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Carleman Estimates For The General Second Order Operators And Applications To Inverse Problems
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Book Synopsis Carleman Estimates for Second Order Partial Differential Operators and Applications by : Xiaoyu Fu
Download or read book Carleman Estimates for Second Order Partial Differential Operators and Applications written by Xiaoyu Fu and published by Springer. This book was released on 2019-11-13 with total page 127 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a brief, self-contained introduction to Carleman estimates for three typical second order partial differential equations, namely elliptic, parabolic, and hyperbolic equations, and their typical applications in control, unique continuation, and inverse problems. There are three particularly important and novel features of the book. First, only some basic calculus is needed in order to obtain the main results presented, though some elementary knowledge of functional analysis and partial differential equations will be helpful in understanding them. Second, all Carleman estimates in the book are derived from a fundamental identity for a second order partial differential operator; the only difference is the choice of weight functions. Third, only rather weak smoothness and/or integrability conditions are needed for the coefficients appearing in the equations. Carleman Estimates for Second Order Partial Differential Operators and Applications will be of interest to all researchers in the field.
Book Synopsis Carleman Estimates for the General Second Order Operators and Applications to Inverse Problems by : Nanhee Kim
Download or read book Carleman Estimates for the General Second Order Operators and Applications to Inverse Problems written by Nanhee Kim and published by . This book was released on 2010 with total page 101 pages. Available in PDF, EPUB and Kindle. Book excerpt: We derive Carleman estimates with two large parameters for a general partial di erential operator of second order under explicit su cient global conditions of pseudo-convexity on the weight function. We use these estimates to derive the most natural Carleman type estimates for the anisotropic system of elasticity with residual stress. Also, we give applications to uniqueness and stability of the continuation, observability, and identi cation of the residual stress from boundary measurements.
Book Synopsis Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems by : Mourad Bellassoued
Download or read book Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems written by Mourad Bellassoued and published by Springer. This book was released on 2017-11-23 with total page 267 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a self-contained account of the method based on Carleman estimates for inverse problems of determining spatially varying functions of differential equations of the hyperbolic type by non-overdetermining data of solutions. The formulation is different from that of Dirichlet-to-Neumann maps and can often prove the global uniqueness and Lipschitz stability even with a single measurement. These types of inverse problems include coefficient inverse problems of determining physical parameters in inhomogeneous media that appear in many applications related to electromagnetism, elasticity, and related phenomena. Although the methodology was created in 1981 by Bukhgeim and Klibanov, its comprehensive development has been accomplished only recently. In spite of the wide applicability of the method, there are few monographs focusing on combined accounts of Carleman estimates and applications to inverse problems. The aim in this book is to fill that gap. The basic tool is Carleman estimates, the theory of which has been established within a very general framework, so that the method using Carleman estimates for inverse problems is misunderstood as being very difficult. The main purpose of the book is to provide an accessible approach to the methodology. To accomplish that goal, the authors include a direct derivation of Carleman estimates, the derivation being based essentially on elementary calculus working flexibly for various equations. Because the inverse problem depends heavily on respective equations, too general and abstract an approach may not be balanced. Thus a direct and concrete means was chosen not only because it is friendly to readers but also is much more relevant. By practical necessity, there is surely a wide range of inverse problems and the method delineated here can solve them. The intention is for readers to learn that method and then apply it to solving new inverse problems.
Book Synopsis Carleman Estimates for Coefficient Inverse Problems and Numerical Applications by : Michael V. Klibanov
Download or read book Carleman Estimates for Coefficient Inverse Problems and Numerical Applications written by Michael V. Klibanov and published by Walter de Gruyter. This book was released on 2012-04-17 with total page 292 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this monograph, the main subject of the author's considerations is coefficient inverse problems. Arising in many areas of natural sciences and technology, such problems consist of determining the variable coefficients of a certain differential operator defined in a domain from boundary measurements of a solution or its functionals. Although the authors pay strong attention to the rigorous justification of known results, they place the primary emphasis on new concepts and developments.
Book Synopsis Inverse Problems and Carleman Estimates by : Michael V. Klibanov
Download or read book Inverse Problems and Carleman Estimates written by Michael V. Klibanov and published by Walter de Gruyter GmbH & Co KG. This book was released on 2021-09-07 with total page 344 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book summarizes the main analytical and numerical results of Carleman estimates. In the analytical part, Carleman estimates for three main types of Partial Differential Equations (PDEs) are derived. In the numerical part, first numerical methods are proposed to solve ill-posed Cauchy problems for both linear and quasilinear PDEs. Next, various versions of the convexification method are developed for a number of Coefficient Inverse Problems.
Book Synopsis Carleman Estimates for Second Order Partial Differential Operators and Applications by : Xiaoyu Fu
Download or read book Carleman Estimates for Second Order Partial Differential Operators and Applications written by Xiaoyu Fu and published by Springer Nature. This book was released on 2019-10-31 with total page 127 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a brief, self-contained introduction to Carleman estimates for three typical second order partial differential equations, namely elliptic, parabolic, and hyperbolic equations, and their typical applications in control, unique continuation, and inverse problems. There are three particularly important and novel features of the book. First, only some basic calculus is needed in order to obtain the main results presented, though some elementary knowledge of functional analysis and partial differential equations will be helpful in understanding them. Second, all Carleman estimates in the book are derived from a fundamental identity for a second order partial differential operator; the only difference is the choice of weight functions. Third, only rather weak smoothness and/or integrability conditions are needed for the coefficients appearing in the equations. Carleman Estimates for Second Order Partial Differential Operators and Applications will be of interest to all researchers in the field.
Book Synopsis Carleman Inequalities by : Nicolas Lerner
Download or read book Carleman Inequalities written by Nicolas Lerner and published by Springer. This book was released on 2019-05-18 with total page 557 pages. Available in PDF, EPUB and Kindle. Book excerpt: Over the past 25 years, Carleman estimates have become an essential tool in several areas related to partial differential equations such as control theory, inverse problems, or fluid mechanics. This book provides a detailed exposition of the basic techniques of Carleman Inequalities, driven by applications to various questions of unique continuation. Beginning with an elementary introduction to the topic, including examples accessible to readers without prior knowledge of advanced mathematics, the book's first five chapters contain a thorough exposition of the most classical results, such as Calderón's and Hörmander's theorems. Later chapters explore a selection of results of the last four decades around the themes of continuation for elliptic equations, with the Jerison-Kenig estimates for strong unique continuation, counterexamples to Cauchy uniqueness of Cohen and Alinhac & Baouendi, operators with partially analytic coefficients with intermediate results between Holmgren's and Hörmander's uniqueness theorems, Wolff's modification of Carleman's method, conditional pseudo-convexity, and more. With examples and special cases motivating the general theory, as well as appendices on mathematical background, this monograph provides an accessible, self-contained basic reference on the subject, including a selection of the developments of the past thirty years in unique continuation.
Book Synopsis Inverse Problems and Related Topics by : Jin Cheng
Download or read book Inverse Problems and Related Topics written by Jin Cheng and published by Springer Nature. This book was released on 2020-02-04 with total page 310 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Carleman Estimates for the General Second Order Operators by : Nanhee Kim
Download or read book Carleman Estimates for the General Second Order Operators written by Nanhee Kim and published by LAP Lambert Academic Publishing. This book was released on 2014-02 with total page 112 pages. Available in PDF, EPUB and Kindle. Book excerpt: We derive Carleman estimates with two large parameters for a general partial differential operator of second order under explicit sufficient global conditions of pseudo-convexity on the weight function. We use these estimates to derive the most natural Carleman type estimates for the anisotropic system of elasticity with residual stress. Also, we give applications to uniqueness and stability of the continuation, observability, and identification of the residual stress from boundary measurements.
Book Synopsis Inverse Problems for Partial Differential Equations by : Victor Isakov
Download or read book Inverse Problems for Partial Differential Equations written by Victor Isakov and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 296 pages. Available in PDF, EPUB and Kindle. Book excerpt: A comprehensive description of the current theoretical and numerical aspects of inverse problems in partial differential equations. Applications include recovery of inclusions from anomalies of their gravity fields, reconstruction of the interior of the human body from exterior electrical, ultrasonic, and magnetic measurement. By presenting the data in a readable and informative manner, the book introduces both scientific and engineering researchers as well as graduate students to the significant work done in this area in recent years, relating it to broader themes in mathematical analysis.
Book Synopsis Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume II by : Jérôme Le Rousseau
Download or read book Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume II written by Jérôme Le Rousseau and published by Springer Nature. This book was released on 2022-04-22 with total page 542 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph explores applications of Carleman estimates in the study of stabilization and controllability properties of partial differential equations, including quantified unique continuation, logarithmic stabilization of the wave equation, and null-controllability of the heat equation. Where the first volume derived these estimates in regular open sets in Euclidean space and Dirichlet boundary conditions, here they are extended to Riemannian manifolds and more general boundary conditions. The book begins with the study of Lopatinskii-Sapiro boundary conditions for the Laplace-Beltrami operator, followed by derivation of Carleman estimates for this operator on Riemannian manifolds. Applications of Carleman estimates are explored next: quantified unique continuation issues, a proof of the logarithmic stabilization of the boundary-damped wave equation, and a spectral inequality with general boundary conditions to derive the null-controllability result for the heat equation. Two additional chapters consider some more advanced results on Carleman estimates. The final part of the book is devoted to exposition of some necessary background material: elements of differential and Riemannian geometry, and Sobolev spaces and Laplace problems on Riemannian manifolds.
Book Synopsis Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume I by : Jérôme Le Rousseau
Download or read book Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume I written by Jérôme Le Rousseau and published by Springer Nature. This book was released on 2022-03-28 with total page 410 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph explores applications of Carleman estimates in the study of stabilization and controllability properties of partial differential equations, including the stabilization property of the damped wave equation and the null-controllability of the heat equation. All analysis is performed in the case of open sets in the Euclidean space; a second volume will extend this treatment to Riemannian manifolds. The first three chapters illustrate the derivation of Carleman estimates using pseudo-differential calculus with a large parameter. Continuation issues are then addressed, followed by a proof of the logarithmic stabilization of the damped wave equation by means of two alternative proofs of the resolvent estimate for the generator of a damped wave semigroup. The authors then discuss null-controllability of the heat equation, its equivalence with observability, and how the spectral inequality allows one to either construct a control function or prove the observability inequality. The final part of the book is devoted to the exposition of some necessary background material: the theory of distributions, invariance under change of variables, elliptic operators with Dirichlet data and associated semigroup, and some elements from functional analysis and semigroup theory.
Book Synopsis Exact Controllability and Stabilization by : V. Komornik
Download or read book Exact Controllability and Stabilization written by V. Komornik and published by Elsevier Masson. This book was released on 1994 with total page 172 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Global Carleman Estimates for Degenerate Parabolic Operators with Applications by : P. Cannarsa
Download or read book Global Carleman Estimates for Degenerate Parabolic Operators with Applications written by P. Cannarsa and published by American Mathematical Soc.. This book was released on 2016-01-25 with total page 225 pages. Available in PDF, EPUB and Kindle. Book excerpt: Degenerate parabolic operators have received increasing attention in recent years because they are associated with both important theoretical analysis, such as stochastic diffusion processes, and interesting applications to engineering, physics, biology, and economics. This manuscript has been conceived to introduce the reader to global Carleman estimates for a class of parabolic operators which may degenerate at the boundary of the space domain, in the normal direction to the boundary. Such a kind of degeneracy is relevant to study the invariance of a domain with respect to a given stochastic diffusion flow, and appears naturally in climatology models.
Book Synopsis Inverse Spectral Problems for Linear Differential Operators and Their Applications by : V A Yurko
Download or read book Inverse Spectral Problems for Linear Differential Operators and Their Applications written by V A Yurko and published by CRC Press. This book was released on 2000-01-18 with total page 272 pages. Available in PDF, EPUB and Kindle. Book excerpt: Aims to construct the inverse problem theory for ordinary non-self-adjoint differential operators of arbitary order on the half-line and on a finite interval. The book consists of two parts: in the first part the author presents a general inverse problem of recovering differential equations with integrable coefficients when the behaviour of the spectrum is arbitrary. The Weyl matrix is introduced and studied as a spectral characteristic. The second part of the book is devoted to solving incomplete inverse problems when a priori information about the operator or its spectrum is available and these problems are significant in applications.
Book Synopsis More Progresses In Analysis - Proceedings Of The 5th International Isaac Congress by : Heinrich G W Begehr
Download or read book More Progresses In Analysis - Proceedings Of The 5th International Isaac Congress written by Heinrich G W Begehr and published by World Scientific. This book was released on 2009-05-12 with total page 1497 pages. Available in PDF, EPUB and Kindle. Book excerpt: International ISAAC (International Society for Analysis, its Applications and Computation) Congresses have been held every second year since 1997. The proceedings report on a regular basis on the progresses of the field in recent years, where the most active areas in analysis, its applications and computation are covered. Plenary lectures also highlight recent results. This volume concentrates mainly on partial differential equations, but also includes function spaces, operator theory, integral transforms and equations, potential theory, complex analysis and generalizations, stochastic analysis, inverse problems, homogenization, continuum mechanics, mathematical biology and medicine. With over 350 participants attending the congress, the book comprises 140 papers from 211 authors.The volume also serves for transferring personal information about the ISAAC and its members. This volume includes citations for O Besov, V Burenkov and R P Gilbert on the occasion of their anniversaries.
Book Synopsis Geometric Methods in Inverse Problems and PDE Control by : Chrisopher B. Croke
Download or read book Geometric Methods in Inverse Problems and PDE Control written by Chrisopher B. Croke and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 334 pages. Available in PDF, EPUB and Kindle. Book excerpt: This IMA Volume in Mathematics and its Applications GEOMETRIC METHODS IN INVERSE PROBLEMS AND PDE CONTROL contains a selection of articles presented at 2001 IMA Summer Program with the same title. We would like to thank Christopher B. Croke (University of Penn sylva nia), Irena Lasiecka (University of Virginia), Gunther Uhlmann (University of Washington), and Michael S. Vogelius (Rutgers University) for their ex cellent work as organizers of the two-week summer workshop and for editing the volume. We also take this opportunity to thank the National Science Founda tion for their support of the IMA. Series Editors Douglas N. Arnold, Director of the IMA Fadil Santosa, Deputy Director of the IMA v PREFACE This volume contains a selected number of articles based on lectures delivered at the IMA 2001 Summer Program on "Geometric Methods in Inverse Problems and PDE Control. " The focus of this program was some common techniques used in the study of inverse coefficient problems and control problems for partial differential equations, with particular emphasis on their strong relation to fundamental problems of geometry. Inverse coef ficient problems for partial differential equations arise in many application areas, for instance in medical imaging, nondestructive testing, and geophys ical prospecting. Control problems involving partial differential equations may arise from the need to optimize a given performance criterion, e. g. , to dampen out undesirable vibrations of a structure , or more generally, to obtain a prescribed behaviour of the dynamics.
