Abstract
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
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)
Darbon, J., Osher, S.: Fast projections onto compact sets in high dimensions using the level set method, Hopf formulas and optimization. (In preparation)
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)
Goldstein, T., Osher, S.: The split Bregman method for L1-regularized problems. SIAM Journal on Imaging Sciences 2 (2), 323–343 (2009)
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)
Lax, P.D.: Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. SIAM, Philadelphia, PA (1990)
Moreau, J.J.: Proximité et dualité dans un espace hilbertien. Bulletin de la Société Mathématique de France 93, 273–299 (1965)
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)
Yin, W., Osher, S.: Error forgetting of Bregman iteration. Journal of Scientific Computing 54 (2–3), 684–695 (2013)
Acknowledgements
Research supported by ONR grants N000141410683, N000141210838 and DOE grant DE-SC00183838.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
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. https://doi.org/10.1007/978-3-319-41589-5_12
Download citation
DOI: https://doi.org/10.1007/978-3-319-41589-5_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-41587-1
Online ISBN: 978-3-319-41589-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)