measurable and must be calibrated using production data. Here, we design and analyze a complete data-to-prediction workflow to quantify post-calibration uncertainties. We also discuss how these uncertainties change under different reservoir conditions