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Mixed \(L^2\)-Wasserstein optimal mapping between prescribed density functions. (English) Zbl 1010.49029

Summary: A time-dependent minimization problem for the computation of a mixed \(L^2\)-Wasserstein distance between two prescribed density functions is introduced in the spirit of [J. D. Benamou and Y. Brenier, Numer. Math. 84, No. 3, 375-393 (2000; Zbl 0968.76069)] for the classical Wasserstein distance. The optimum of the cost function corresponds to an optimal mapping between prescribed initial and final densities. We enforce the final density conditions through a penalization term added to our cost function. A conjugate gradient method is used to solve this relaxed problem. We obtain an algorithm which computes an interpolated \(L^2\)-Wasserstein distance between two densities and the corresponding optimal mapping.

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
49M30 Other numerical methods in calculus of variations (MSC2010)
65K10 Numerical optimization and variational techniques
49J20 Existence theories for optimal control problems involving partial differential equations

Citations:

Zbl 0968.76069
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References:

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