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Implicit Functions and Solution Mappings: A View from Variational Analysis de Asen L. Dontchev

Implicit Functions and Solution Mappings: A View from Variational Analysis: Springer Series in Operations Research and Financial Engineering

Autor Asen L. Dontchev, R. Tyrrell Rockafellar
en Limba Engleză Paperback – 3 sep 2016

Descoperim aici o lucrare fundamentală care extinde limitele analizei matematice clasice către teritoriul complex al analizei variaționale. Implicit Functions and Solution Mappings nu este doar un tratat despre teorema funcțiilor implicite, ci o explorare interdisciplinară care conectează analiza matematică, ecuațiile cu derivate parțiale, optimizarea și analiza numerică. Credem că această a doua ediție, publicată de Springer, reușește să unifice rezultate care anterior erau dispersate în literatura de specialitate, oferind o perspectivă coerentă asupra stabilității și regularității soluțiilor.

Notăm cu interes structura riguroasă a volumului. Autorii pornesc de la fundamentele rezolvării ecuațiilor și avansează rapid către analiza mulțimilor de soluții pentru probleme variaționale (solution mappings). Capitolele dedicate analizei set-valued și derivatelor generalizate pregătesc terenul pentru conceptele de regularitate metrică în dimensiuni infinite, culminând cu aplicații practice în analiza numerică variațională. Această progresie transformă volumul dintr-o referință teoretică într-un instrument de lucru esențial pentru algoritmii de optimizare.

Acoperă aceeași arie tematică precum The Implicit Function Theorem de Steven G. Krantz, dar cu o abordare mult mai tehnică și orientată spre analiza non-smooth și variațională, spre deosebire de perspectiva mai geometrică și clasică a lui Krantz. În contextul operei autorilor, cartea se situează ca o continuare naturală și aprofundată a temelor din Lectures on Variational Analysis de Asen L. Dontchev, rafinând conceptele de multifuncții și stabilitate. Stilul este precis, dens, specific seriei Springer Series in Operations Research and Financial Engineering, fiind destinat celor care au deja o bază solidă în analiză funcțională.

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Specificații

ISBN-13: 9781493939695
ISBN-10: 1493939696
Pagini: 494
Ilustrații: XXVIII, 466 p. 14 illus., 1 illus. in color.
Dimensiuni: 155 x 235 x 25 mm
Greutate: 0.69 kg
Ediția:Softcover reprint of the original 2nd ed. 2014
Editura: Springer
Colecția Springer
Seria Springer Series in Operations Research and Financial Engineering

Locul publicării:New York, NY, United States

De ce să citești această carte

Această lucrare este esențială pentru cercetătorii și studenții avansați în matematică aplicată și optimizare. Cititorul câștigă o înțelegere profundă a modului în care soluțiile problemelor de optimizare reacționează la perturbații, un instrument critic în modelarea financiară și inginerie. Este recomandată pentru rigoarea cu care tratează regularitatea metrică și pentru plusul de claritate adus de noile exerciții și exemple incluse în această ediție.

Despre autor

Asen L. Dontchev a fost un matematician renumit pentru contribuțiile sale în control optim și analiză variațională, fiind un cercetător proeminent la Mathematical Reviews. R. Tyrrell Rockafellar este considerat unul dintre părinții optimizării moderne și ai analizei convexe, profesor emerit la University of Washington. Colaborarea lor în cadrul acestui volum reprezintă sinteza a zeci de ani de cercetare de vârf, Rockafellar aducând viziunea sistemică asupra convexității, în timp ce Dontchev completează tabloul cu expertiza sa în stabilitate și metode numerice.

Cuprins

Introduction and equation-solving background.- Solution mappings for variational problems.- Set-valued analysis of solution mappings.- Regularity properties through generalized derivatives.- Metric regularity in infinite dimensions.- Applications in numerical variational analysis.

Recenzii

“The book represents the state of the art of the modern theory of inverse and implicit functions and provides an important source for studies of numerical methods and applications in this area. It can be warmly recommended to all specialists and advanced students working in optimization, analysis, numerical mathematics, and other mathematical fields, as well as to all those who apply variational analysis in engineering, physics, operations research, economics, finance, and more.” (Diethard Klatte, SIAM Review, Vol. 57 (2), June, 2015)
“The book commences with a helpful context-setting preface followed by six chapters. Each chapter starts with a useful preamble and concludes with a careful and instructive commentary, while a good set of references, a notation guide and a somewhat brief index complete this study. … I unreservedly recommended this book to all practitioners and graduate students interested in modern optimization theory or control theory or to those just engaged bybeautiful analysis cleanly described.” (Jonathan Michael Browein, IEEE Control Systems Magazine, February, 2012).
“This book is devoted to the theory of inverse and implicit functions and some of its modifications for solution mappings in variational problems. … The book is targeted to a broad audience of researchers, teachers and graduate students. It can be used as well as a textbook as a reference book on the topic. Undoubtedly, it will be used by mathematicians dealing with functional and numerical analysis, optimization, adjacent branches and also by specialists in mechanics, physics, engineering, economics and so on.” (Peter Zabreiko, Zentralblatt MATH, Vol. 1178, 2010).
“The present monograph will be a most welcome and valuable addition. … This book will save much time and effort, both for those doing research in variational analysis and for students learning the field. This important contribution fills a gap in the existing literature.” (Stephen M. Robinson, Mathematical Reviews, Issue 2010).

Notă biografică

Asen L. Dontchev is a Professor at the Department of Mathematics at the University of Michigan. R. Tyrrell Rockafellar is a Professor at the Department of Mathematics at the University of Washington and at the Department of Industrial and Systems Engineering at the University of Florida.

Textul de pe ultima copertă

The implicit function theorem is one of the most important theorems in analysis and its many variants are basic tools in partial differential equations and numerical analysis.
This second edition of Implicit Functions and Solution Mappings presents an updated and more complete picture of the field by including solutions of problems that have been solved since the first edition was published, and places old and new results in a broader perspective. The purpose of this self-contained work is to provide a reference on the topic and to provide a unified collection of a number of results which are currently scattered throughout the literature. Updates to this edition include new sections in almost all chapters, new exercises and examples, updated commentaries to chapters and an enlarged index and references section.
From reviews of the first edition:
“The book commences with a helpful context-setting preface followed by six chapters. Each chapter starts with a useful preamble and concludes with a careful and instructive commentary, while a good set of references, a notation guide, and a somewhat brief index complete this study. … I unreservedly recommended this book to all practitioners and graduate students interested in modern optimization theory or control theory or to those just engaged by beautiful analysis cleanly described.” (Jonathan Michael Borwein, IEEE Control Systems Magazine, February, 2012)
“This book is devoted to the theory of inverse and implicit functions and some of its modifications for solution mappings in variational problems. … The book is targeted to a broad audience of researchers, teachers and graduate students. It can be used as well as a textbook as a reference book on the topic. Undoubtedly, it will be used by mathematicians dealing with functional and numerical analysis, optimization, adjacent branches and also by specialists in mechanics, physics, engineering, economics, and so on.” (Peter Zabreiko, Zentralblatt MATH, Vol. 1178, 2010)
“The present monograph will be a most welcome and valuable addition. … This book will save much time and effort, both for those doing research in variational analysis and for students learning the field. This important contribution fills a gap in the existing literature.” (Stephen M. Robinson, Mathematical Reviews, Issue 2010)

Caracteristici

Covers major developments of the last five years Complements and extends significantly the material covered in the first edition Includes new solutions of problems that were open when the first edition was published