M.Sc Thesis | |

M.Sc Student | Meisler Dmitry |
---|---|

Subject | Point Allocations and Transport Algorithms |

Department | Department of Applied Mathematics |

Supervisor | PROFESSOR EMERITUS Gershon Wolansky |

This
work is concerned with the Monge problem of optimal mass transportation between
two probability measures *mu*_{0} and *mu* on a state space *Omega*,
where *mu*_{0}=*rho*_{0}*(x)dx* is a continuous
measure (with respect to Lebesgue measure) and *mu=m _{1}delta_y_{1}*

Let
*C(mu _{0},mu)* be the cost of transportation of the measure

* min_{mu in B ^{1}(Gamma)}[C(mu_{0},mu)(mu)]*,

In order to find the optimal partition we use the Monge-Kantorovich duality. We show that the dual problem is concave and is strictly concave near the solution, is differentiable, and we find its gradient and Hessian. We prove that it has a unique global maximum where the gradient vanishes. This allows us to use optimization methods in order to find the optimal partition.

The
contribution of our work is in the solution of the second optimization problem
since it has an additional interpretation: it approximates the solution of the
Monge problem for the continuous measures *mu _{0}*,