Optimal Experimental Design For Large Scale Bayesian Inverse Problems
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| Author | : Keyi Wu (Ph. D.) |
| Publisher | : |
| Total Pages | : 0 |
| Release | : 2022 |
| Genre | : |
| ISBN | : |
Bayesian optimal experimental design (BOED)—including active learning, Bayesian optimization, and sensor placement—provides a probabilistic framework to maximize the expected information gain (EIG) or mutual information (MI) for uncertain parameters or quantities of interest with limited experimental data. However, evaluating the EIG remains prohibitive for largescale complex models due to the need to compute double integrals with respect to both the parameter and data distributions. In this work, we develop a fast and scalable computational framework to solve Bayesian optimal experimental design (OED) problems governed by partial differential equations (PDEs) with application to optimal sensor placement by maximizing the EIG. We (1) exploit the low-rank structure of the Jacobian of the parameter-to-observable map to extract the intrinsic low-dimensional data-informed subspace, and (2) employ a series of approximations of the EIG that reduce the number of PDE solves while retaining a high correlation with the true EIG. This allows us to propose an efficient offline–online decomposition for the optimization problem, using a new swapping greedy algorithm for both OED problems and goal-oriented linear OED problems. The offline stage dominates the cost and entails precomputing all components requiring PDE solusion. The online stage optimizes sensor placement and does not require any PDE solves. We provide a detailed error analysis with an upper bound for the approximation error in evaluating the EIG for OED and goal-oriented OED linear cases. Finally, we evaluate the EIG with a derivative-informed projected neural network (DIPNet) surrogate for parameter-to-observable maps. With this surrogate, no further PDE solves are required to solve the optimization problem. We provided an analysis of the error propagated from the DIPNet approximation to the approximation of the normalization constant and the EIG under suitable assumptions. We demonstrate the efficiency and scalability of the proposed methods for both linear inverse problems, in which one seeks to infer the initial condition for an advection–diffusion equation, and nonlinear inverse problems, in which one seeks to infer coefficients for a Poisson problem, an acoustic Helmholtz problem and an advection–diffusion–reaction problem. This dissertation is based on the following articles: A fast and scalable computational framework for large-scale and high-dimensional Bayesian optimal experimental design by Keyi Wu, Peng Chen, and Omar Ghattas [88]; An efficient method for goal-oriented linear Bayesian optimal experimental design: Application to optimal sensor placement by Keyi Wu, Peng Chen, and Omar Ghattas [89]; and Derivative-informed projected neural network for large-scale Bayesian optimal experimental design by Keyi Wu, Thomas O’Leary-Roseberry, Peng Chen, and Omar Ghattas [90]. This material is based upon work partially funded by DOE ASCR DE-SC0019303 and DESC0021239, DOD MURI FA9550-21-1-0084, and NSF DMS-2012453
| Author | : Juan Chiachio-Ruano |
| Publisher | : CRC Press |
| Total Pages | : 289 |
| Release | : 2021-11-10 |
| Genre | : Mathematics |
| ISBN | : 1351869655 |
This book is devoted to a special class of engineering problems called Bayesian inverse problems. These problems comprise not only the probabilistic Bayesian formulation of engineering problems, but also the associated stochastic simulation methods needed to solve them. Through this book, the reader will learn how this class of methods can be useful to rigorously address a range of engineering problems where empirical data and fundamental knowledge come into play. The book is written for a non-expert audience and it is contributed to by many of the most renowned academic experts in this field.
| Author | : Christian Levi Lucero |
| Publisher | : |
| Total Pages | : 93 |
| Release | : 2013 |
| Genre | : Bayes theorem |
| ISBN | : |
| Author | : Peter Goos |
| Publisher | : John Wiley & Sons |
| Total Pages | : 249 |
| Release | : 2011-06-28 |
| Genre | : Science |
| ISBN | : 1119976162 |
"This is an engaging and informative book on the modern practice of experimental design. The authors' writing style is entertaining, the consulting dialogs are extremely enjoyable, and the technical material is presented brilliantly but not overwhelmingly. The book is a joy to read. Everyone who practices or teaches DOE should read this book." - Douglas C. Montgomery, Regents Professor, Department of Industrial Engineering, Arizona State University "It's been said: 'Design for the experiment, don't experiment for the design.' This book ably demonstrates this notion by showing how tailor-made, optimal designs can be effectively employed to meet a client's actual needs. It should be required reading for anyone interested in using the design of experiments in industrial settings." —Christopher J. Nachtsheim, Frank A Donaldson Chair in Operations Management, Carlson School of Management, University of Minnesota This book demonstrates the utility of the computer-aided optimal design approach using real industrial examples. These examples address questions such as the following: How can I do screening inexpensively if I have dozens of factors to investigate? What can I do if I have day-to-day variability and I can only perform 3 runs a day? How can I do RSM cost effectively if I have categorical factors? How can I design and analyze experiments when there is a factor that can only be changed a few times over the study? How can I include both ingredients in a mixture and processing factors in the same study? How can I design an experiment if there are many factor combinations that are impossible to run? How can I make sure that a time trend due to warming up of equipment does not affect the conclusions from a study? How can I take into account batch information in when designing experiments involving multiple batches? How can I add runs to a botched experiment to resolve ambiguities? While answering these questions the book also shows how to evaluate and compare designs. This allows researchers to make sensible trade-offs between the cost of experimentation and the amount of information they obtain.
| Author | : Anthony Atkinson |
| Publisher | : OUP Oxford |
| Total Pages | : 528 |
| Release | : 2007-05-24 |
| Genre | : Mathematics |
| ISBN | : 9780199296590 |
This text focuses on optimum experimental design using SAS, a powerful software package that provides a complete set of statistical tools including analysis of variance, regression, categorical data analysis, and multivariate analysis. SAS codes, results, plots, numerous figures and tables are provided, along with a fully supported website.
| Author | : Jesús López-Fidalgo |
| Publisher | : Springer Nature |
| Total Pages | : 228 |
| Release | : 2023-10-14 |
| Genre | : Mathematics |
| ISBN | : 3031359186 |
This textbook provides a concise introduction to optimal experimental design and efficiently prepares the reader for research in the area. It presents the common concepts and techniques for linear and nonlinear models as well as Bayesian optimal designs. The last two chapters are devoted to particular themes of interest, including recent developments and hot topics in optimal experimental design, and real-world applications. Numerous examples and exercises are included, some of them with solutions or hints, as well as references to the existing software for computing designs. The book is primarily intended for graduate students and young researchers in statistics and applied mathematics who are new to the field of optimal experimental design. Given the applications and the way concepts and results are introduced, parts of the text will also appeal to engineers and other applied researchers.
| Author | : |
| Publisher | : |
| Total Pages | : 90 |
| Release | : 2015 |
| Genre | : |
| ISBN | : |
The optimal selection of experimental conditions is essential to maximizing the value of data for inference and prediction. We propose an information theoretic framework and algorithms for robust optimal experimental design with simulation-based models, with the goal of maximizing information gain in targeted subsets of model parameters, particularly in situations where experiments are costly. Our framework employs a Bayesian statistical setting, which naturally incorporates heterogeneous sources of information. An objective function reflects expected information gain from proposed experimental designs. Monte Carlo sampling is used to evaluate the expected information gain, and stochastic approximation algorithms make optimization feasible for computationally intensive and high-dimensional problems. A key aspect of our framework is the introduction of model calibration discrepancy terms that are used to "relax" the model so that proposed optimal experiments are more robust to model error or inadequacy. We illustrate the approach via several model problems and misspecification scenarios. In particular, we show how optimal designs are modified by allowing for model error, and we evaluate the performance of various designs by simulating "real-world" data from models not considered explicitly in the optimization objective.
| Author | : Xun Huan |
| Publisher | : |
| Total Pages | : 186 |
| Release | : 2015 |
| Genre | : |
| ISBN | : |
Experimental data play a crucial role in developing and refining models of physical systems. Some experiments can be more valuable than others, however. Well-chosen experiments can save substantial resources, and hence optimal experimental design (OED) seeks to quantify and maximize the value of experimental data. Common current practice for designing a sequence of experiments uses suboptimal approaches: batch (open-loop) design that chooses all experiments simultaneously with no feedback of information, or greedy (myopic) design that optimally selects the next experiment without accounting for future observations and dynamics. In contrast, sequential optimal experimental design (sOED) is free of these limitations. With the goal of acquiring experimental data that are optimal for model parameter inference, we develop a rigorous Bayesian formulation for OED using an objective that incorporates a measure of information gain. This framework is first demonstrated in a batch design setting, and then extended to sOED using a dynamic programming (DP) formulation. We also develop new numerical tools for sOED to accommodate nonlinear models with continuous (and often unbounded) parameter, design, and observation spaces. Two major techniques are employed to make solution of the DP problem computationally feasible. First, the optimal policy is sought using a one-step lookahead representation combined with approximate value iteration. This approximate dynamic programming method couples backward induction and regression to construct value function approximations. It also iteratively generates trajectories via exploration and exploitation to further improve approximation accuracy in frequently visited regions of the state space. Second, transport maps are used to represent belief states, which reflect the intermediate posteriors within the sequential design process. Transport maps offer a finite-dimensional representation of these generally non-Gaussian random variables, and also enable fast approximate Bayesian inference, which must be performed millions of times under nested combinations of optimization and Monte Carlo sampling. The overall sOED algorithm is demonstrated and verified against analytic solutions on a simple linear-Gaussian model. Its advantages over batch and greedy designs are then shown via a nonlinear application of optimal sequential sensing: inferring contaminant source location from a sensor in a time-dependent convection-diffusion system. Finally, the capability of the algorithm is tested for multidimensional parameter and design spaces in a more complex setting of the source inversion problem.
| Author | : Lorenz Biegler |
| Publisher | : John Wiley & Sons |
| Total Pages | : 403 |
| Release | : 2011-06-24 |
| Genre | : Mathematics |
| ISBN | : 1119957583 |
This book focuses on computational methods for large-scale statistical inverse problems and provides an introduction to statistical Bayesian and frequentist methodologies. Recent research advances for approximation methods are discussed, along with Kalman filtering methods and optimization-based approaches to solving inverse problems. The aim is to cross-fertilize the perspectives of researchers in the areas of data assimilation, statistics, large-scale optimization, applied and computational mathematics, high performance computing, and cutting-edge applications. The solution to large-scale inverse problems critically depends on methods to reduce computational cost. Recent research approaches tackle this challenge in a variety of different ways. Many of the computational frameworks highlighted in this book build upon state-of-the-art methods for simulation of the forward problem, such as, fast Partial Differential Equation (PDE) solvers, reduced-order models and emulators of the forward problem, stochastic spectral approximations, and ensemble-based approximations, as well as exploiting the machinery for large-scale deterministic optimization through adjoint and other sensitivity analysis methods. Key Features: Brings together the perspectives of researchers in areas of inverse problems and data assimilation. Assesses the current state-of-the-art and identify needs and opportunities for future research. Focuses on the computational methods used to analyze and simulate inverse problems. Written by leading experts of inverse problems and uncertainty quantification. Graduate students and researchers working in statistics, mathematics and engineering will benefit from this book.
| Author | : Andrej Pázman |
| Publisher | : Springer |
| Total Pages | : 256 |
| Release | : 1986-01-31 |
| Genre | : Computers |
| ISBN | : |
Introductory remarks about the experiment and its disign. The regression model and methods of estimation. The ordering of designs and the properties of variaces of estimates. Optimality critaria in the regression model. Iterative computation of optimum desings Design of experiments in particular cases. The functional model and measurements of physical fields.
