Variational Source Conditions Quadratic Inverse Problems Sparsity Promoting Regularization

Author : Jens Flemming
Publisher : Springer
Page : 182 pages
File Size : 51,7 Mb
Release : 2018-09-08
Category : Mathematics
ISBN : 9783319952642

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Variational Source Conditions, Quadratic Inverse Problems, Sparsity Promoting Regularization by Jens Flemming Pdf

The book collects and contributes new results on the theory and practice of ill-posed inverse problems. Different notions of ill-posedness in Banach spaces for linear and nonlinear inverse problems are discussed not only in standard settings but also in situations up to now not covered by the literature. Especially, ill-posedness of linear operators with uncomplemented null spaces is examined.Tools for convergence rate analysis of regularization methods are extended to a wider field of applicability. It is shown that the tool known as variational source condition always yields convergence rate results. A theory for nonlinear inverse problems with quadratic structure is developed as well as corresponding regularization methods. The new methods are applied to a difficult inverse problem from laser optics.Sparsity promoting regularization is examined in detail from a Banach space point of view. Extensive convergence analysis reveals new insights into the behavior of Tikhonov-type regularization with sparsity enforcing penalty.

Author : Jin Cheng,Shuai Lu,Masahiro Yamamoto
Publisher : Springer Nature
Page : 310 pages
File Size : 42,5 Mb
Release : 2020-02-04
Category : Mathematics
ISBN : 9789811515927

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Inverse Problems and Related Topics by Jin Cheng,Shuai Lu,Masahiro Yamamoto Pdf

This volume contains 13 chapters, which are extended versions of the presentations at International Conference on Inverse Problems at Fudan University, Shanghai, China, October 12-14, 2018, in honor of Masahiro Yamamoto on the occasion of his 60th anniversary. The chapters are authored by world-renowned researchers and rising young talents, and are updated accounts of various aspects of the researches on inverse problems. The volume covers theories of inverse problems for partial differential equations, regularization methods, and related topics from control theory. This book addresses a wide audience of researchers and young post-docs and graduate students who are interested in mathematical sciences as well as mathematics.

Author : Andreas Kirsch
Publisher : Springer Nature
Page : 412 pages
File Size : 49,7 Mb
Release : 2021-02-15
Category : Mathematics
ISBN : 9783030633431

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An Introduction to the Mathematical Theory of Inverse Problems by Andreas Kirsch Pdf

This graduate-level textbook introduces the reader to the area of inverse problems, vital to many fields including geophysical exploration, system identification, nondestructive testing, and ultrasonic tomography. It aims to expose the basic notions and difficulties encountered with ill-posed problems, analyzing basic properties of regularization methods for ill-posed problems via several simple analytical and numerical examples. The book also presents three special nonlinear inverse problems in detail: the inverse spectral problem, the inverse problem of electrical impedance tomography (EIT), and the inverse scattering problem. The corresponding direct problems are studied with respect to existence, uniqueness, and continuous dependence on parameters. Ultimately, the text discusses theoretical results as well as numerical procedures for the inverse problems, including many exercises and illustrations to complement coursework in mathematics and engineering. This updated text includes a new chapter on the theory of nonlinear inverse problems in response to the field’s growing popularity, as well as a new section on the interior transmission eigenvalue problem which complements the Sturm-Liouville problem and which has received great attention since the previous edition was published.

Author : Dennis Trede
Publisher : Logos Verlag Berlin GmbH
Page : 137 pages
File Size : 47,6 Mb
Release : 2010
Category : Computers
ISBN : 9783832524661

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Inverse Problems with Sparsity Constraints by Dennis Trede Pdf

This thesis contributes to the field of inverse problems with sparsity constraints. Since the pioneering work by Daubechies, Defries and De Mol in 2004, methods for solving operator equations with sparsity constraints play a central role in the field of inverse problems. This can be explained by the fact that the solutions of many inverse problems have a sparse structure, in other words, they can be represented using only finitely many elements of a suitable basis or dictionary. Generally, to stably solve an ill-posed inverse problem one needs additional assumptions on the unknown solution--the so-called source condition. In this thesis, the sparseness stands for the source condition, and with that in mind, stability results for two different approximation methods are deduced, namely, results for the Tikhonov regularization with a sparsity-enforcing penalty and for the orthogonal matching pursuit. The practical relevance of the theoretical results is shown with two examples of convolution type, namely, an example from mass spectrometry and an example from digital holography of particles.

Author : Ito Kazufumi,Jin Bangti
Publisher : World Scientific
Page : 332 pages
File Size : 41,7 Mb
Release : 2014-08-28
Category : Mathematics
ISBN : 9789814596213

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Inverse Problems: Tikhonov Theory And Algorithms by Ito Kazufumi,Jin Bangti Pdf

Inverse problems arise in practical applications whenever one needs to deduce unknowns from observables. This monograph is a valuable contribution to the highly topical field of computational inverse problems. Both mathematical theory and numerical algorithms for model-based inverse problems are discussed in detail. The mathematical theory focuses on nonsmooth Tikhonov regularization for linear and nonlinear inverse problems. The computational methods include nonsmooth optimization algorithms, direct inversion methods and uncertainty quantification via Bayesian inference.The book offers a comprehensive treatment of modern techniques, and seamlessly blends regularization theory with computational methods, which is essential for developing accurate and efficient inversion algorithms for many practical inverse problems.It demonstrates many current developments in the field of computational inversion, such as value function calculus, augmented Tikhonov regularization, multi-parameter Tikhonov regularization, semismooth Newton method, direct sampling method, uncertainty quantification and approximate Bayesian inference. It is written for graduate students and researchers in mathematics, natural science and engineering.

Author : Tim Salditt,Alexander Egner,D. Russell Luke
Publisher : Springer Nature
Page : 634 pages
File Size : 55,7 Mb
Release : 2020-06-09
Category : Science
ISBN : 9783030344139

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Nanoscale Photonic Imaging by Tim Salditt,Alexander Egner,D. Russell Luke Pdf

This open access book, edited and authored by a team of world-leading researchers, provides a broad overview of advanced photonic methods for nanoscale visualization, as well as describing a range of fascinating in-depth studies. Introductory chapters cover the most relevant physics and basic methods that young researchers need to master in order to work effectively in the field of nanoscale photonic imaging, from physical first principles, to instrumentation, to mathematical foundations of imaging and data analysis. Subsequent chapters demonstrate how these cutting edge methods are applied to a variety of systems, including complex fluids and biomolecular systems, for visualizing their structure and dynamics, in space and on timescales extending over many orders of magnitude down to the femtosecond range. Progress in nanoscale photonic imaging in Göttingen has been the sum total of more than a decade of work by a wide range of scientists and mathematicians across disciplines, working together in a vibrant collaboration of a kind rarely matched. This volume presents the highlights of their research achievements and serves as a record of the unique and remarkable constellation of contributors, as well as looking ahead at the future prospects in this field. It will serve not only as a useful reference for experienced researchers but also as a valuable point of entry for newcomers.

Author : Richard Huber
Publisher : Unknown
Page : 136 pages
File Size : 54,7 Mb
Release : 2019
Category : Computer science
ISBN : 3658253916

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Variational Regularization for Systems of Inverse Problems by Richard Huber Pdf

Tikhonov regularization is a cornerstone technique in solving inverse problems with applications in countless scientific fields. Richard Huber discusses a multi-parameter Tikhonov approach for systems of inverse problems in order to take advantage of their specific structure. Such an approach allows to choose the regularization weights of each subproblem individually with respect to the corresponding noise levels and degrees of ill-posedness. Contents General Tikhonov Regularization Specific Discrepancies Regularization Functionals Application to STEM Tomography Reconstruction Target Groups Researchers and students in the field of mathematics Experts in the areas of mathematics, imaging, computer vision and nanotechnology The Author Richard Huber wrote his master's thesis under the supervision of Prof. Dr. Kristian Bredies at the Institute for Mathematics and Scientific Computing at Graz University, Austria.

Author : Roland Glowinski,Stanley J. Osher,Wotao Yin
Publisher : Springer
Page : 820 pages
File Size : 46,9 Mb
Release : 2017-01-05
Category : Mathematics
ISBN : 9783319415895

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Splitting Methods in Communication, Imaging, Science, and Engineering by Roland Glowinski,Stanley J. Osher,Wotao Yin Pdf

This book is about computational methods based on operator splitting. It consists of twenty-three chapters written by recognized splitting method contributors and practitioners, and covers a vast spectrum of topics and application areas, including computational mechanics, computational physics, image processing, wireless communication, nonlinear optics, and finance. Therefore, the book presents very versatile aspects of splitting methods and their applications, motivating the cross-fertilization of ideas.

Author : Per Christian Hansen
Publisher : SIAM
Page : 220 pages
File Size : 49,8 Mb
Release : 2010-01-01
Category : Mathematics
ISBN : 9780898718836

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Discrete Inverse Problems by Per Christian Hansen Pdf

This book gives an introduction to the practical treatment of inverse problems by means of numerical methods, with a focus on basic mathematical and computational aspects. To solve inverse problems, we demonstrate that insight about them goes hand in hand with algorithms.

Author : Francis Bach,Rodolphe Jenatton,Julien Mairal
Publisher : Unknown
Page : 124 pages
File Size : 53,6 Mb
Release : 2011-12-23
Category : Computers
ISBN : 160198510X

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Optimization with Sparsity-Inducing Penalties by Francis Bach,Rodolphe Jenatton,Julien Mairal Pdf

Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. They were first dedicated to linear variable selection but numerous extensions have now emerged such as structured sparsity or kernel selection. It turns out that many of the related estimation problems can be cast as convex optimization problems by regularizing the empirical risk with appropriate nonsmooth norms. Optimization with Sparsity-Inducing Penalties presents optimization tools and techniques dedicated to such sparsity-inducing penalties from a general perspective. It covers proximal methods, block-coordinate descent, reweighted ?2-penalized techniques, working-set and homotopy methods, as well as non-convex formulations and extensions, and provides an extensive set of experiments to compare various algorithms from a computational point of view. The presentation of Optimization with Sparsity-Inducing Penalties is essentially based on existing literature, but the process of constructing a general framework leads naturally to new results, connections and points of view. It is an ideal reference on the topic for anyone working in machine learning and related areas.

Author : Michael S Zhdanov
Publisher : Elsevier
Page : 635 pages
File Size : 41,9 Mb
Release : 2002-04-24
Category : Science
ISBN : 9780080532509

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Geophysical Inverse Theory and Regularization Problems by Michael S Zhdanov Pdf

This book presents state-of-the-art geophysical inverse theory developed in modern mathematical terminology. The book brings together fundamental results developed by the Russian mathematical school in regularization theory and combines them with the related research in geophysical inversion carried out in the West. It presents a detailed exposition of the methods of regularized solution of inverse problems based on the ideas of Tikhonov regularization, and shows the different forms of their applications in both linear and nonlinear methods of geophysical inversion. This text is the first to treat many kinds of inversion and imaging techniques in a unified mathematical manner. The book is divided in five parts covering the foundations of the inversion theory and its applications to the solution of different geophysical inverse problems, including potential field, electromagnetic, and seismic methods. The first part is an introduction to inversion theory. The second part contains a description of the basic methods of solution of the linear and nonlinear inverse problems using regularization. The following parts treat the application of regularization methods in gravity and magnetic, electromagnetic, and seismic inverse problems. The key connecting idea of these applied parts of the book is the analogy between the solutions of the forward and inverse problems in different geophysical methods. The book also includes chapters related to the modern technology of geophysical imaging, based on seismic and electromagnetic migration. This volume is unique in its focus on providing a link between the methods used in gravity, electromagnetic, and seismic imaging and inversion, and represents an exhaustive treatise on inversion theory.

Author : Charles Byrne
Publisher : CRC Press
Page : 298 pages
File Size : 52,9 Mb
Release : 2014-02-12
Category : Business & Economics
ISBN : 9781482222340

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Iterative Optimization in Inverse Problems by Charles Byrne Pdf

Iterative Optimization in Inverse Problems brings together a number of important iterative algorithms for medical imaging, optimization, and statistical estimation. It incorporates recent work that has not appeared in other books and draws on the author's considerable research in the field, including his recently developed class of SUMMA algorithms

Author : A.N. Tikhonov
Publisher : Springer
Page : 0 pages
File Size : 50,7 Mb
Release : 2014-08-23
Category : Mathematics
ISBN : 9401751692

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Nonlinear Ill-Posed Problems by A.N. Tikhonov Pdf

Author : Otmar Scherzer,Markus Grasmair,Harald Grossauer,Markus Haltmeier,Frank Lenzen
Publisher : Springer Science & Business Media
Page : 323 pages
File Size : 46,8 Mb
Release : 2008-09-26
Category : Mathematics
ISBN : 9780387692777

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Variational Methods in Imaging by Otmar Scherzer,Markus Grasmair,Harald Grossauer,Markus Haltmeier,Frank Lenzen Pdf

This book is devoted to the study of variational methods in imaging. The presentation is mathematically rigorous and covers a detailed treatment of the approach from an inverse problems point of view. Many numerical examples accompany the theory throughout the text. It is geared towards graduate students and researchers in applied mathematics. Researchers in the area of imaging science will also find this book appealing. It can serve as a main text in courses in image processing or as a supplemental text for courses on regularization and inverse problems at the graduate level.

Author : Thomas Schuster,Barbara Kaltenbacher,Bernd Hofmann,Kamil S. Kazimierski
Publisher : Walter de Gruyter
Page : 296 pages
File Size : 40,7 Mb
Release : 2012-07-30
Category : Mathematics
ISBN : 9783110255720

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Regularization Methods in Banach Spaces by Thomas Schuster,Barbara Kaltenbacher,Bernd Hofmann,Kamil S. Kazimierski Pdf

Regularization methods aimed at finding stable approximate solutions are a necessary tool to tackle inverse and ill-posed problems. Inverse problems arise in a large variety of applications ranging from medical imaging and non-destructive testing via finance to systems biology. Many of these problems belong to the class of parameter identification problems in partial differential equations (PDEs) and thus are computationally demanding and mathematically challenging. Hence there is a substantial need for stable and efficient solvers for this kind of problems as well as for a rigorous convergence analysis of these methods. This monograph consists of five parts. Part I motivates the importance of developing and analyzing regularization methods in Banach spaces by presenting four applications which intrinsically demand for a Banach space setting and giving a brief glimpse of sparsity constraints. Part II summarizes all mathematical tools that are necessary to carry out an analysis in Banach spaces. Part III represents the current state-of-the-art concerning Tikhonov regularization in Banach spaces. Part IV about iterative regularization methods is concerned with linear operator equations and the iterative solution of nonlinear operator equations by gradient type methods and the iteratively regularized Gauß-Newton method. Part V finally outlines the method of approximate inverse which is based on the efficient evaluation of the measured data with reconstruction kernels.