Download or read book Generalized Solutions of Operator Equations and Extreme Elements written by D.A. Klyushin and published by Springer Science & Business Media. This book was released on 2011-10-05 with total page 219 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract models for many problems in science and engineering take the form of an operator equation. The resolution of these problems often requires determining the existence and uniqueness of solutions to these equations. "Generalized Solutions of Operator Equations and Extreme Elements" presents recently obtained results in the study of the generalized solutions of operator equations and extreme elements in linear topological spaces. The presented results offer new methods of identifying these solutions and studying their properties. These new methods involve the application of a priori estimations and a general topological approach to construct generalized solutions of linear and nonlinear operator equations. The monograph is intended for mathematicians, graduate students and researchers studying functional analysis, operator theory, and the theory of optimal control.

Download or read book Generalized Difference Methods for Differential Equations written by Ronghua Li and published by CRC Press. This book was released on 2000-01-03 with total page 472 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text presents a comprehensive mathematical theory for elliptic, parabolic, and hyperbolic differential equations. It compares finite element and finite difference methods and illustrates applications of generalized difference methods to elastic bodies, electromagnetic fields, underground water pollution, and coupled sound-heat flows.

Download or read book Well Set Variational Problems and Generalized Solutions for Multivalued Operator Equations written by George Dincă and published by . This book was released on 1987 with total page 17 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Approximate Solution of Operator Equations written by Mark Aleksandrovich Krasnoselʹskiĭ and published by . This book was released on 1969 with total page 484 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book KWIC Index for Numerical Algebra written by Alston Scott Householder and published by . This book was released on 1972 with total page 552 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Positive Solutions of Operator Equations written by Mark Aleksandrovich Krasnoselʹskiĭ and published by . This book was released on 1964 with total page 392 pages. Available in PDF, EPUB and Kindle. Book excerpt: "This book is devoted to a systematic study of an important aspect of non-linear functional analysis. The methods developed are for the study of equations containing essential non-linearities and in particular, of equations which can have many solutions and have found various applications to problems in wave theory, loss of stability of elastic systems, problems of geometry in the large, theory of periodic solutions of equations of non-linear mechanics, theory of non-linear boundary value problems and others" - annotation.

Download or read book Harmonic and Applied Analysis written by Filippo De Mari and published by Springer Nature. This book was released on 2021-12-13 with total page 316 pages. Available in PDF, EPUB and Kindle. Book excerpt: Deep connections exist between harmonic and applied analysis and the diverse yet connected topics of machine learning, data analysis, and imaging science. This volume explores these rapidly growing areas and features contributions presented at the second and third editions of the Summer Schools on Applied Harmonic Analysis, held at the University of Genova in 2017 and 2019. Each chapter offers an introduction to essential material and then demonstrates connections to more advanced research, with the aim of providing an accessible entrance for students and researchers. Topics covered include ill-posed problems; concentration inequalities; regularization and large-scale machine learning; unitarization of the radon transform on symmetric spaces; and proximal gradient methods for machine learning and imaging.

Download or read book Matrix and Operator Equations and Applications written by Mohammad Sal Moslehian and published by Springer Nature. This book was released on 2023-07-29 with total page 763 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book concerns matrix and operator equations that are widely applied in various disciplines of science to formulate challenging problems and solve them in a faithful way. The main aim of this contributed book is to study several important matrix and operator equalities and equations in a systematic and self-contained fashion. Some powerful methods have been used to investigate some significant equations in functional analysis, operator theory, matrix analysis, and numerous subjects in the last decades. The book is divided into two parts: (I) Matrix Equations and (II) Operator Equations. In the first part, the state-of-the-art of systems of matrix equations is given and generalized inverses are used to find their solutions. The semi-tensor product of matrices is used to solve quaternion matrix equations. The contents of some chapters are related to the relationship between matrix inequalities, matrix means, numerical range, and matrix equations. In addition, quaternion algebras and their applications are employed in solving some famous matrix equations like Sylvester, Stein, and Lyapunov equations. A chapter devoted to studying Hermitian polynomial matrix equations, which frequently arise from linear-quadratic control problems. Moreover, some classical and recently discovered inequalities for matrix exponentials are reviewed. In the second part, the latest developments in solving several equations appearing in modern operator theory are demonstrated. These are of interest to a wide audience of pure and applied mathematicians. For example, the Daugavet equation in the linear and nonlinear setting, iterative processes and Volterra-Fredholm integral equations, semicircular elements induced by connected finite graphs, free probability, singular integral operators with shifts, and operator differential equations closely related to the properties of the coefficient operators in some equations are discussed. The chapters give a comprehensive account of their subjects. The exhibited chapters are written in a reader-friendly style and can be read independently. Each chapter contains a rich bibliography. This book is intended for use by both researchers and graduate students of mathematics, physics, and engineering.

Download or read book Computational Electromagnetism written by Alain Bossavit and published by Academic Press. This book was released on 1998-02-04 with total page 375 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computational Electromagnetism refers to the modern concept of computer-aided analysis, and design, of virtually all electric devices such as motors, machines, transformers, etc., as well as of the equipment inthe currently booming field of telecommunications, such as antennas, radars, etc. The present book is uniquely written to enable the reader-- be it a student, a scientist, or a practitioner-- to successfully perform important simulation techniques and to design efficient computer software for electromagnetic device analysis. Numerous illustrations, solved exercises, original ideas, and an extensive and up-to-date bibliography make it a valuable reference for both experts and beginners in the field. A researcher and practitioner will find in it information rarely available in other sources, such as on symmetry, bilateral error bounds by complimentarity, edge and face elements, treatment of infinite domains, etc. At the same time, the book is a useful teaching tool for courses in computational techniques in certain fields of physics and electrical engineering. As a self-contained text, it presents an extensive coverage of the most important concepts from Maxwells equations to computer-solvable algebraic systems-- for both static, quasi-static, and harmonic high-frequency problems. Benefits To the Engineer A sound background necessary not only to understand the principles behind variational methods and finite elements, but also to design pertinent and well-structured software. To the Specialist in Numerical Modeling The book offers new perspectives of practical importance on classical issues: the underlying symmetry of Maxwell equations, their interaction with other fields of physics in real-life modeling, the benefits of edge and face elements, approaches to error analysis, and "complementarity." To the Teacher An expository strategy that will allow you to guide the student along a safe and easy route through otherwise difficult concepts: weak formulations and their relation to fundamental conservation principles of physics, functional spaces, Hilbert spaces, approximation principles, finite elements, and algorithms for solving linear systems. At a higher level, the book provides a concise and self-contained introduction to edge elements and their application to mathematical modeling of the basic electromagnetic phenomena, and static problems, such as eddy-current problems and microwaves in cavities. To the Student Solved exercises, with "hint" and "full solution" sections, will both test and enhance the understanding of the material. Numerous illustrations will help in grasping difficult mathematical concepts.

Download or read book Generalization of a Method for the Series Solution of Nonlinear Operator Equations written by Andrew Maxwell Olson and published by . This book was released on 1969 with total page 394 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Homotopy Analysis Method in Nonlinear Differential Equations written by Shijun Liao and published by Springer Science & Business Media. This book was released on 2012-06-22 with total page 566 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Homotopy Analysis Method in Nonlinear Differential Equations" presents the latest developments and applications of the analytic approximation method for highly nonlinear problems, namely the homotopy analysis method (HAM). Unlike perturbation methods, the HAM has nothing to do with small/large physical parameters. In addition, it provides great freedom to choose the equation-type of linear sub-problems and the base functions of a solution. Above all, it provides a convenient way to guarantee the convergence of a solution. This book consists of three parts. Part I provides its basic ideas and theoretical development. Part II presents the HAM-based Mathematica package BVPh 1.0 for nonlinear boundary-value problems and its applications. Part III shows the validity of the HAM for nonlinear PDEs, such as the American put option and resonance criterion of nonlinear travelling waves. New solutions to a number of nonlinear problems are presented, illustrating the originality of the HAM. Mathematica codes are freely available online to make it easy for readers to understand and use the HAM. This book is suitable for researchers and postgraduates in applied mathematics, physics, nonlinear mechanics, finance and engineering. Dr. Shijun Liao, a distinguished professor of Shanghai Jiao Tong University, is a pioneer of the HAM.

Download or read book Applied Mechanics Reviews written by and published by . This book was released on 1971 with total page 776 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Differential Equations An Introduction To Basic Concepts Results And Applications Third Edition written by Ioan I Vrabie and published by World Scientific Publishing Company. This book was released on 2016-05-30 with total page 529 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents, in a unitary frame and from a new perspective, the main concepts and results of one of the most fascinating branches of modern mathematics, namely differential equations, and offers the reader another point of view concerning a possible way to approach the problems of existence, uniqueness, approximation, and continuation of the solutions to a Cauchy problem. In addition, it contains simple introductions to some topics which are not usually included in classical textbooks: the exponential formula, conservation laws, generalized solutions, Caratheodory solutions, differential inclusions, variational inequalities, viability, invariance, and gradient systems.In this new edition, some typos have been corrected and two new topics have been added: Delay differential equations and differential equations subjected to nonlocal initial conditions. The bibliography has also been updated and expanded.

Download or read book Elementary Operator Theory written by Marat V. Markin and published by Walter de Gruyter GmbH & Co KG. This book was released on 2020-04-06 with total page 425 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book is intended as a text for a one-semester graduate course in operator theory to be taught "from scratch'', not as a sequel to a functional analysis course, with the basics of the spectral theory of linear operators taking the center stage. The book consists of six chapters and appendix, with the material flowing from the fundamentals of abstract spaces (metric, vector, normed vector, and inner product), the Banach Fixed-Point Theorem and its applications, such as Picard's Existence and Uniqueness Theorem, through the basics of linear operators, two of the three fundamental principles (the Uniform Boundedness Principle and the Open Mapping Theorem and its equivalents: the Inverse Mapping and Closed Graph Theorems), to the elements of the spectral theory, including Gelfand's Spectral Radius Theorem and the Spectral Theorem for Compact Self-Adjoint Operators, and its applications, such as the celebrated Lyapunov Stability Theorem. Conceived as a text to be used in a classroom, the book constantly calls for the student's actively mastering the knowledge of the subject matter. There are problems at the end of each chapter, starting with Chapter 2 and totaling at 150. Many important statements are given as problems and frequently referred to in the main body. There are also 432 Exercises throughout the text, including Chapter 1 and the Appendix, which require of the student to prove or verify a statement or an example, fill in certain details in a proof, or provide an intermediate step or a counterexample. They are also an inherent part of the material. More difficult problems are marked with an asterisk, many problems and exercises are supplied with "existential'' hints. The book is generous on Examples and contains numerous Remarks accompanying definitions, examples, and statements to discuss certain subtleties, raise questions on whether the converse assertions are true, whenever appropriate, or whether the conditions are essential. With carefully chosen material, proper attention given to applications, and plenty of examples, problems, and exercises, this well-designed text is ideal for a one-semester Master's level graduate course in operator theory with emphasis on spectral theory for students majoring in mathematics, physics, computer science, and engineering. Contents Preface Preliminaries Metric Spaces Vector Spaces, Normed Vector Spaces, and Banach Spaces Linear Operators Elements of Spectral Theory in a Banach Space Setting Elements of Spectral Theory in a Hilbert Space Setting Appendix: The Axiom of Choice and Equivalents Bibliography Index

Download or read book Optimization and Regularization for Computational Inverse Problems and Applications written by Yanfei Wang and published by Springer Science & Business Media. This book was released on 2011-06-29 with total page 354 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Optimization and Regularization for Computational Inverse Problems and Applications" focuses on advances in inversion theory and recent developments with practical applications, particularly emphasizing the combination of optimization and regularization for solving inverse problems. This book covers both the methods, including standard regularization theory, Fejer processes for linear and nonlinear problems, the balancing principle, extrapolated regularization, nonstandard regularization, nonlinear gradient method, the nonmonotone gradient method, subspace method and Lie group method; and the practical applications, such as the reconstruction problem for inverse scattering, molecular spectra data processing, quantitative remote sensing inversion, seismic inversion using the Lie group method, and the gravitational lensing problem. Scientists, researchers and engineers, as well as graduate students engaged in applied mathematics, engineering, geophysics, medical science, image processing, remote sensing and atmospheric science will benefit from this book. Dr. Yanfei Wang is a Professor at the Institute of Geology and Geophysics, Chinese Academy of Sciences, China. Dr. Sc. Anatoly G. Yagola is a Professor and Assistant Dean of the Physical Faculty, Lomonosov Moscow State University, Russia. Dr. Changchun Yang is a Professor and Vice Director of the Institute of Geology and Geophysics, Chinese Academy of Sciences, China.

Download or read book What is the Complexity of Solution restricted Operator Equations written by Columbia University. Dept. of Computer Science and published by . This book was released on 1995 with total page 94 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: "We study the worst case complexity of operator equations Lu = f, where L: G -> X is a bounded linear injection of normed linear spaces. Past work on the complexity of such problems has generally assumed that the class F of problem elements f is specified in advance, typically as the unit ball of X. However, there are many problems for which this not good enough a priori knowledge of F. Mixed elliptic- hyperbolic problems are one example, the difficulty being that we do not have strong enough technical tools at our disposal, so that we are unable to find good bounds on the complexity. Ill-posed problems are another example, because we know that the complexity of computing finite-error approximations is infinite if F is a ball in X. In this paper, we pursue another idea. Rather than restrict the class F of problem elements f, we will consider problems that are solution-restricted, i.e., we restrict the class U of solution elements u. In particular, we assume that U is the unit ball of a normed linear space W continuously embedded in G. First, we consider solution-restricted problems in full generality. We show how optimal information is determined by subspaces that are optimal with respect to Gelfand widths. Using this result, we can characterize convergent problems (i.e., those such that for any [epsilon]> 0, there exists an algorithm whose cost is finite and whose error is at most [epsilon]. We find that a solution-restricted problem is convergent iff the embedding of W into G is compact. Note that this characterization of convergent problems is independent of L; it applies equally to well-posed and ill-posed problems. Next, we restrict our attention to the Hilbert case, in which G, W, and X are all Hilbert spaces. We show that linear algorithms are optimal (or nearly-optimal) in the Hilbert case, and give a more precise characterization of optimal information. Then, we consider specific applications. The first application we consider is any problem for which G and W are standard Sobolev Hilbert spaces; we call this the 'standard problem' since it includes many problems of practical interest. We show that finite element information and generalized Galerkin methods are nearly-optimal for standard problems. We then look at elliptic boundary-value problems, Fredholm integral equations of the second kind, the Tricomi problem (a mixed hyperbolic-elliptic problem arising in the study of transonic flow), the inverse finite Laplace transform, and the backwards heat equation. (Note that with the exception of the backwards heat equation, all of these are standard problems. Moreover, the inverse finite Laplace transform and the backwards heat equation are ill-posed problems.) We determine the problem complexity and derive nearly-optimal algorithms for all these problems."

Download or read book Large Scale Inverse Problems written by Mike Cullen and published by Walter de Gruyter. This book was released on 2013-08-29 with total page 216 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is thesecond volume of a three volume series recording the "Radon Special Semester 2011 on Multiscale Simulation & Analysis in Energy and the Environment" that took placein Linz, Austria, October 3-7, 2011. This volume addresses the common ground in the mathematical and computational procedures required for large-scale inverse problems and data assimilation in forefront applications. The solution of inverse problems is fundamental to a wide variety of applications such as weather forecasting, medical tomography, and oil exploration. Regularisation techniques are needed to ensure solutions of sufficient quality to be useful, and soundly theoretically based. This book addresses the common techniques required for all the applications, and is thus truly interdisciplinary. Thiscollection of surveyarticlesfocusses onthe large inverse problems commonly arising in simulation and forecasting in the earth sciences. For example, operational weather forecasting models have between 107 and 108 degrees of freedom. Even so, these degrees of freedom represent grossly space-time averaged properties of the atmosphere. Accurate forecasts require accurate initial conditions. With recent developments in satellite data, there are between 106 and 107 observations each day. However, while these also represent space-time averaged properties, the averaging implicit in the measurements is quite different from that used in the models. In atmosphere and ocean applications, there is a physically-based model available which can be used to regularise the problem. We assume that there is a set of observations with known error characteristics available over a period of time. The basic deterministic technique is to fit a model trajectory to the observations over a period of time to within the observation error. Since the model is not perfect the model trajectory has to be corrected, which defines the data assimilation problem. The stochastic view can be expressed by using an ensemble of model trajectories, and calculating corrections to both the mean value and the spread which allow the observations to be fitted by each ensemble member. In other areas of earth science, only the structure of the model formulation itself is known and the aim is to use the past observation history to determine the unknown model parameters. The book records the achievements of Workshop2 "Large-Scale Inverse Problems and Applications in the Earth Sciences". Itinvolves experts in the theory of inverse problems together with experts working on both theoretical and practical aspects of the techniques by which large inverse problems arise in the earth sciences.