Multiresolution analysis applied to the Monge-Kantorovich problem. (English) Zbl 1470.49070

Summary: We give a scheme of approximation of the MK problem based on the symmetries of the underlying spaces. We take a Haar type MRA constructed according to the geometry of our spaces. Thus, applying the Haar type MRA based on symmetries to the MK problem, we obtain a sequence of transportation problem that approximates the original MK problem for each of MRA. Moreover, the optimal solutions of each level solution converge in the weak sense to the optimal solution of original problem.


49Q20 Variational problems in a geometric measure-theoretic setting
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
65T60 Numerical methods for wavelets
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[1] Hernández-Lerma, O.; Lasserre, J. B., Approximation schemes for infinite linear programs, SIAM Journal on Optimization, 8, 4, 973-988, (1998) · Zbl 0912.90219 · doi:10.1137/S1052623497315768
[2] González-Hernández, J.; Gabriel, J. R.; Hernández-Lerma, O., On solutions to the mass transfer problem, SIAM Journal on Optimization, 17, 2, 485-499, (2006) · Zbl 1165.49313 · doi:10.1137/050623991
[3] Gabriel, J. R.; González-Hernández, J.; López-Martínez, R. R., Numerical approximations to the mass transfer problem on compact spaces, IMA Journal of Numerical Analysis, 30, 4, 1121-1136, (2010) · Zbl 1210.65109 · doi:10.1093/imanum/drn076
[4] Gröchenig, K.; Madych, W. R., Multiresolution analysis, Haar bases, and self-similar tilings of Rn, Institute of Electrical and Electronics Engineers Transactions on Information Theory, 38, 2, 556-568, (1992) · Zbl 0742.42012 · doi:10.1109/18.119723
[5] Guo, K.; Labate, D.; Lim, W.-Q.; Weiss, G.; Wilson, E., Wavelets with composite dilations and their MRA properties, Applied and Computational Harmonic Analysis, 20, 2, 202-236, (2006) · Zbl 1086.42026 · doi:10.1016/j.acha.2005.07.002
[6] Krishtal, I. A.; Robinson, B. D.; Weiss, G. L.; Wilson, E. N., Some simple Haar-type wavelets in higher dimensions, The Journal of Geometric Analysis, 17, 1, 87-96, (2007) · Zbl 1124.42026 · doi:10.1007/BF02922084
[7] Bazaraa Mokhtar, S.; John, J.; Hanif, D., Linear Programming and Network Flows, (2009), Wiley · Zbl 0722.90042
[8] Billingsley, P., Convergence of Probability Measures, (1968), New York, NY, USA: Wiley, New York, NY, USA · Zbl 0172.21201
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