Stable Methods for III-Posed Variational Problems

Stable Methods for III-Posed Variational Problems

Author: Alexander Kaplan

Publisher: Wiley-VCH

Published: 1994-09-13

Total Pages: 448

ISBN-13:

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Iterative prox-regularization methods for solving ill-posed convex variational problems in Hilbert spaces are subject of this book. A general framework is developed to analyse simultaneously procedures of regularization and successively refined discretization in connection with specific optimization methods for solving the discrete problems. This allows an efficient control of the solution process as a whole. In the first part of the book various methods for treating ill-posed problems are presented, including a study of the regularizing properties of a number of specific optimization algorithms. In the second part, a new class of multi-step methods is introduced which is based on a generalization of the iterative prox-regularization concept. Compared with former methods these new methods permit a more effective use of rough approximations of the infinite dimensional problems and consequently an acceleration of the numerical process. Special versions of these methods are given for ill-posed convex semi-infinite optimization problems and elliptic variational inequalities with weakly coercive operators, including some problems in elasticity theory.


Book Synopsis Stable Methods for III-Posed Variational Problems by : Alexander Kaplan

Download or read book Stable Methods for III-Posed Variational Problems written by Alexander Kaplan and published by Wiley-VCH. This book was released on 1994-09-13 with total page 448 pages. Available in PDF, EPUB and Kindle. Book excerpt: Iterative prox-regularization methods for solving ill-posed convex variational problems in Hilbert spaces are subject of this book. A general framework is developed to analyse simultaneously procedures of regularization and successively refined discretization in connection with specific optimization methods for solving the discrete problems. This allows an efficient control of the solution process as a whole. In the first part of the book various methods for treating ill-posed problems are presented, including a study of the regularizing properties of a number of specific optimization algorithms. In the second part, a new class of multi-step methods is introduced which is based on a generalization of the iterative prox-regularization concept. Compared with former methods these new methods permit a more effective use of rough approximations of the infinite dimensional problems and consequently an acceleration of the numerical process. Special versions of these methods are given for ill-posed convex semi-infinite optimization problems and elliptic variational inequalities with weakly coercive operators, including some problems in elasticity theory.

Stable Methods for III-Posed Variational Problems - Prox-Regularization of Elliptic Variational Inequalities

Stable Methods for III-Posed Variational Problems - Prox-Regularization of Elliptic Variational Inequalities

Author: Alexander Kaplan

Publisher:

Published: 1994-09-01

Total Pages: 438

ISBN-13: 9783527400423

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Book Synopsis Stable Methods for III-Posed Variational Problems - Prox-Regularization of Elliptic Variational Inequalities by : Alexander Kaplan

Download or read book Stable Methods for III-Posed Variational Problems - Prox-Regularization of Elliptic Variational Inequalities written by Alexander Kaplan and published by . This book was released on 1994-09-01 with total page 438 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Ill-posed Variational Problems and Regularization Techniques

Ill-posed Variational Problems and Regularization Techniques

Author: Michel Thera

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 281

ISBN-13: 3642457800

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This book presents recent developments in the field of ill-posed variational problems and variational inequalities, covering a large range of theoretical, numerical and practical aspects. The main topics are: - Regularization techniques for equilibrium and fixed point problems, variational inequalities and complementary problems, - Links between approximation, penalization and regularization, - Bundle methods, nonsmooth optimization and regularization, - Error Bounds for regularized optimization problems.


Book Synopsis Ill-posed Variational Problems and Regularization Techniques by : Michel Thera

Download or read book Ill-posed Variational Problems and Regularization Techniques written by Michel Thera and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 281 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents recent developments in the field of ill-posed variational problems and variational inequalities, covering a large range of theoretical, numerical and practical aspects. The main topics are: - Regularization techniques for equilibrium and fixed point problems, variational inequalities and complementary problems, - Links between approximation, penalization and regularization, - Bundle methods, nonsmooth optimization and regularization, - Error Bounds for regularized optimization problems.

Convergence Analysis of Proximal-like Methods for Variational Inequalities and Fixed Point Problems

Convergence Analysis of Proximal-like Methods for Variational Inequalities and Fixed Point Problems

Author: Nils Langenberg

Publisher: Logos Verlag Berlin GmbH

Published: 2011

Total Pages: 255

ISBN-13: 3832528903

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Several regularization methods for variational inequalities and fixed point problems are studied. Known convergence results especially require some kind of monotonicity of the problem data as well as, especially for Bregman-function-based algorithms, some additional assumption known as the cutting plane property. Unfortunately, these assumptions may be considered as rather restrictive e.g. in the framework of Nash equilibrium problems. This motivates the development of convergence results under weaker hypotheses which constitute the major subject of the present book. Studied methods include the Bregman-function-based Proximal Point Algorithm (BPPA), Cohen's Auxiliary Problem Principle and an extragradient algorithm.Moreover, this work also contains the first numerical comparison of stopping criteria in the framework of the BPPA. Although such conditions are the subject of theoretical investigations frequently, their numerical effectiveness and a deducible preference were still unknown. This gives rise to the necessity of the presented numerical experiments.


Book Synopsis Convergence Analysis of Proximal-like Methods for Variational Inequalities and Fixed Point Problems by : Nils Langenberg

Download or read book Convergence Analysis of Proximal-like Methods for Variational Inequalities and Fixed Point Problems written by Nils Langenberg and published by Logos Verlag Berlin GmbH. This book was released on 2011 with total page 255 pages. Available in PDF, EPUB and Kindle. Book excerpt: Several regularization methods for variational inequalities and fixed point problems are studied. Known convergence results especially require some kind of monotonicity of the problem data as well as, especially for Bregman-function-based algorithms, some additional assumption known as the cutting plane property. Unfortunately, these assumptions may be considered as rather restrictive e.g. in the framework of Nash equilibrium problems. This motivates the development of convergence results under weaker hypotheses which constitute the major subject of the present book. Studied methods include the Bregman-function-based Proximal Point Algorithm (BPPA), Cohen's Auxiliary Problem Principle and an extragradient algorithm.Moreover, this work also contains the first numerical comparison of stopping criteria in the framework of the BPPA. Although such conditions are the subject of theoretical investigations frequently, their numerical effectiveness and a deducible preference were still unknown. This gives rise to the necessity of the presented numerical experiments.

Encyclopedia of Optimization

Encyclopedia of Optimization

Author: Christodoulos A. Floudas

Publisher: Springer Science & Business Media

Published: 2008-09-04

Total Pages: 4646

ISBN-13: 0387747583

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The goal of the Encyclopedia of Optimization is to introduce the reader to a complete set of topics that show the spectrum of research, the richness of ideas, and the breadth of applications that has come from this field. The second edition builds on the success of the former edition with more than 150 completely new entries, designed to ensure that the reference addresses recent areas where optimization theories and techniques have advanced. Particularly heavy attention resulted in health science and transportation, with entries such as "Algorithms for Genomics", "Optimization and Radiotherapy Treatment Design", and "Crew Scheduling".


Book Synopsis Encyclopedia of Optimization by : Christodoulos A. Floudas

Download or read book Encyclopedia of Optimization written by Christodoulos A. Floudas and published by Springer Science & Business Media. This book was released on 2008-09-04 with total page 4646 pages. Available in PDF, EPUB and Kindle. Book excerpt: The goal of the Encyclopedia of Optimization is to introduce the reader to a complete set of topics that show the spectrum of research, the richness of ideas, and the breadth of applications that has come from this field. The second edition builds on the success of the former edition with more than 150 completely new entries, designed to ensure that the reference addresses recent areas where optimization theories and techniques have advanced. Particularly heavy attention resulted in health science and transportation, with entries such as "Algorithms for Genomics", "Optimization and Radiotherapy Treatment Design", and "Crew Scheduling".

Regularization Algorithms for Ill-Posed Problems

Regularization Algorithms for Ill-Posed Problems

Author: Anatoly B. Bakushinsky

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2018-02-05

Total Pages: 342

ISBN-13: 3110556383

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This specialized and authoritative book contains an overview of modern approaches to constructing approximations to solutions of ill-posed operator equations, both linear and nonlinear. These approximation schemes form a basis for implementable numerical algorithms for the stable solution of operator equations arising in contemporary mathematical modeling, and in particular when solving inverse problems of mathematical physics. The book presents in detail stable solution methods for ill-posed problems using the methodology of iterative regularization of classical iterative schemes and the techniques of finite dimensional and finite difference approximations of the problems under study. Special attention is paid to ill-posed Cauchy problems for linear operator differential equations and to ill-posed variational inequalities and optimization problems. The readers are expected to have basic knowledge in functional analysis and differential equations. The book will be of interest to applied mathematicians and specialists in mathematical modeling and inverse problems, and also to advanced students in these fields. Contents Introduction Regularization Methods For Linear Equations Finite Difference Methods Iterative Regularization Methods Finite-Dimensional Iterative Processes Variational Inequalities and Optimization Problems


Book Synopsis Regularization Algorithms for Ill-Posed Problems by : Anatoly B. Bakushinsky

Download or read book Regularization Algorithms for Ill-Posed Problems written by Anatoly B. Bakushinsky and published by Walter de Gruyter GmbH & Co KG. This book was released on 2018-02-05 with total page 342 pages. Available in PDF, EPUB and Kindle. Book excerpt: This specialized and authoritative book contains an overview of modern approaches to constructing approximations to solutions of ill-posed operator equations, both linear and nonlinear. These approximation schemes form a basis for implementable numerical algorithms for the stable solution of operator equations arising in contemporary mathematical modeling, and in particular when solving inverse problems of mathematical physics. The book presents in detail stable solution methods for ill-posed problems using the methodology of iterative regularization of classical iterative schemes and the techniques of finite dimensional and finite difference approximations of the problems under study. Special attention is paid to ill-posed Cauchy problems for linear operator differential equations and to ill-posed variational inequalities and optimization problems. The readers are expected to have basic knowledge in functional analysis and differential equations. The book will be of interest to applied mathematicians and specialists in mathematical modeling and inverse problems, and also to advanced students in these fields. Contents Introduction Regularization Methods For Linear Equations Finite Difference Methods Iterative Regularization Methods Finite-Dimensional Iterative Processes Variational Inequalities and Optimization Problems

Nonlinear Ill-posed Problems of Monotone Type

Nonlinear Ill-posed Problems of Monotone Type

Author: Yakov Alber

Publisher: Springer Science & Business Media

Published: 2006-02-23

Total Pages: 422

ISBN-13: 1402043961

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Interest in regularization methods for ill-posed nonlinear operator equations and variational inequalities of monotone type in Hilbert and Banach spaces has grown rapidly over recent years. Results in the field over the last three decades, previously only available in journal articles, are comprehensively explored with particular attention given to applications of regularization methods as well as to practical methods used in computational analysis.


Book Synopsis Nonlinear Ill-posed Problems of Monotone Type by : Yakov Alber

Download or read book Nonlinear Ill-posed Problems of Monotone Type written by Yakov Alber and published by Springer Science & Business Media. This book was released on 2006-02-23 with total page 422 pages. Available in PDF, EPUB and Kindle. Book excerpt: Interest in regularization methods for ill-posed nonlinear operator equations and variational inequalities of monotone type in Hilbert and Banach spaces has grown rapidly over recent years. Results in the field over the last three decades, previously only available in journal articles, are comprehensively explored with particular attention given to applications of regularization methods as well as to practical methods used in computational analysis.

Theory of Linear Ill-Posed Problems and its Applications

Theory of Linear Ill-Posed Problems and its Applications

Author: Valentin K. Ivanov

Publisher: Walter de Gruyter

Published: 2013-02-18

Total Pages: 296

ISBN-13: 3110944820

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This monograph is a revised and extended version of the Russian edition from 1978. It includes the general theory of linear ill-posed problems concerning e. g. the structure of sets of uniform regularization, the theory of error estimation, and the optimality method. As a distinguishing feature the book considers ill-posed problems not only in Hilbert but also in Banach spaces. It is natural that since the appearance of the first edition considerable progress has been made in the theory of inverse and ill-posed problems as wall as in ist applications. To reflect these accomplishments the authors included additional material e. g. comments to each chapter and a list of monographs with annotations.


Book Synopsis Theory of Linear Ill-Posed Problems and its Applications by : Valentin K. Ivanov

Download or read book Theory of Linear Ill-Posed Problems and its Applications written by Valentin K. Ivanov and published by Walter de Gruyter. This book was released on 2013-02-18 with total page 296 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph is a revised and extended version of the Russian edition from 1978. It includes the general theory of linear ill-posed problems concerning e. g. the structure of sets of uniform regularization, the theory of error estimation, and the optimality method. As a distinguishing feature the book considers ill-posed problems not only in Hilbert but also in Banach spaces. It is natural that since the appearance of the first edition considerable progress has been made in the theory of inverse and ill-posed problems as wall as in ist applications. To reflect these accomplishments the authors included additional material e. g. comments to each chapter and a list of monographs with annotations.

Recent Advances in Optimization

Recent Advances in Optimization

Author: Peter Gritzmann

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 388

ISBN-13: 364259073X

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This book presents recent theoretical and practical aspects in the field of optimization and convex analysis. The topics covered in this volume include: - Equilibrium models in economics. - Control theory and semi-infinite programming. - Ill-posed variational problems. - Global optimization. - Variational methods in image restoration. - Nonsmooth optimization. - Duality theory in convex and nonconvex optimization. - Methods for large scale problems.


Book Synopsis Recent Advances in Optimization by : Peter Gritzmann

Download or read book Recent Advances in Optimization written by Peter Gritzmann and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 388 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents recent theoretical and practical aspects in the field of optimization and convex analysis. The topics covered in this volume include: - Equilibrium models in economics. - Control theory and semi-infinite programming. - Ill-posed variational problems. - Global optimization. - Variational methods in image restoration. - Nonsmooth optimization. - Duality theory in convex and nonconvex optimization. - Methods for large scale problems.

From Convexity to Nonconvexity

From Convexity to Nonconvexity

Author: R.P. Gilbert

Publisher: Springer Science & Business Media

Published: 2013-12-01

Total Pages: 395

ISBN-13: 1461302870

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This collection of papers is dedicated to the memory of Gaetano Fichera, a great mathematician and also a good friend to the editors. Regrettably it took an unusual amount of time to bring this collection out. This was primarily due to the fact that the main editor who had collected all of the materials, for this volume, P. D. Panagiotopoulos, died unexpectedly during the period when we were editing the manuscript. The other two editors in appreciation of Panagiotopoulos' contribution to this field, believe it is therefore fitting that this collection be dedicated to his memory also. The theme of the collection is centered around the seminal research of G. Fichera on the Signorini problem. Variants on this idea enter in different ways. For example, by bringing in friction the problem is no longer self-adjoint and the minimization formulation is not valid. A large portion of this collection is devoted to survey papers concerning hemivariational methods, with a main point of its application to nonsmooth mechanics. Hemivariational inequali ties, which are a generalization of variational inequalities, were pioneered by Panagiotopoulos. There are many applications of this theory to the study of non convex energy functionals occurring in many branches of mechanics. An area of concentration concerns contact problems, in particular, quasistatic and dynamic contact problems with friction and damage. Nonsmooth optimization methods which may be divided into the main groups of subgradient methods and bundle methods are also discussed in this collection.


Book Synopsis From Convexity to Nonconvexity by : R.P. Gilbert

Download or read book From Convexity to Nonconvexity written by R.P. Gilbert and published by Springer Science & Business Media. This book was released on 2013-12-01 with total page 395 pages. Available in PDF, EPUB and Kindle. Book excerpt: This collection of papers is dedicated to the memory of Gaetano Fichera, a great mathematician and also a good friend to the editors. Regrettably it took an unusual amount of time to bring this collection out. This was primarily due to the fact that the main editor who had collected all of the materials, for this volume, P. D. Panagiotopoulos, died unexpectedly during the period when we were editing the manuscript. The other two editors in appreciation of Panagiotopoulos' contribution to this field, believe it is therefore fitting that this collection be dedicated to his memory also. The theme of the collection is centered around the seminal research of G. Fichera on the Signorini problem. Variants on this idea enter in different ways. For example, by bringing in friction the problem is no longer self-adjoint and the minimization formulation is not valid. A large portion of this collection is devoted to survey papers concerning hemivariational methods, with a main point of its application to nonsmooth mechanics. Hemivariational inequali ties, which are a generalization of variational inequalities, were pioneered by Panagiotopoulos. There are many applications of this theory to the study of non convex energy functionals occurring in many branches of mechanics. An area of concentration concerns contact problems, in particular, quasistatic and dynamic contact problems with friction and damage. Nonsmooth optimization methods which may be divided into the main groups of subgradient methods and bundle methods are also discussed in this collection.