An Analytical Approach to Monge’s Problem

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Rezaei, Nima | 2020

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 53007 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Ranjbar Motlagh, Alireza
Abstract:
Monge’s problem was solved by Brenier in 1990. In general, the problem remained unresolved for a long time. some of its cases were solved by assumptions, but the general case and its analytical solution for the first time by Ian Bernier The French mathematician was introduced. He wrote his famous article Solved this nonlinear problem with technical assumptions. With the help of convex analysis and fundamental theorems in functions with vector values, he proved the existence and unity of this nonlinear problem. Monge’s problem, also known as optimal transport, suggests whether a stablesized mapping can be done by having two probabilistic spaces and one cost function. (Inverted image of any measurable set in the range has the same measure in the domain) So that the cost is minimized. In this Thesis, the Monge’s problem is analyzed in detail As well as a conceptual explanation called polar analysis Paid
Keywords:
Real Measure ; Edelstein Theorem ; Monge’s Problem ; Vector Valued Functions ; Polar Factorization Vector Valued Functions