Data Driven Models In Inverse Problems
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Author | : Tatiana A. Bubba |
Publisher | : Walter de Gruyter GmbH & Co KG |
Total Pages | : 664 |
Release | : 2024-11-18 |
Genre | : Mathematics |
ISBN | : 3111251292 |
Advances in learning-based methods are revolutionizing several fields in applied mathematics, including inverse problems, resulting in a major paradigm shift towards data-driven approaches. This volume, which is inspired by this cutting-edge area of research, brings together contributors from the inverse problem community and shows how to successfully combine model- and data-driven approaches to gain insight into practical and theoretical issues.
Author | : Tatiana A Bubba |
Publisher | : |
Total Pages | : 0 |
Release | : 2024-11-20 |
Genre | : Computers |
ISBN | : 9783111250038 |
Advances in learning-based methods are revolutionizing several fields in applied mathematics, including inverse problems, resulting in a major paradigm shift towards data-driven approaches. This volume, which is inspired by this cutting-edge area of research, brings together contributors from the inverse problem community and shows how to successfully combine model- and data-driven approaches to gain insight into practical and theoretical issues.
Author | : H. T. Banks |
Publisher | : CRC Press |
Total Pages | : 403 |
Release | : 2014-04-01 |
Genre | : Mathematics |
ISBN | : 1482206439 |
Modeling and Inverse Problems in the Presence of Uncertainty collects recent research-including the authors' own substantial projects-on uncertainty propagation and quantification. It covers two sources of uncertainty: where uncertainty is present primarily due to measurement errors and where uncertainty is present due to the modeling formulation i
Author | : Luis Tenorio |
Publisher | : SIAM |
Total Pages | : 275 |
Release | : 2017-07-06 |
Genre | : Mathematics |
ISBN | : 1611974917 |
Inverse problems are found in many applications, such as medical imaging, engineering, astronomy, and geophysics, among others. To solve an inverse problem is to recover an object from noisy, usually indirect observations. Solutions to inverse problems are subject to many potential sources of error introduced by approximate mathematical models, regularization methods, numerical approximations for efficient computations, noisy data, and limitations in the number of observations; thus it is important to include an assessment of the uncertainties as part of the solution. Such assessment is interdisciplinary by nature, as it requires, in addition to knowledge of the particular application, methods from applied mathematics, probability, and statistics. This book bridges applied mathematics and statistics by providing a basic introduction to probability and statistics for uncertainty quantification in the context of inverse problems, as well as an introduction to statistical regularization of inverse problems. The author covers basic statistical inference, introduces the framework of ill-posed inverse problems, and explains statistical questions that arise in their applications. An Introduction to Data Analysis and Uncertainty Quantification for Inverse Problems?includes many examples that explain techniques which are useful to address general problems arising in uncertainty quantification, Bayesian and non-Bayesian statistical methods and discussions of their complementary roles, and analysis of a real data set to illustrate the methodology covered throughout the book.
Author | : Maximilian Schmidt |
Publisher | : |
Total Pages | : 0 |
Release | : 2022 |
Genre | : |
ISBN | : |
Artificial neural networks from the field of deep learning are increasingly becoming the state of the art in more and more applications. Their success is based on learning complex relationships in a system purely from data. For this, the data-driven networks often require hundreds of thousands of reference examples. They are contrasted by model-based approaches that use mathematical methods to describe the processes in a system. They work without large amounts of data but often cannot cover all the nuances of an application. In inverse problems, model-based approaches have been the standard so far. Here, the necessary amount of data to use purely data-driven deep learning is usually unavailable. In addition, requirements are placed on the model properties that cannot always be proven for classical neural networks. Hybrid deep learning models that combine data-driven and model-based approaches can solve these challenges. In recent years, their research has steadily gained importance. In this thesis, several hybrid deep learning approaches for solving inverse problems are presented and further developed. These include the deep image prior (DIP) and conditional invertible neural networks (CINN). The reconstruction problem in computed tomography (CT) serves as a central example to compare the models with each other, as well as to reveal their strengths and weaknesses. This is done in particular concerning the unique challenges in inverse problems, such as lack of data and ill-posedness. For this purpose, a realistic medical CT dataset is presented and used. The performed comparison for medical and industrial data clearly shows that the hybrid approaches are superior to the classical, model-based methods in many areas. Countless applications from inverse problems can thus already benefit from hybrid deep learning approaches.
Author | : Mark Asch |
Publisher | : SIAM |
Total Pages | : 857 |
Release | : 2022-08-04 |
Genre | : Mathematics |
ISBN | : 1611976979 |
This book brings together the mathematical and numerical frameworks needed for developing digital twins. Starting from the basics—probability, statistics, numerical methods, optimization, and machine learning—and moving on to data assimilation, inverse problems, and Bayesian uncertainty quantification, the book provides a comprehensive toolbox for digital twins. Emphasis is also placed on the design process, denoted as the “inference cycle,” the aim of which is to propose a global methodology for complex problems. Readers will find guidelines and decision trees to help them choose the right tools for the job; a comprehensive reference section with all recent methods, covering both model-based and data-driven approaches; a vast selection of examples and all accompanying code; and a companion website containing updates, case studies, and extended material. A Toolbox for Digital Twins: From Model-Based to Data-Driven is for researchers and engineers, engineering students, and scientists in any domain where data and models need to be coupled to produce digital twins.
Author | : Ellen Brooke Le |
Publisher | : |
Total Pages | : 266 |
Release | : 2018 |
Genre | : |
ISBN | : |
A persistent central challenge in computational science and engineering (CSE), with both national and global security implications, is the efficient solution of large-scale Bayesian inverse problems. These problems range from estimating material parameters in subsurface simulations to estimating phenomenological parameters in climate models. Despite recent progress, our ability to quantify uncertainties and solve large-scale inverse problems lags well behind our ability to develop the governing forward simulations. Inverse problems present unique computational challenges that are only magnified as we include larger observational data sets and demand higher-resolution parameter estimates. Even with the current state-of-the-art, solving deterministic large-scale inverse problems is prohibitively expensive. Large-scale uncertainty quantification (UQ), cast in the Bayesian inversion framework, is thus rendered intractable. To conquer these challenges, new methods that target the root causes of computational complexity are needed. In this dissertation, we propose data driven strategies for overcoming this "curse of di- mensionality." First, we address the computational complexity induced in large-scale inverse problems by high-dimensional observational data. We propose a randomized misfit approach (RMA), which uses random projections--quasi-orthogonal, information-preserving transformations--to map the high-dimensional data-misfit vector to a low dimensional space. We provide the first theoretical explanation for why randomized misfit methods are successful in practice with a small reduced data-misfit dimension (n = O(1)). Next, we develop the randomized geostatistical approach (RGA) for Bayesian sub- surface inverse problems with high-dimensional data. We show that the RGA is able to resolve transient groundwater inverse problems with noisy observed data dimensions up to 107, whereas a comparison method fails due to out-of-memory errors. Finally, we address the solution of Bayesian inverse problems with spatially localized data. The motivation is CSE applications that would gain from high-fidelity estimation over a smaller data-local domain, versus expensive and uncertain estimation over the full simulation domain. We propose several truncated domain inversion methods using domain decomposition theory to build model-informed artificial boundary conditions. Numerical investigations of MAP estimation and sampling demonstrate improved fidelity and fewer partial differential equation (PDE) solves with our truncated methods.
Author | : Steve Pressé |
Publisher | : Cambridge University Press |
Total Pages | : 433 |
Release | : 2023-07-31 |
Genre | : Science |
ISBN | : 1009098500 |
A self-contained and accessible guide to probabilistic data modeling, ideal for students and researchers in the natural sciences.
Author | : Albert Tarantola |
Publisher | : SIAM |
Total Pages | : 349 |
Release | : 2005-01-01 |
Genre | : Mathematics |
ISBN | : 9780898717921 |
While the prediction of observations is a forward problem, the use of actual observations to infer the properties of a model is an inverse problem. Inverse problems are difficult because they may not have a unique solution. The description of uncertainties plays a central role in the theory, which is based on probability theory. This book proposes a general approach that is valid for linear as well as for nonlinear problems. The philosophy is essentially probabilistic and allows the reader to understand the basic difficulties appearing in the resolution of inverse problems. The book attempts to explain how a method of acquisition of information can be applied to actual real-world problems, and many of the arguments are heuristic.
Author | : Alik Ismail-Zadeh |
Publisher | : Springer |
Total Pages | : 113 |
Release | : 2016-05-17 |
Genre | : Science |
ISBN | : 3319278010 |
This book describes the methods and numerical approaches for data assimilation in geodynamical models and presents several applications of the described methodology in relevant case studies. The book starts with a brief overview of the basic principles in data-driven geodynamic modelling, inverse problems, and data assimilation methods, which is then followed by methodological chapters on backward advection, variational (or adjoint), and quasi-reversibility methods. The chapters are accompanied by case studies presenting the applicability of the methods for solving geodynamic problems; namely, mantle plume evolution; lithosphere dynamics in and beneath two distinct geological domains – the south-eastern Carpathian Mountains and the Japanese Islands; salt diapirism in sedimentary basins; and volcanic lava flow. Applications of data-driven modelling are of interest to the industry and to experts dealing with geohazards and risk mitigation. Explanation of the sedimentary basin evolution complicated by deformations due to salt tectonics can help in oil and gas exploration; better understanding of the stress-strain evolution in the past and stress localization in the present can provide an insight into large earthquake preparation processes; volcanic lava flow assessments can advise on risk mitigation in the populated areas. The book is an essential tool for advanced courses on data assimilation and numerical modelling in geodynamics.