Oberman, Adam M. Computing the convex envelope using a nonlinear partial differential equation. (English) Zbl 1154.35056 Math. Models Methods Appl. Sci. 18, No. 5, 759-780 (2008). The paper is concerned with the study of the numerical solving with a finite difference method, of the convex envelope using a fully nonlinear partial differential equation. In one dimension, this equation is reduced to the classical obstacle problem. It provides a local characterization of the convex envelope, that allows numerical methods to be constructed which involve only local conditions, eliminating costly global constrains. The scheme is shown to converge and the computational results for smooth and nonsmooth data are presented. Reviewer: Nicolae Pop (Baia Mare) Cited in 1 ReviewCited in 18 Documents MSC: 35J70 Degenerate elliptic equations 26B25 Convexity of real functions of several variables, generalizations 52A41 Convex functions and convex programs in convex geometry 65N06 Finite difference methods for boundary value problems involving PDEs Keywords:convex envelope; partial differential equation; finite difference method Software:na13 PDFBibTeX XMLCite \textit{A. M. Oberman}, Math. Models Methods Appl. Sci. 18, No. 5, 759--780 (2008; Zbl 1154.35056) Full Text: DOI References: [1] Barles G., Asympt. Anal. 4 pp 271– [2] Bertsekas D. P., Convex Analysis and Optimization (2003) · Zbl 1140.90001 [3] DOI: 10.1017/CBO9780511804441 · doi:10.1017/CBO9780511804441 [4] Brenier Y., C. R. Acad. Sci. Paris Sér. I Math. 308 pp 587– [5] DOI: 10.1137/0731007 · Zbl 0796.65009 · doi:10.1137/0731007 [6] DOI: 10.1007/BF03024318 · Zbl 0800.49038 · doi:10.1007/BF03024318 [7] DOI: 10.1007/PL00005446 · doi:10.1007/PL00005446 [8] DOI: 10.1090/S0273-0979-1992-00266-5 · Zbl 0755.35015 · doi:10.1090/S0273-0979-1992-00266-5 [9] DOI: 10.1007/978-3-662-04245-8 · doi:10.1007/978-3-662-04245-8 [10] DOI: 10.1137/S0036142997325581 · Zbl 0941.65062 · doi:10.1137/S0036142997325581 [11] DOI: 10.1090/S0002-9947-1990-0986024-2 · doi:10.1090/S0002-9947-1990-0986024-2 [12] DOI: 10.1007/s00607-005-0119-1 · Zbl 1098.49022 · doi:10.1007/s00607-005-0119-1 [13] DOI: 10.1023/A:1017591716397 · Zbl 1007.90049 · doi:10.1023/A:1017591716397 [14] DOI: 10.1016/S0764-4442(01)02117-6 · Zbl 1053.49013 · doi:10.1016/S0764-4442(01)02117-6 [15] DOI: 10.1002/1522-2616(200106)226:1<153::AID-MANA153>3.0.CO;2-2 · Zbl 1048.49011 · doi:10.1002/1522-2616(200106)226:1<153::AID-MANA153>3.0.CO;2-2 [16] DOI: 10.1137/040608039 · Zbl 1104.65056 · doi:10.1137/040608039 [17] DOI: 10.1016/S0764-4442(98)80397-2 · Zbl 0924.46059 · doi:10.1016/S0764-4442(98)80397-2 [18] DOI: 10.1007/BF00248008 · Zbl 0852.90117 · doi:10.1007/BF00248008 [19] DOI: 10.1023/A:1019191114493 · Zbl 0909.65037 · doi:10.1023/A:1019191114493 [20] DOI: 10.1007/PL00009331 · Zbl 0892.68102 · doi:10.1007/PL00009331 [21] DOI: 10.1007/s00211-004-0566-1 · Zbl 1070.65082 · doi:10.1007/s00211-004-0566-1 [22] DOI: 10.1090/S0025-5718-04-01688-6 · Zbl 1094.65110 · doi:10.1090/S0025-5718-04-01688-6 [23] DOI: 10.1090/S0002-9939-07-08887-9 · Zbl 1190.35107 · doi:10.1090/S0002-9939-07-08887-9 [24] DOI: 10.2307/2999574 · Zbl 1015.91515 · doi:10.2307/2999574 [25] Rockafellar R. T., Convex Analysis (1997) · Zbl 0932.90001 [26] DOI: 10.1080/03605309908821476 · Zbl 0935.35087 · doi:10.1080/03605309908821476 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.