Bayesian Approach to Inverse Problems

The focus of this book is on «ill-posed inverse problems». These problems cannot be solved only on the basis of observed data. The building of solutions involves the recognition of other pieces of a priori information. These solutions are then specific to the pieces of information taken into account. Clarifying and taking these pieces of information into account is necessary for grasping the domain of validity and the field of application for the solutions built. For too long, the interest in these problems has remained very limited in the signal-image community. However, the community has since recognized that these matters are more interesting and they have become the subject of much greater enthusiasm. From the application field’s point of view, a significant part of the book is devoted to conventional subjects in the field of inversion: biological and medical imaging, astronomy, non-destructive evaluation, processing of video sequences, target tracking, sensor networks and digital communications. The variety of chapters is also clear, when we examine the acquisition modalities at stake: conventional modalities, such as tomography and NMR, visible or infrared optical imaging, or more recent modalities such as atomic force imaging and polarized light imaging.

This book studies methods to concretely address inverse problems. An inverse problem arises when the causes that produced a given effect must be determined or when one seeks to indirectly estimate the parameters of a physical system. The author uses practical examples to illustrate inverse problems in physical sciences. He presents the techniques and specific methods chosen to solve inverse problems in a general domain of application, choosing to focus on a small number of methods that can be used in most applications. This book is aimed at readers with a mathematical and scientific computing background. Despite this, it is a book with a practical perspective. The methods described are applicable, have been applied, and are often illustrated by numerical examples.

Discrete Signals and Inverse Problems examines fundamental concepts necessary to engineers and scientists working with discrete signal processing and inverse problem solving, and places emphasis on the clear understanding of algorithms within the context of application needs. Based on the original ‘Introduction to Discrete Signals and Inverse Problems in Civil Engineering’, this expanded and enriched version: combines discrete signal processing and inverse problem solving in one book covers the most versatile tools that are needed to process engineering and scientific data presents step-by-step ‘implementation procedures’ for the most relevant algorithms provides instructive figures, solved examples and insightful exercises Discrete Signals and Inverse Problems is essential reading for experimental researchers and practicing engineers in civil, mechanical and electrical engineering, non-destructive testing and instrumentation. This book is also an excellent reference for advanced undergraduate students and graduate students in engineering and science.

This book provides researchers and engineers in the imaging field with the skills they need to effectively deal with nonlinear inverse problems associated with different imaging modalities, including impedance imaging, optical tomography, elastography, and electrical source imaging. Focusing on numerically implementable methods, the book bridges the gap between theory and applications, helping readers tackle problems in applied mathematics and engineering. Complete, self-contained coverage includes basic concepts, models, computational methods, numerical simulations, examples, and case studies. Provides a step-by-step progressive treatment of topics for ease of understanding. Discusses the underlying physical phenomena as well as implementation details of image reconstruction algorithms as prerequisites for finding solutions to non linear inverse problems with practical significance and value. Includes end of chapter problems, case studies and examples with solutions throughout the book. Companion website will provide further examples and solutions, experimental data sets, open problems, teaching material such as PowerPoint slides and software including MATLAB m files. Essential reading for Graduate students and researchers in imaging science working across the areas of applied mathematics, biomedical engineering, and electrical engineering and specifically those involved in nonlinear imaging techniques, impedance imaging, optical tomography, elastography, and electrical source imaging

Comparing and contrasting the reality of subjectivity in the work of history's great scientists and the modern Bayesian approach to statistical analysis Scientists and researchers are taught to analyze their data from an objective point of view, allowing the data to speak for themselves rather than assigning them meaning based on expectations or opinions. But scientists have never behaved fully objectively. Throughout history, some of our greatest scientific minds have relied on intuition, hunches, and personal beliefs to make sense of empirical data-and these subjective influences have often aided in humanity's greatest scientific achievements. The authors argue that subjectivity has not only played a significant role in the advancement of science, but that science will advance more rapidly if the modern methods of Bayesian statistical analysis replace some of the classical twentieth-century methods that have traditionally been taught. To accomplish this goal, the authors examine the lives and work of history's great scientists and show that even the most successful have sometimes misrepresented findings or been influenced by their own preconceived notions of religion, metaphysics, and the occult, or the personal beliefs of their mentors. Contrary to popular belief, our greatest scientific thinkers approached their data with a combination of subjectivity and empiricism, and thus informally achieved what is more formally accomplished by the modern Bayesian approach to data analysis. Yet we are still taught that science is purely objective. This innovative book dispels that myth using historical accounts and biographical sketches of more than a dozen great scientists, including Aristotle, Galileo Galilei, Johannes Kepler, William Harvey, Sir Isaac Newton, Antoine Levoisier, Alexander von Humboldt, Michael Faraday, Charles Darwin, Louis Pasteur, Gregor Mendel, Sigmund Freud, Marie Curie, Robert Millikan, Albert Einstein, Sir Cyril Burt, and Margaret Mead. Also included is a detailed treatment of the modern Bayesian approach to data analysis. Up-to-date references to the Bayesian theoretical and applied literature, as well as reference lists of the primary sources of the principal works of all the scientists discussed, round out this comprehensive treatment of the subject. Readers will benefit from this cogent and enlightening view of the history of subjectivity in science and the authors' alternative vision of how the Bayesian approach should be used to further the cause of science and learning well into the twenty-first century.

"…this edition is useful and effective in teaching Bayesian inference at both elementary and intermediate levels. It is a well-written book on elementary Bayesian inference, and the material is easily accessible. It is both concise and timely, and provides a good collection of overviews and reviews of important tools used in Bayesian statistical methods." There is a strong upsurge in the use of Bayesian methods in applied statistical analysis, yet most introductory statistics texts only present frequentist methods. Bayesian statistics has many important advantages that students should learn about if they are going into fields where statistics will be used. In this third Edition, four newly-added chapters address topics that reflect the rapid advances in the field of Bayesian statistics. The authors continue to provide a Bayesian treatment of introductory statistical topics, such as scientific data gathering, discrete random variables, robust Bayesian methods, and Bayesian approaches to inference for discrete random variables, binomial proportions, Poisson, and normal means, and simple linear regression. In addition, more advanced topics in the field are presented in four new chapters: Bayesian inference for a normal with unknown mean and variance; Bayesian inference for a Multivariate Normal mean vector; Bayesian inference for the Multiple Linear Regression Model; and Computational Bayesian Statistics including Markov Chain Monte Carlo. The inclusion of these topics will facilitate readers' ability to advance from a minimal understanding of Statistics to the ability to tackle topics in more applied, advanced level books. Minitab macros and R functions are available on the book's related website to assist with chapter exercises. Introduction to Bayesian Statistics, Third Edition also features: Topics including the Joint Likelihood function and inference using independent Jeffreys priors and join conjugate prior The cutting-edge topic of computational Bayesian Statistics in a new chapter, with a unique focus on Markov Chain Monte Carlo methods Exercises throughout the book that have been updated to reflect new applications and the latest software applications Detailed appendices that guide readers through the use of R and Minitab software for Bayesian analysis and Monte Carlo simulations, with all related macros available on the book's website Introduction to Bayesian Statistics, Third Edition is a textbook for upper-undergraduate or first-year graduate level courses on introductory statistics course with a Bayesian emphasis. It can also be used as a reference work for statisticians who require a working knowledge of Bayesian statistics.

The concept of an inverse problem is a familiar one to most scientists and engineers, particularly in the field of signal and image processing, imaging systems (medical, geophysical, industrial non-destructive testing, etc.) and computer vision. In imaging systems, the aim is not just to estimate unobserved images, but also their geometric characteristics from observed quantities that are linked to these unobserved quantities through the forward problem. This book focuses on imagery and vision problems that can be clearly written in terms of an inverse problem where an estimate for the image and its geometrical attributes (contours and regions) is sought. The chapters of this book use a consistent methodology to examine inverse problems such as: noise removal; restoration by deconvolution; 2D or 3D reconstruction in X-ray, tomography or microwave imaging; reconstruction of the surface of a 3D object using X-ray tomography or making use of its shading; reconstruction of the surface of a 3D landscape based on several satellite photos; super-resolution; motion estimation in a sequence of images; separation of several images mixed using instruments with different sensitivities or transfer functions; and more.

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.

This book deals with the various aspects of stochastic dynamics, the resolution of large mechanical systems, and inverse problems. It integrates the most recent ideas from research and industry in the field of stochastic dynamics and optimization in structural mechanics over 11 chapters. These chapters provide an update on the various tools for dealing with uncertainties, stochastic dynamics, reliability and optimization of systems. The optimization–reliability coupling in structures dynamics is approached in order to take into account the uncertainties in the modeling and the resolution of the problems encountered. Accompanied by detailed examples of uncertainties, optimization, reliability, and model reduction, this book presents the newest design tools. It is intended for students and engineers and is a valuable support for practicing engineers and teacher-researchers.

This book is a new edition of a title originally published in1992. No other book has been published that treats inverse spectral and inverse scattering results by using the so called Poisson summation formula and the related study of singularities. This book presents these in a closed and comprehensive form, and the exposition is based on a combination of different tools and results from dynamical systems, microlocal analysis, spectral and scattering theory. The content of the first edition is still relevant, however the new edition will include several new results established after 1992; new text will comprise about a third of the content of the new edition. The main chapters in the first edition in combination with the new chapters will provide a better and more comprehensive presentation of importance for the applications inverse results. These results are obtained by modern mathematical techniques which will be presented together in order to give the readers the opportunity to completely understand them. Moreover, some basic generic properties established by the authors after the publication of the first edition establishing the wide range of applicability of the Poison relation will be presented for first time in the new edition of the book.