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Splitting Enables Overcoming the Curse of Dimensionality

Part of the Scientific Computation book series (SCIENTCOMP)


In this chapter we briefly outline a new and remarkably fast algorithm for solving a large class of high dimensional Hamilton-Jacobi (H-J) initial value problems arising in optimal control and elsewhere [1]. This is done without the use of grids or numerical approximations. Moreover, by using the level set method [8] we can rapidly compute projections of a point in \(\mathbb{R}^{n}\), n large to a fairly arbitrary compact set [2]. The method seems to generalize widely beyond what will we present here to some nonconvex Hamiltonians, new linear programming algorithms, differential games, and perhaps state dependent Hamiltonians.


  • Nonconvex Hamiltonians
  • Linear Programming Algorithm
  • Differential Games
  • Initial Value Problem
  • Hopf Formula

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  • DOI: 10.1007/978-3-319-41589-5_12
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Fig. 12.1


  1. Darbon, J., Osher, S.: Algorithms for overcoming the curse of dimensionality for certain Hamilton-Jacobi equations arising in control theory and elsewhere. Research in the Mathematical Sciences (to appear)

    Google Scholar 

  2. Darbon, J., Osher, S.: Fast projections onto compact sets in high dimensions using the level set method, Hopf formulas and optimization. (In preparation)

    Google Scholar 

  3. Glowinski, R., Marroco, A.: Sur l’approximation, par éléments finis d’ordre un, et la résolution, par pénalisation-dualité d’une classe de problèmes de Dirichlet non linéaires. ESAIM: Mathematical Modelling and Numerical Analysis 9 (R2), 41–76 (1975)

    MATH  Google Scholar 

  4. Goldstein, T., Osher, S.: The split Bregman method for L1-regularized problems. SIAM Journal on Imaging Sciences 2 (2), 323–343 (2009)

    CrossRef  MATH  MathSciNet  Google Scholar 

  5. Hopf, E.: Generalized solutions of non-linear equations of first order (First order nonlinear partial differential equation discussing global locally-Lipschitzian solutions via Jacoby theorem extension). Journal of Mathematics and Mechanics 14, 951–973 (1965)

    Google Scholar 

  6. Lax, P.D.: Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. SIAM, Philadelphia, PA (1990)

    Google Scholar 

  7. Moreau, J.J.: Proximité et dualité dans un espace hilbertien. Bulletin de la Société Mathématique de France 93, 273–299 (1965)

    MATH  MathSciNet  Google Scholar 

  8. Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations. Journal of Computational Physics 79 (1), 12–49 (1988)

    CrossRef  MATH  MathSciNet  Google Scholar 

  9. Yin, W., Osher, S.: Error forgetting of Bregman iteration. Journal of Scientific Computing 54 (2–3), 684–695 (2013)

    CrossRef  MATH  MathSciNet  Google Scholar 

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Research supported by ONR grants N000141410683, N000141210838 and DOE grant DE-SC00183838.

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Correspondence to Jérôme Darbon .

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Darbon, J., Osher, S.J. (2016). Splitting Enables Overcoming the Curse of Dimensionality. In: Glowinski, R., Osher, S., Yin, W. (eds) Splitting Methods in Communication, Imaging, Science, and Engineering. Scientific Computation. Springer, Cham.

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