HANDLING CONVEXITY-LIKE CONSTRAINTS IN VARIATIONAL PROBLEMS

We provide a general framework to construct finite-dimensional approximations of the space of convex functions, which also applies to the space of c-convex functions and to the space of support functions of convex bodies. We give precise estimates of the distance between the approximation space and...

Full description

Saved in:
Bibliographic Details
Published inSIAM journal on numerical analysis Vol. 52; no. 5; pp. 2466 - 2487
Main Authors Merigot, Quentin, Oudet, Edouard
Format Journal Article
LanguageEnglish
Published Society for Industrial and Applied Mathematics 01.01.2014
Subjects
Algorithms
Analysis of PDEs
Approximation
Calculus of variations
Convexity
Discretization
Interpolation
Mathematical analysis
Mathematical constants
Mathematical functions
Mathematical models
Mathematical vectors
Mathematics
Noise reduction
Numerical Analysis
Principal agent problem
Three dimensional
Triangulation
convergence analysis
support function
second-order constraints
convexity
Online AccessGet full text

Cover

More Information
Summary:We provide a general framework to construct finite-dimensional approximations of the space of convex functions, which also applies to the space of c-convex functions and to the space of support functions of convex bodies. We give precise estimates of the distance between the approximation space and the admissible set. This framework applies to the approximation of convex functions by piecewise-linear functions on a mesh of the domain and by other finite-dimensional spaces such as tensor-product splines. We show how these discretizations are well suited for the numerical solution of problems of calculus of variations under convexity constraints. Our implementation relies on proximal algorithms and can be easily parallelized, thus making it applicable to large-scale problems in dimension two and three. We illustrate the versatility and the efficiency of our approach on the numerical solution of three problems in calculus of variation: three-dimensional denoising, the principal agent problem, and optimization within the class of convex bodies.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 23
ISSN:0036-1429
1095-7170
DOI:10.1137/130938359