Some numerical methods for the study of the convexity notions arising in the calculus of variations
ESAIM: Modélisation mathématique et analyse numérique, Tome 32 (1998) no. 2, pp. 153-175.
@article{M2AN_1998__32_2_153_0,
     author = {Dacorogna, Bernard and Haeberly, Jean-Pierre},
     title = {Some numerical methods for the study of the convexity notions arising in the calculus of variations},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {153--175},
     publisher = {Elsevier},
     volume = {32},
     number = {2},
     year = {1998},
     mrnumber = {1622606},
     zbl = {0905.65075},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_1998__32_2_153_0/}
}
TY  - JOUR
AU  - Dacorogna, Bernard
AU  - Haeberly, Jean-Pierre
TI  - Some numerical methods for the study of the convexity notions arising in the calculus of variations
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 1998
SP  - 153
EP  - 175
VL  - 32
IS  - 2
PB  - Elsevier
UR  - http://www.numdam.org/item/M2AN_1998__32_2_153_0/
LA  - en
ID  - M2AN_1998__32_2_153_0
ER  - 
%0 Journal Article
%A Dacorogna, Bernard
%A Haeberly, Jean-Pierre
%T Some numerical methods for the study of the convexity notions arising in the calculus of variations
%J ESAIM: Modélisation mathématique et analyse numérique
%D 1998
%P 153-175
%V 32
%N 2
%I Elsevier
%U http://www.numdam.org/item/M2AN_1998__32_2_153_0/
%G en
%F M2AN_1998__32_2_153_0
Dacorogna, Bernard; Haeberly, Jean-Pierre. Some numerical methods for the study of the convexity notions arising in the calculus of variations. ESAIM: Modélisation mathématique et analyse numérique, Tome 32 (1998) no. 2, pp. 153-175. http://www.numdam.org/item/M2AN_1998__32_2_153_0/

[1] J. J. Aubert and B. Dacorogna: An example of a quasiconvex function that is not polyconvex in two dimensions, Arch. Rational Mech. Anal., 117, 155-166 (1992). | MR | Zbl

[2] F. Aluffi-Pentini, V. Parisi and F. Zirilli: Global optimization and stochastic differential equations, JOTA 47, 1-16 (1985). | MR | Zbl

[3] G. Aubert: Contribution aux problèmes du calcul des variations et applications à l'élasticité non linéaire, Thèse de doctorat, Paris VI (1986).

[4] J. M. Ball: Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 64, 337-403 (1977). | MR | Zbl

[5] J. M. Ball and F. Murat: W1,p quasiconvexity and variational problems for multiple integrals, J. Funct. Anal. 58, 225-253 (1984). | MR | Zbl

[6] B. Brighi and M. Chipot: Approximated Convex Envelope of a Function, SIAM J. Numer. Anal. 31, 128-148 (1994). | MR | Zbl

[7] M. Chipot: Numerical analysis of oscillations in nonconvex problems, Numer. Math. 59, 747-767 (1991). | MR | Zbl

[8] M. Chipot and C. Collins: Numerical approximations in variational problems with potential wells, SIAM J. Numer. Anal. 29, 1002-1019 (1992). | MR | Zbl

[9] M. Chipot and V. Lécuyer: Analysis and computations in the four-well problem, Preprint. | MR | Zbl

[10] C. Collins, D. Kinderlehrer and M. Luskin: Numerical approximation of the solution of a variational problem with a double well potential, SIAM J. Numer. Anal. 28, 321-332 (1991). | MR | Zbl

[11] C. Collins and M. Luskin: Optimal order error estimates for the finite element approximation of the solution of a nonconvex variational problem, Math. Comp. 57, 621-637 (1991). | MR | Zbl

[12] B. Dacorogna: Direct Methods in the Calculus of Variations, Berlin : Springer 1989. | MR | Zbl

[13] B. Dacorogna, J. Douchet, W. Gangbo and J. Rappaz: Some examples of rank one convex functions in dimension two, Proc. of Royal Soc. Edinburgh 114A, 135-150 (1990). | MR | Zbl

[14] B. Dacorogna and J. P. Haeberly: Remarks on a Numerical Study of Convexity, Quasiconvexity and Rank One Convexity, in Progress in Nonlinear Differential Equations and Their Applications, vol. 25, pp. 143-154, R. Serapioni and F. Tomarelli, Eds., Basel: Birkhäuser 1996. | MR | Zbl

[15] B. Dacorogna and J.-P. Haeberly: On Convexity Properties of Homogeneous Functions of Degree One, Proc. of Royal Soc. Edinburgh 126, 947-956 (1996). | MR | Zbl

[16] B. Dacorogna and P. Marcellini: A counterexample in the vectorial calculus of variations, in Material instabilities in continuum mechanics, pp. 77-83, proceedings edited by J.-M. Ball, Oxford : Oxford Science Publc. 1988. | MR | Zbl

[17] P. Gill, W. Murray and M. Wright: Practical Optimization, London : Academic Press 1981. | MR | Zbl

[18] P. A. Gremaud: Numerical optimization and quasiconvexity, IMA Preprint Series #1133, University of Minnesota, April 1993. | MR | Zbl

[19] H. Hartwig: A polyconvexity condition in dimension two, Proc. of Royal Soc. Edinburgh 125A, 901-910 (1995). | MR | Zbl

[20] T. Iwaniec and A. Lutoborski: Integral estimates for null Lagrangian, Arch. Rational Mech. Anal.125, 25-79 (1993). | MR | Zbl

[21] D. C. Liu and J. Nocedal: On the limited memory BFGS method for large scale optimization, Math. Program. 45, 503-528 (1989). | MR | Zbl

[22] J. J. Moré and S. J. Wright: Optimization Software Guide, Frontiers in Applied Mathematics 14, Philadelphia : SIAM 1993. | MR | Zbl

[23] C. B. Morrey: Quasiconvexity and the semicontinuity of multiple integrals, Pacific J. Math. 2, 25-53 (1952). | MR | Zbl

[24] J. Nocedal: Theory of algorithms for unconstrained optimization, Acta Numerica, 199-242 (1991). | MR | Zbl

[25] M. Overton: On minimizing the maximum eigenvalue of a symmetric matrix, SIAM J. Matrix Anal. Appl. 9, 256-268 (1988). | MR | Zbl

[26] M. Overton: Large-scale optimization of eigenvalues, SIAM J. Optim. 2, 88-120 (1992). | MR | Zbl

[27] W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery: Numerical Recipes in C, 2nd ed., Cambridge: Cambridge University Press. 1992. | MR | Zbl

[28] F. F. Shanno: Conjugate gradient methods with inexact searches, Math. Oper. Res. 3, 244-236 (1978). | MR | Zbl

[29] D. F. Shanno and K. H. Phua : Remark on algorithm 500 : minimization of unconstrained multivariate functions, ACM Trans. on Math. Software 6, 618-622 (1980).

[30] V. Sverak : Rank one convexity does not imply quasiconvexity, Proc. of Royal Soc. Edinburgh, 120A, 185-189 (1992). | MR | Zbl

[31] M. Tomasslni : A survey of genetic algorithms, to appear in Annual Reviews of Computational Physics, 3, World Scientific. | MR

[32] J. L. Zhou and A. L. Tits : User's guide for FSQP version 3.4 (released December 1994), Systems Research Center TR-92-107r4, University of Maryland.