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Newton’s method and secant method for set-valued mappings

Title data

Baier, Robert ; Hessel-von Molo, Mirko:
Newton’s method and secant method for set-valued mappings.
In: Lirkov, Ivan ; Margenov, Svetozar D. ; Waśniewski, Jerzy (ed.): Large-scale scientific computing : 8th international conference, LSSC 2011, Sozopol, Bulgaria, June 6 - 10, 2011 ; revised selected papers. - Berlin , 2012 . - pp. 91-98 . - (Lecture Notes in Computer Science ; 7116 )
ISBN 978-3-642-29842-4
DOI: https://doi.org/10.1007/978-3-642-29843-1_9

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Abstract in another language

For finding zeros or fixed points of set-valued maps, the fact that the space of convex, compact, nonempty sets of ℝ n is not a vector space presents a major disadvantage. Therefore, fixed point iterations or variants of Newton’s method, in which the derivative is applied only to a smooth single-valued part of the set-valued map, are often applied for calculations. We will embed the set-valued map with convex, compact images (i.e. by embedding its images) and shift the problem to the Banach space of directed sets. This Banach space extends the arithmetic operations of convex sets and allows to consider the Fréchet-derivative or divided differences of maps that have embedded convex images. For the transformed problem, Newton’s method and the secant method in Banach spaces are applied via directed sets. The results can be visualized as usual nonconvex sets in ℝ n .

Further data

Item Type: Article in a book Yes CONTENTS: 1. Introduction 1.1 Directed Sets 2. Fréchet-Derivative for Set-Valued Maps 3. Newton's Method and Secant Method 3.1 Newton's Method for Directed Sets 3.2 Secant Method Based on Directed Sets 4. Examples set-valued Newton's method; set-valued secant method; Gauß-Newton method; directed sets; embedding of convex compact sets Mathematics Subject Classification Code: 65J15 (52A20 65H10 90C56 26E25 54C60) Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics V (Applied Mathematics) > Chair Mathematics V (Applied Mathematics) - Univ.-Prof. Dr. Lars GrüneFacultiesFaculties > Faculty of Mathematics, Physics und Computer ScienceFaculties > Faculty of Mathematics, Physics und Computer Science > Department of MathematicsFaculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics V (Applied Mathematics) Yes 500 Science 22 Feb 2021 12:47 28 May 2021 10:31 https://eref.uni-bayreuth.de/id/eprint/63245