We recall the history of the transportation (Kantorovich) metric and the Monge–Kantorovich problem. We also describe several little-known applications: the first one concerns the theory of decreasing sequences of partitions (tower of measures and iterated metric), the second one relates to Ornstein's theory of Bernoulli automorphisms (¯ d-metric), and the third one is the formulation of the str...

We present a new, simple, and elegant algorithm for computing the optimal mapping for the Monge-Kantorovich problem with quadratic cost. The method arises from a reformulation of the dual problem into an unconstrained minimization of a convex, continuous functional, for which the derivative can be explicitly found. The Monge-Kantorovich problem has applications in many fields; examples from ima...

When restricted to cost arrays possessing the sum Monge property, many combinatorial optimization problems with sum objective functions become significantly easier to solve. The more general algebraic assignment and transportation problems are similarly easier to solve given cost arrays possessing the corresponding algebraic Monge property. We show that Monge-array results for two sum-of-edge-c...

We consider the problem of designing a reflector that transforms a spherical wave front with a given intensity into an output front illuminating a prespecified region of the far-sphere with prescribed intensity. In earlier approaches, it was shown that in the geometric optics approximation this problem is reduced to solving a second order nonlinear elliptic partial differential equation of Mong...

We report here two recent developments in the theory of optimal transportation. The first is a proof for the existence of optimal mappings to the Monge mass transportation problem [16]. The other is an application of optimal transportation in geometric optics [19]. The optimal transportation problem, in general, deals with the redistribution of materials in the most economical way. The original...

We present an optimal data structure for submatrix maximum queries in n×nMonge matrices. Our result is a two-way reduction showing that the problem is equivalent to the classical predecessor problem in a universe of polynomial size. This gives a data structure of O(n) space that answers submatrix maximum queries in O(log logn) time, as well as a matching lower bound, showing that O(log log n) q...

In the present paper, we discuss Monge-Ampère equations from viewpoint of differential geometry. It is known that a Monge--Ampère equation corresponds to special exterior system on 1-jet space. this generalize and prove $(k+1)$st order generalized $k$-jet space its solution naturally an integral manifold corresponding system. Moreover, show Korteweg-de Vries (KdV) Cauchy--Riemann are examples o...

This paper is concerned with developing and analyzing convergent semi-Lagrangian methods for the fully nonlinear elliptic Monge–Ampère equation on general triangular grids. This is done by establishing an equivalent (in the viscosity sense) Hamilton–Jacobi–Bellman formulation of the Monge–Ampère equation. A significant benefit of the reformulation is the removal of the convexity constraint from...

In this paper, we construct several types of multi-valued solutions to the Monge–Ampère equation in higher dimensions. Recently, there has been considerable interest in understanding the behavior of a metric generated by a solution of the Monge–Ampère equation at a singularity. See, for instance Loftin, Yau and Zaslow [14] and Leung [12]. To explain our results let us refer first to the theory ...

We give algorithmic results for combinatorial problems with cost arrays possessing certain algebraic Monge properties. We extend Monge-array results for two shortest path problems to a general algebraic setting, with values in an ordered commutative semigroup, if the semigroup operator is strictly compatible with the order relation. We show how our algorithms can be modified to solve bottleneck...