Approximating Viability Kernels and Resilience Values: Algorithms and Practical Issues Illustrated with KAVIAR Software

  • Laetitia Chapel
  • Guillaume Deffuant
Part of the Understanding Complex Systems book series (UCS)


In Chap.2, we presented the definition of resilience based on viability theory (Martin 2004), and we argued that this definition is more general than the equilibrium based definition, and fits better the usual meaning of resilience. Several chapters illustrate this approach on individual based models (language dynamics, bacteria, savanna). In each of these case studies, a preliminary work is to approximate the individual based model with more of a synthetic model, because the tools for computing viability kernels cannot deal with dynamical systems with a state space of high dimension.


Support Vector Machine Classification Procedure Optimal Controller Control Space Outer Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. .
    Aubin JP (1991) Viability Theory. Birkhäuser, BaselGoogle Scholar
  2. .
    Aubin JP (2001) Viability kernels and capture basins of sets under differential inclusions. SIAM J Contr Optim 40(3):853–881CrossRefGoogle Scholar
  3. .
    Aubin JP, Frankowska H (1996) The Viability Kernel Algorithm for Computing Value Functions of Infinite Horizon Optimal Control Problems J Math Anal Appl 201(2):555–576Google Scholar
  4. .
    Bellman R (1961) Adaptive Control Processes: A Guided Tour. Princeton University Press, PrincetonGoogle Scholar
  5. .
    Bokanowski O, Martin S, Munos R, Zidani H (2006) An anti-diffusive scheme for viability problems. Appl Numer Math 56(9): 1147–1162CrossRefGoogle Scholar
  6. .
    Bonneuil N (2006) Computing the viability kernel in large state dimension. J Math Anal Appl 323:1444–1454CrossRefGoogle Scholar
  7. .
    Boser BE, Guyon I, Vapnik V (1992) A training algorithm for optimal margin classifiers. In: Proceedings of the Fifth Annual Workshop on Computational Learning Theory, Pittsburgh, USA, pp 144–152Google Scholar
  8. .
    Chapel L, Deffuant G (2007) SVM viability controller active learning: application to bike control. In: Proceedings of the IEEE International Symposium on Approximate Dynamic Programming and Reinforcement Learning (ADPRL’07), Honolulu, USA, April 1–5, pp 193–200Google Scholar
  9. .
    Chapel L, Deffuant G (2011) Inner and Outer Capture Basin Approximation with Support Vector Machines. To appear in: Proceedings of the 8th International Conference on Informatics in Control, Automation and Robotics, Noordwikerhout, The NetherlandsGoogle Scholar
  10. .
    Cristianini N, Shawe-Taylor J (2000) Support Vector Machines and other kernel-based learning methods. Cambridge University Press, CambridgeGoogle Scholar
  11. .
    Deffuant G, Chapel L, Martin S (2007) Approximating viability kernels with support vector machines. IEEE Trans Automat Contr 52(5):933–937CrossRefGoogle Scholar
  12. .
    Loosli G, Deffuant G, Canu S (2008) BALK : bandwidth autosetting for SVM with local kernels. Application to data on incomplete grids. Conférence d’apprentissageGoogle Scholar
  13. .
    Martin S (2004) The cost of restoration as a way of defining resilience: a viability approach applied to a model of lake eutrophication. Ecol Soc 9(2)8Google Scholar
  14. .
    Platt JC (1999) Fast training of support vector machines using sequential minimal optimization. In: Schölkopf B, Burges CJC, Smola A (eds) Advances in kernel methods – support vector learning. MIT, Cambridge, pp 185–208Google Scholar
  15. .
    Saint-Pierre P (1994) Approximation of the viability kernel. Appl Math Optim 29:187–209CrossRefGoogle Scholar
  16. .
    Scholkopf B, Smola A (2002) Learning with Kernels: support vector machines, regularization, optimization, and beyond. MIT, Cambridge, MA, USAGoogle Scholar
  17. .
    Vapnik V (1995) The nature of statistical learning theory. Springer, New YorkGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Cemagref - LISCAubièreFrance
  2. 2.Lab-STICCUniversité de Bretagne sudVannes CedexFrance

Personalised recommendations