Ivar Ekeland

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Picture of the Julia set
Ivar Ekeland has written popular books about chaos theory and about fractals,[1][2] such as the Julia set (animated). Ekeland's exposition provided mathematical inspiration to Michael Crichton's discussion of chaos in Jurassic Park.[3]

Ivar I. Ekeland (born 2 July 1944, Paris) is a French mathematician of Norwegian descent. Ekeland has written influential monographs and textbooks on nonlinear

Ian Malcom appearing in Michael Crichton's 1990 novel Jurassic Park.[3]

Biography

Ekeland studied at the

École Spéciale Militaire de Saint-Cyr, and the University of British Columbia in Vancouver
. He was the chairman of Paris-Dauphine University from 1989 to 1994.

Ekeland is a recipient of the D'Alembert Prize and the Jean Rostand prize. He is also a member of the Norwegian Academy of Science and Letters.[5]

Popular science: Jurassic Park by Crichton and Spielberg

Picture of Jeff Goldblum
Actor Jeff Goldblum consulted Ekeland while preparing to play a mathematician specializing in chaos theory in Spielberg's Jurassic Park.[6]

Ekeland has written several books on popular science, in which he has explained parts of dynamical systems, chaos theory, and probability theory.[1][7][8] These books were first written in French and then translated into English and other languages, where they received praise for their mathematical accuracy as well as their value as literature and as entertainment.[1]

Through these writings, Ekeland had an influence on

Jurassic Park, on both the novel and film. Ekeland's Mathematics and the unexpected and James Gleick's Chaos inspired the discussions of chaos theory in the novel Jurassic Park by Michael Crichton.[3] When the novel was adapted for the film Jurassic Park by Steven Spielberg, Ekeland and Gleick were consulted by the actor Jeff Goldblum as he prepared to play the mathematician specializing in chaos theory.[6]

Research

Ekeland has contributed to

variational calculus and mathematical optimization
.

Variational principle

In mathematical analysis, Ekeland's variational principle, discovered by Ivar Ekeland,[9][10][11] is a theorem that asserts that there exists a nearly optimal solution to a class of optimization problems.[12]

Ekeland's variational principle can be used when the lower

compact, so that the Bolzano–Weierstrass theorem can not be applied. Ekeland's principle relies on the completeness of the metric space.[13]

Ekeland's principle leads to a quick proof of the

Caristi fixed point theorem.[13][14]

Ekeland was associated with the University of Paris when he proposed this theorem.[9]

Variational theory of Hamiltonian systems

Ivar Ekeland is an expert on

periodic solutions of Hamiltonian systems and particularly to the theory of Kreĭn indices for linear systems (Floquet theory) was described in his monograph.[4]

Additive optimization problems

The Shapley–Folkman lemma depicted by a diagram with two panes, one on the left and the other on the right. The left-hand pane displays four sets, which are displayed in a two-by-two array. Each of the sets contains exactly two points, which are displayed in red. In each set, the two points are joined by a pink line-segment, which is the convex hull of the original set. Each set has exactly one point that is indicated with a plus-symbol. In the top row of the two-by-two array, the plus-symbol lies in the interior of the line segment; in the bottom row, the plus-symbol coincides with one of the red-points. This completes the description of the left-hand pane of the diagram. The right-hand pane displays the Minkowski sum of the sets, which is the union of the sums having exactly one point from each summand-set; for the displayed sets, the sixteen sums are distinct points, which are displayed in red: The right-hand red sum-points are the sums of the left-hand red summand-points. The convex hull of the sixteen red-points is shaded in pink. In the pink interior of the right-hand sumset lies exactly one plus-symbol, which is the (unique) sum of the plus-symbols from the right-hand side. Comparing the left array and the right pane, one confirms that the right-hand plus-symbol is indeed the sum of the four plus-symbols from the left-hand sets, precisely two points from the original non-convex summand-sets and two points from the convex hulls of the remaining summand-sets.
Ivar Ekeland applied the Shapley–Folkman lemma to explain Claude Lemarechal's success with Lagrangian relaxation on non-convex minimization problems. This lemma concerns the Minkowski addition of four sets. The point (+) in the convex hull of the Minkowski sum of the four non-convex sets (right) is the sum of four points (+) from the (left-hand) sets—two points in two non-convex sets plus two points in the convex hulls of two sets. The convex hulls are shaded pink. The original sets each have exactly two points (shown in red).

Ekeland explained the success of methods of convex minimization on large problems that appeared to be non-convex. In many optimization problems, the

objective function
 f are separable, that is, the sum of many summand-functions each with its own argument:

For example, problems of linear optimization are separable. For a separable problem, we consider an optimal solution

with the minimum value f(xmin). For a separable problem, we consider an optimal solution (xmin, f(xmin)) to the "convexified problem", where convex hulls are taken of the graphs of the summand functions. Such an optimal solution is the limit of a sequence of points in the convexified problem

[15][16] An application of the Shapley–Folkman lemma represents the given optimal-point as a sum of points in the graphs of the original summands and of a small number of convexified summands.

This analysis was published by Ivar Ekeland in 1974 to explain the apparent convexity of separable problems with many summands, despite the non-convexity of the summand problems. In 1973, the young mathematician Claude Lemaréchal was surprised by his success with convex minimization methods on problems that were known to be non-convex.[17][15][18] Ekeland's analysis explained the success of methods of convex minimization on large and separable problems, despite the non-convexities of the summand functions.[15][18][19] The Shapley–Folkman lemma has encouraged the use of methods of convex minimization on other applications with sums of many functions.[15][20][21][22]

Bibliography

Research

  • Ekeland, Ivar;
    MR463993
    ) ed.)
The book is cited over 500 times in MathSciNet.

Exposition for a popular audience

logistic function system was described as an example of chaos theory in Ekeland's Mathematics and the unexpected.[1]

See also

Notes

  1. ^
    ISBN 978-0-226-19990-0
    .
  2. MR945956
    )
  3. ^
    ISBN 9780345418951
    . Retrieved 2011-04-19.
  4. ^
    MR 1051888
    .
  5. ^ "Group 1: Mathematical studies". Norwegian Academy of Science and Letters. Archived from the original on 27 September 2011. Retrieved 12 April 2011.
  6. ^
    ASIN B002FZISIO
    . Retrieved 2011-04-12.
  7. MR 1243636
    .
  8. MR 2259005
    .
  9. ^
    ISSN 0022-247X
    .
  10. MR 0526967
    .
  11. ^ Ekeland & Temam 1999, pp. 357–373.
  12. MR 2303896
    .
  13. ^
    ISBN 978-0-521-38289-2
    .
  14. ISBN 978-0-691-11768-3
    . Retrieved January 31, 2009.
  15. ^ a b c d (Ekeland & Temam 1999, pp. 357–359): Published in the first English edition of 1976, Ekeland's appendix proves the Shapley–Folkman lemma, also acknowledging Lemaréchal's experiments on page 373.
  16. ^ The limit of a sequence is a member of the closure of the original set, which is the smallest closed set that contains the original set. The Minkowski sum of two closed sets need not be closed, so the following inclusion can be strict
    Clos(P) + Clos(Q) ⊆ Clos( Clos(P) + Clos(Q) );
    the inclusion can be strict even for two convex closed summand-sets, according to Rockafellar (1997, pp. 49 and 75). Ensuring that the Minkowski sum of sets be closed requires the closure operation, which appends limits of convergent sequences.
    MR 1451876
    .
  17. MR 1295240
    .
  18. ^
    MR 0395844
    .
  19. MR 0484418
    .
  20. MR 2449499
    .
  21. ^ Bertsekas (1996, pp. 364–381)acknowledging Ekeland & Temam (1999) on page 374 and Aubin & Ekeland (1976) on page 381:
    MR 0690767
    .

    S2CID 6329622
    . Retrieved 2 February 2011.
  22. ISBN 978-1-886529-00-7
    .

External links

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