Inverse Problems for Partial Differential Equations

Author: Victor Isakov

Publisher: Springer

ISBN: 3319516582

Category: Mathematics

Page: 408

View: 8494

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.
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Methods for Solving Inverse Problems in Mathematical Physics

Author: Global Express Ltd. Co.,Aleksey I. Prilepko,Dmitry G. Orlovsky,Igor A. Vasin

Publisher: CRC Press

ISBN: 9780824719876

Category: Mathematics

Page: 744

View: 6428

Developing an approach to the question of existence, uniqueness and stability of solutions, this work presents a systematic elaboration of the theory of inverse problems for all principal types of partial differential equations. It covers up-to-date methods of linear and nonlinear analysis, the theory of differential equations in Banach spaces, applications of functional analysis, and semigroup theory.
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An Introduction to the Mathematical Theory of Inverse Problems

Author: Andreas Kirsch

Publisher: Springer Science & Business Media

ISBN: 9781441984746

Category: Mathematics

Page: 310

View: 3400

This book introduces the reader to the area of inverse problems. The study of inverse problems is of vital interest to many areas of science and technology such as geophysical exploration, system identification, nondestructive testing and ultrasonic tomography. The aim of this book is twofold: in the first part, the reader is exposed to the basic notions and difficulties encountered with ill-posed problems. Basic properties of regularization methods for linear ill-posed problems are studied by means of several simple analytical and numerical examples. The second part of the book presents two special nonlinear inverse problems in detail - the inverse spectral problem and the inverse scattering problem. The corresponding direct problems are studied with respect to existence, uniqueness and continuous dependence on parameters. Then some theoretical results as well as numerical procedures for the inverse problems are discussed. The choice of material and its presentation in the book are new, thus making it particularly suitable for graduate students. Basic knowledge of real analysis is assumed. In this new edition, the Factorization Method is included as one of the prominent members in this monograph. Since the Factorization Method is particularly simple for the problem of EIT and this field has attracted a lot of attention during the past decade a chapter on EIT has been added in this monograph as Chapter 5 while the chapter on inverse scattering theory is now Chapter 6.The main changes of this second edition compared to the first edition concern only Chapters 5 and 6 and the Appendix A. Chapter 5 introduces the reader to the inverse problem of electrical impedance tomography.
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Inverse Problems in the Mathematical Sciences

Author: Charles W. Groetsch

Publisher: Springer Science & Business Media

ISBN: 3322992020

Category: Technology & Engineering

Page: 154

View: 4764

Inverse problems are immensely important in modern science and technology. However, the broad mathematical issues raised by inverse problems receive scant attention in the university curriculum. This book aims to remedy this state of affairs by supplying an accessible introduction, at a modest mathematical level, to the alluring field of inverse problems. Many models of inverse problems from science and engineering are dealt with and nearly a hundred exercises, of varying difficulty, involving mathematical analysis, numerical treatment, or modelling of inverse problems, are provided. The main themes of the book are: causation problem modeled as integral equations; model identification problems, posed as coefficient determination problems in differential equations; the functional analytic framework for inverse problems; and a survey of the principal numerical methods for inverse problems. An extensive annotated bibliography furnishes leads on the history of inverse problems and a guide to the frontiers of current research.
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Partial Differential Equations of Applied Mathematics

Author: Erich Zauderer

Publisher: Wiley-Interscience

ISBN: 9780471315162

Category: Mathematics

Page: 894

View: 1074

The only comprehensive guide to modeling, characterizing, and solving partial differential equations This classic text by Erich Zauderer provides a comprehensive account of partial differential equations and their applications. Dr. Zauderer develops mathematical models that give rise to partial differential equations and describes classical and modern solution techniques. With an emphasis on practical applications, he makes liberal use of real-world examples, explores both linear and nonlinear problems, and provides approximate as well as exact solutions. He also describes approximation methods for simplifying complicated solutions and for solving linear and nonlinear problems not readily solved by standard methods. The book begins with a demonstration of how the three basic types of equations (parabolic, hyperbolic, and elliptic) can be derived from random walk models. It continues in a less statistical vein to cover an exceptionally broad range of topics, including stabilities, singularities, transform methods, the use of Green's functions, and perturbation and asymptotic treatments. Features that set Partial Differential Equations of Applied Mathematics, Second Edition above all other texts in the field include: Coverage of random walk problems, discontinuous and singular solutions, and perturbation and asymptotic methods More than 800 practice exercises, many of which are fully worked out Numerous up-to-date examples from engineering and the physical sciences Partial Differential Equations of Applied Mathematics, Second Edition is a superior advanced-undergraduate to graduate-level text for students in engineering, the sciences, and applied mathematics. The title is also a valuable working resource for professionals in these fields. Dr. Zauderer received his doctorate in mathematics from the New York University-Courant Institute. Prior to joining the staff of Polytechnic University, he was a Senior Weitzmann Fellow of the Weitzmann Institute of Science in Rehovot, Israel.
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Variational Methods in Imaging

Author: Otmar Scherzer,Markus Grasmair,Harald Grossauer,Markus Haltmeier,Frank Lenzen

Publisher: Springer Science & Business Media

ISBN: 0387692770

Category: Mathematics

Page: 320

View: 9271

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.
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An Invitation to Applied Mathematics

Differential Equations, Modeling, and Computation

Author: Carmen Chicone

Publisher: Academic Press

ISBN: 0128041544

Category: Mathematics

Page: 878

View: 7718

An Invitation to Applied Mathematics: Differential Equations, Modeling, and Computation introduces the reader to the methodology of modern applied mathematics in modeling, analysis, and scientific computing with emphasis on the use of ordinary and partial differential equations. Each topic is introduced with an attractive physical problem, where a mathematical model is constructed using physical and constitutive laws arising from the conservation of mass, conservation of momentum, or Maxwell's electrodynamics. Relevant mathematical analysis (which might employ vector calculus, Fourier series, nonlinear ODEs, bifurcation theory, perturbation theory, potential theory, control theory, or probability theory) or scientific computing (which might include Newton's method, the method of lines, finite differences, finite elements, finite volumes, boundary elements, projection methods, smoothed particle hydrodynamics, or Lagrangian methods) is developed in context and used to make physically significant predictions. The target audience is advanced undergraduates (who have at least a working knowledge of vector calculus and linear ordinary differential equations) or beginning graduate students. Readers will gain a solid and exciting introduction to modeling, mathematical analysis, and computation that provides the key ideas and skills needed to enter the wider world of modern applied mathematics. Presents an integrated wealth of modeling, analysis, and numerical methods in one volume Provides practical and comprehensible introductions to complex subjects, for example, conservation laws, CFD, SPH, BEM, and FEM Includes a rich set of applications, with more appealing problems and projects suggested
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Differential Equations

Inverse and Direct Problems

Author: Angelo Favini,Alfredo Lorenzi

Publisher: CRC Press

ISBN: 9781420011135

Category: Mathematics

Page: 304

View: 8720

With contributions from some of the leading authorities in the field, the work in Differential Equations: Inverse and Direct Problems stimulates the preparation of new research results and offers exciting possibilities not only in the future of mathematics but also in physics, engineering, superconductivity in special materials, and other scientific fields. Exploring the hypotheses and numerical approaches that relate to pure and applied mathematics, this collection of research papers and surveys extends the theories and methods of differential equations. The book begins with discussions on Banach spaces, linear and nonlinear theory of semigroups, integrodifferential equations, the physical interpretation of general Wentzell boundary conditions, and unconditional martingale difference (UMD) spaces. It then proceeds to deal with models in superconductivity, hyperbolic partial differential equations (PDEs), blowup of solutions, reaction-diffusion equation with memory, and Navier-Stokes equations. The volume concludes with analyses on Fourier-Laplace multipliers, gradient estimates for Dirichlet parabolic problems, a nonlinear system of PDEs, and the complex Ginzburg-Landau equation. By combining direct and inverse problems into one book, this compilation is a useful reference for those working in the world of pure or applied mathematics.
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Linear Inverse Problems and Tikhonov Regularization

Author: Mark Gockenbach

Publisher: The Mathematical Association of America

ISBN: 0883851415

Category: Mathematics

Page: 333

View: 8009

Inverse problems occur frequently in science and technology, whenever we need to infer causes from effects that we can measure. Mathematically, they are difficult problems because they are unstable: small bits of noise in the measurement can completely throw off the solution. Nevertheless, there are methods for finding good approximate solutions. Linear Inverse Problems and Tikhonov Regularization examines one such method: Tikhonov regularization for linear inverse problems defined on Hilbert spaces. This is a clear example of the power of applying deep mathematical theory to solve practical problems. Beginning with a basic analysis of Tikhonov regularization, this book introduces the singular value expansion for compact operators, and uses it to explain why and how the method works. Tikhonov regularization with seminorms is also analyzed, which requires introducing densely defined unbounded operators and their basic properties. Some of the relevant background is included in appendices, making the book accessible to a wide range of readers.
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Introduction to Inverse Problems for Differential Equations

Author: Alemdar Hasanov Hasanoğlu,Vladimir G. Romanov

Publisher: Springer

ISBN: 331962797X

Category: Mathematics

Page: 261

View: 906

This book presents a systematic exposition of the main ideas and methods in treating inverse problems for PDEs arising in basic mathematical models, though it makes no claim to being exhaustive. Mathematical models of most physical phenomena are governed by initial and boundary value problems for PDEs, and inverse problems governed by these equations arise naturally in nearly all branches of science and engineering. The book’s content, especially in the Introduction and Part I, is self-contained and is intended to also be accessible for beginning graduate students, whose mathematical background includes only basic courses in advanced calculus, PDEs and functional analysis. Further, the book can be used as the backbone for a lecture course on inverse and ill-posed problems for partial differential equations. In turn, the second part of the book consists of six nearly-independent chapters. The choice of these chapters was motivated by the fact that the inverse coefficient and source problems considered here are based on the basic and commonly used mathematical models governed by PDEs. These chapters describe not only these inverse problems, but also main inversion methods and techniques. Since the most distinctive features of any inverse problems related to PDEs are hidden in the properties of the corresponding solutions to direct problems, special attention is paid to the investigation of these properties.
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Linear Integral Equations

Author: Rainer Kress

Publisher: Springer Science & Business Media

ISBN: 1461495938

Category: Mathematics

Page: 412

View: 9404

This book combines theory, applications, and numerical methods, and covers each of these fields with the same weight. In order to make the book accessible to mathematicians, physicists, and engineers alike, the author has made it as self-contained as possible, requiring only a solid foundation in differential and integral calculus. The functional analysis which is necessary for an adequate treatment of the theory and the numerical solution of integral equations is developed within the book itself. Problems are included at the end of each chapter. For this third edition in order to make the introduction to the basic functional analytic tools more complete the Hahn–Banach extension theorem and the Banach open mapping theorem are now included in the text. The treatment of boundary value problems in potential theory has been extended by a more complete discussion of integral equations of the first kind in the classical Holder space setting and of both integral equations of the first and second kind in the contemporary Sobolev space setting. In the numerical solution part of the book, the author included a new collocation method for two-dimensional hypersingular boundary integral equations and a collocation method for the three-dimensional Lippmann-Schwinger equation. The final chapter of the book on inverse boundary value problems for the Laplace equation has been largely rewritten with special attention to the trilogy of decomposition, iterative and sampling methods Reviews of earlier editions: "This book is an excellent introductory text for students, scientists, and engineers who want to learn the basic theory of linear integral equations and their numerical solution." (Math. Reviews, 2000) "This is a good introductory text book on linear integral equations. It contains almost all the topics necessary for a student. The presentation of the subject matter is lucid, clear and in the proper modern framework without being too abstract." (ZbMath, 1999)
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Inverse Problem Theory and Methods for Model Parameter Estimation

Author: Albert Tarantola

Publisher: SIAM

ISBN: 9780898717921

Category: Engineering

Page: 342

View: 5675

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.
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Partial Differential Equations

Modelling and Numerical Simulation

Author: Roland Glowinski,Pekka Neittaanmäki

Publisher: Springer Science & Business Media

ISBN: 1402087586

Category: Science

Page: 292

View: 9841

For more than 250 years partial di?erential equations have been clearly the most important tool available to mankind in order to understand a large variety of phenomena, natural at ?rst and then those originating from - man activity and technological development. Mechanics, physics and their engineering applications were the ?rst to bene?t from the impact of partial di?erential equations on modeling and design, but a little less than a century ago the Schr ̈ odinger equation was the key opening the door to the application of partial di?erential equations to quantum chemistry, for small atomic and molecular systems at ?rst, but then for systems of fast growing complexity. The place of partial di?erential equations in mathematics is a very particular one: initially, the partial di?erential equations modeling natural phenomena were derived by combining calculus with physical reasoning in order to - press conservation laws and principles in partial di?erential equation form, leading to the wave equation, the heat equation, the equations of elasticity, the Euler and Navier–Stokes equations for ?uids, the Maxwell equations of electro-magnetics, etc. It is in order to solve ‘constructively’ the heat equation that Fourier developed the series bearing his name in the early 19th century; Fourier series (and later integrals) have played (and still play) a fundamental roleinbothpureandappliedmathematics,includingmanyareasquiteremote from partial di?erential equations. On the other hand, several areas of mathematics such as di?erential ge- etry have bene?ted from their interactions with partial di?erential equations.
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Handbook of Linear Partial Differential Equations for Engineers and Scientists, Second Edition

Author: Andrei D. Polyanin,Vladimir E. Nazaikinskii

Publisher: CRC Press

ISBN: 1466581492

Category: Mathematics

Page: 1609

View: 7609

Includes nearly 4,000 linear partial differential equations (PDEs) with solutions Presents solutions of numerous problems relevant to heat and mass transfer, wave theory, hydrodynamics, aerodynamics, elasticity, acoustics, electrodynamics, diffraction theory, quantum mechanics, chemical engineering sciences, electrical engineering, and other fields Outlines basic methods for solving various problems in science and engineering Contains much more linear equations, problems, and solutions than any other book currently available Provides a database of test problems for numerical and approximate analytical methods for solving linear PDEs and systems of coupled PDEs New to the Second Edition More than 700 pages with 1,500+ new first-, second-, third-, fourth-, and higher-order linear equations with solutions Systems of coupled PDEs with solutions Some analytical methods, including decomposition methods and their applications Symbolic and numerical methods for solving linear PDEs with Maple, Mathematica, and MATLAB® Many new problems, illustrative examples, tables, and figures To accommodate different mathematical backgrounds, the authors avoid wherever possible the use of special terminology, outline some of the methods in a schematic, simplified manner, and arrange the material in increasing order of complexity.
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Partial Differential Equations

An Introduction

Author: David Colton

Publisher: Courier Corporation

ISBN: 0486438341

Category: Mathematics

Page: 308

View: 6398

This text offers students in mathematics, engineering, and the applied sciences a solid foundation for advanced studies in mathematics. Classical topics presented in a modern context include coverage of integral equations and basic scattering theory. Includes examples of inverse problems arising from improperly posed applications as well as exercises, many with answers. 1988 edition.
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Inverse Boundary Spectral Problems

Author: Alexander Kachalov,Yaroslav Kurylev,Matti Lassas

Publisher: CRC Press

ISBN: 142003622X

Category: Mathematics

Page: 260

View: 8107

Inverse boundary problems are a rapidly developing area of applied mathematics with applications throughout physics and the engineering sciences. However, the mathematical theory of inverse problems remains incomplete and needs further development to aid in the solution of many important practical problems. Inverse Boundary Spectral Problems develop a rigorous theory for solving several types of inverse problems exactly. In it, the authors consider the following: "Can the unknown coefficients of an elliptic partial differential equation be determined from the eigenvalues and the boundary values of the eigenfunctions?" Along with this problem, many inverse problems for heat and wave equations are solved. The authors approach inverse problems in a coordinate invariant way, that is, by applying ideas drawn from differential geometry. To solve them, they apply methods of Riemannian geometry, modern control theory, and the theory of localized wave packets, also known as Gaussian beams. The treatment includes the relevant background of each of these areas. Although the theory of inverse boundary spectral problems has been in development for at least 10 years, until now the literature has been scattered throughout various journals. This self-contained monograph summarizes the relevant concepts and the techniques useful for dealing with them.
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Real-time PDE-constrained Optimization

Author: Lorenz T. Biegler,Omar Ghattas,Matthias Heinkenschloss,David Keyes,Bart van Bloemen Waanders

Publisher: SIAM

ISBN: 9780898718935

Category: Differential equations, Partial

Page: 312

View: 1460

Many engineering and scientific problems in design, control, and parameter estimation can be formulated as optimization problems that are governed by partial differential equations (PDEs). The complexities of the PDEs--and the requirement for rapid solution--pose significant difficulties. A particularly challenging class of PDE-constrained optimization problems is characterized by the need for real-time solution, i.e., in time scales that are sufficiently rapid to support simulation-based decision making. Real-Time PDE-Constrained Optimization, the first book devoted to real-time optimization for systems governed by PDEs, focuses on new formulations, methods, and algorithms needed to facilitate real-time, PDE-constrained optimization. In addition to presenting state-of-the-art algorithms and formulations, the text illustrates these algorithms with a diverse set of applications that includes problems in the areas of aerodynamics, biology, fluid dynamics, medicine, chemical processes, homeland security, and structural dynamics. Audience: readers who have expertise in simulation and are interested in incorporating optimization into their simulations, who have expertise in numerical optimization and are interested in adapting optimization methods to the class of infinite-dimensional simulation problems, or who have worked in "offline" optimization contexts and are interested in moving to "online" optimization.
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Mathematical Methods for Wave Phenomena

Author: Norman Bleistein

Publisher: Academic Press

ISBN: 0080916953

Category: Mathematics

Page: 341

View: 2278

Computer Science and Applied Mathematics: Mathematical Methods for Wave Phenomena focuses on the methods of applied mathematics, including equations, wave fronts, boundary value problems, and scattering problems. The publication initially ponders on first-order partial differential equations, Dirac delta function, Fourier transforms, asymptotics, and second-order partial differential equations. Discussions focus on prototype second-order equations, asymptotic expansions, asymptotic expansions of Fourier integrals with monotonic phase, method of stationary phase, propagation of wave fronts, and variable index of refraction. The text then examines wave equation in one space dimension, as well as initial boundary value problems, characteristics for the wave equation in one space dimension, and asymptotic solution of the Klein-Gordon equation. The manuscript offers information on wave equation in two and three dimensions and Helmholtz equation and other elliptic equations. Topics include energy integral, domain of dependence, and uniqueness, scattering problems, Green's functions, and problems in unbounded domains and the Sommerfeld radiation condition. The asymptotic techniques for direct scattering problems and the inverse methods for reflector imaging are also elaborated. The text is a dependable reference for computer science experts and mathematicians pursuing studies on the mathematical methods of wave phenomena.
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Inverse Problems in Ordinary Differential Equations and Applications

Author: Jaume Llibre,Rafael Ramírez

Publisher: Birkhäuser

ISBN: 3319263390

Category: Mathematics

Page: 266

View: 2769

This book is dedicated to study the inverse problem of ordinary differential equations, that is it focuses in finding all ordinary differential equations that satisfy a given set of properties. The Nambu bracket is the central tool in developing this approach. The authors start characterizing the ordinary differential equations in R^N which have a given set of partial integrals or first integrals. The results obtained are applied first to planar polynomial differential systems with a given set of such integrals, second to solve the 16th Hilbert problem restricted to generic algebraic limit cycles, third for solving the inverse problem for constrained Lagrangian and Hamiltonian mechanical systems, fourth for studying the integrability of a constrained rigid body. Finally the authors conclude with an analysis on nonholonomic mechanics, a generalization of the Hamiltonian principle, and the statement an solution of the inverse problem in vakonomic mechanics.
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