Author : Tatiana A. Bubba
Publisher : Walter de Gruyter GmbH & Co KG
Page : 664 pages
File Size : 45,84 MB
Release : 2024-11-18
Category : 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 : H. T. Banks
Publisher : CRC Press
Page : 403 pages
File Size : 16,89 MB
Release : 2014-04-01
Category : 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 : Maximilian Schmidt
Publisher :
Page : 0 pages
File Size : 15,98 MB
Release : 2022
Category :
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 : Luis Tenorio
Publisher : SIAM
Page : 275 pages
File Size : 12,32 MB
Release : 2017-07-06
Category : 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 : Mark Asch
Publisher : SIAM
Page : 857 pages
File Size : 21,76 MB
Release : 2022-08-04
Category : 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 : Albert Tarantola
Publisher : SIAM
Page : 349 pages
File Size : 25,7 MB
Release : 2005-01-01
Category : 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 : Steve Pressé
Publisher : Cambridge University Press
Page : 433 pages
File Size : 31,86 MB
Release : 2023-07-31
Category : Science
ISBN : 1009098500
A self-contained and accessible guide to probabilistic data modeling, ideal for students and researchers in the natural sciences.
Author : Alik Ismail-Zadeh
Publisher : Springer
Page : 113 pages
File Size : 45,77 MB
Release : 2016-05-17
Category : 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.
Author : H. T. Banks
Publisher : CRC Press
Page : 408 pages
File Size : 25,84 MB
Release : 2014-04-01
Category : Mathematics
ISBN : 1482206420
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 itself. After a useful review of relevant probability and statistical concepts, the book summarizes mathematical and statistical aspects of inverse problem methodology, including ordinary, weighted, and generalized least-squares formulations. It then discusses asymptotic theories, bootstrapping, and issues related to the evaluation of correctness of assumed form of statistical models. The authors go on to present methods for evaluating and comparing the validity of appropriateness of a collection of models for describing a given data set, including statistically based model selection and comparison techniques. They also explore recent results on the estimation of probability distributions when they are embedded in complex mathematical models and only aggregate (not individual) data are available. In addition, they briefly discuss the optimal design of experiments in support of inverse problems for given models. The book concludes with a focus on uncertainty in model formulation itself, covering the general relationship of differential equations driven by white noise and the ones driven by colored noise in terms of their resulting probability density functions. It also deals with questions related to the appropriateness of discrete versus continuum models in transitions from small to large numbers of individuals. With many examples throughout addressing problems in physics, biology, and other areas, this book is intended for applied mathematicians interested in deterministic and/or stochastic models and their interactions. It is also suitable for scientists in biology, medicine, engineering, and physics working on basic modeling and inverse problems, uncertainty in modeling, propagation of uncertainty, and statistical modeling.
Author : Ke Chen
Publisher : Springer Nature
Page : 1981 pages
File Size : 37,74 MB
Release : 2023-02-24
Category : Mathematics
ISBN : 3030986616
This handbook gathers together the state of the art on mathematical models and algorithms for imaging and vision. Its emphasis lies on rigorous mathematical methods, which represent the optimal solutions to a class of imaging and vision problems, and on effective algorithms, which are necessary for the methods to be translated to practical use in various applications. Viewing discrete images as data sampled from functional surfaces enables the use of advanced tools from calculus, functions and calculus of variations, and nonlinear optimization, and provides the basis of high-resolution imaging through geometry and variational models. Besides, optimization naturally connects traditional model-driven approaches to the emerging data-driven approaches of machine and deep learning. No other framework can provide comparable accuracy and precision to imaging and vision. Written by leading researchers in imaging and vision, the chapters in this handbook all start with gentle introductions, which make this work accessible to graduate students. For newcomers to the field, the book provides a comprehensive and fast-track introduction to the content, to save time and get on with tackling new and emerging challenges. For researchers, exposure to the state of the art of research works leads to an overall view of the entire field so as to guide new research directions and avoid pitfalls in moving the field forward and looking into the next decades of imaging and information services. This work can greatly benefit graduate students, researchers, and practitioners in imaging and vision; applied mathematicians; medical imagers; engineers; and computer scientists.
