A Primer on Generated Jacobian Equations: Geometry, Optics, Economics

Guillen, Nestor

doi:10.1090/noti1956

Cited by 12 publications

(11 citation statements)

References 19 publications

(39 reference statements)

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“…These algorithms share a common feature, in that they are all derived from Kantorovich duality. Some of them have been adapted to variants of optimal transport problems, such as multi-marginal optimal transport problems problems [82], barycenters with respect to optimal transport metrics [2], partial [25] and unbalanced optimal transport [31,66], gradient flows in the Wasserstein space [62,4], generated Jacobian equations [56,95]. However, we consider these extensions to be out of the scope of this chapter.…”

Section: Introductionmentioning

confidence: 99%

Optimal transport: discretization and algorithms

Mérigot

Thibert

2021

Handbook of Numerical Analysis

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Section: Introductionmentioning

confidence: 99%

Optimal transport: discretization and algorithms

Mérigot

Thibert

2021

Handbook of Numerical Analysis

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“…Here we give the definitions required for the generalised convexity theory. More detailed introductions can be found in [9,7,27]. We begin with the definition of generating functions.…”

Section: G -mentioning

confidence: 99%

“…The dual generating function is well defined because < 0. We note if (•, 0 , 0 ) is a support at 0 by (7) we have 0 = * ( 0 , 0 , ( 0 )) and subsequently the support is (•, 0 , * ( 0 , 0 , ( 0 ))). Differentiating (10) we obtain the identities…”

Section: Definition 5 the Dual Generating Function Is The Unique Func...mentioning

confidence: 99%

Strict $g$-convexity for generated Jacobian equations with applications to global regularity

Rankin¹

2021

Preprint

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A. This article has two purposes. The first is to prove solutions of the second boundary value problem for generated Jacobian equations (GJEs) are strictly -convex. The second is to prove the global 3 regularity of Aleksandrov solutions to the same problem. In particular, Aleksandrov solutions are classical solutions. These are related because the strict -convexity is essential for the proof of the global regularity. The assumptions for the strict -convexity are the natural extension of those used by Chen and Wang in the optimal transport case. They are the Loeper maximum principle condition, a positively pinched right-hand side, a * -convex target, and a source domain strictly contained in aconvex domain. This improves the existing domain conditions though at the expense of requiring a 3 generating function. This is appropriate for global regularity where existence is proved assuming a 4 generating function. We prove the global regularity under the hypothesis that Jiang and Trudinger recently used to obtain the existence of a globally smooth solution and an additional condition on the height of solutions. Our proof of global regularity is by modifying Jiang and Trudinger's existence result to construct a globally 3 solution intersecting the Aleksandrov solution. Then the strict convexity yields the interior regularity to apply the author's uniqueness results.

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“…The material below is largely due to Trudinger [Tru14] with other presentations in [GK17, Jeo20, Jha17, JT18, LT16, Ran20]. Guillen's survey article [Gui19] is a good introduction and lists the 2D theory we develop here as an open problem. Throughout it is helpful to keep in mind the cases ( , , ) = • − and ( , , ) = ( , ) − where is a cost function from optimal transport.…”

Section: G Gjementioning

confidence: 99%

Strict convexity and $C^1$ regularity of solutions to generated Jacobian equations in dimension two

Rankin

2020

Preprint

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A. We present a proof of strict -convexity in 2D for solutions of generated Jacobian equations with a -Monge-Ampère measure bounded away from 0. Subsequently this implies 1 differentiability in the case of a -Monge-Ampère measure bounded from above. Our proof follows one given by Trudinger and Wang in the Monge-Ampère case. Thus, like theirs, our argument is local and yields a quantitative estimate on the -convexity. As a result our differentiability result is new even in the optimal transport case: we weaken previously required domain convexity conditions. Moreover in the optimal transport case and the Monge-Ampère case our key assumptions, namely A3w and domain convexity, are necessary.

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A Primer on Generated Jacobian Equations: Geometry, Optics, Economics

Cited by 12 publications

References 19 publications

Optimal transport: discretization and algorithms

Optimal transport: discretization and algorithms

Strict $g$-convexity for generated Jacobian equations with applications to global regularity

Strict convexity and $C^1$ regularity of solutions to generated Jacobian equations in dimension two

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