New Direct Numerical Methods for Some Multidimensional Problems of the Calculus of Variations. (12th March 2018)

• Record Type: Journal Article
• Authors: Dolgopolik, Maksim V.
• Is Part Of: Numerical functional analysis and optimization. Volume 39:Number 4(2018)
• Page Start: 467
• Page End: 490
• Publication Date: 2018-03-12
• Subjects: Calculus of variations -- direct numerical method -- direction of steepest descent -- method of steepest descent -- method of gradient descent; 49M05 -- 49M30; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2017.1378231
• Languages: English
• ISSNs: 0163-0563

Abstract

ABSTRACT: In this paper, we develop a new approach to the design of direct numerical methods for multidimensional problems of the calculus of variations. The approach is based on a transformation of the problem with the use of a new class of Sobolev-like spaces that is studied in the article. This transformation allows one to analytically compute the direction of steepest descent of the main functional of the calculus of variations with respect to a certain inner product, and, in turn, to construct new direct numerical methods for multidimensional problems of the calculus of variations. In the end of the paper, we point out how the approach developed in the article can be extended to the case of problems with more general boundary conditions, problems for functionals depending on higher order derivatives, and problems with isoperimetric and/or pointwise constraints.