Symmetric Monge–Kantorovich problems and polar decompositions of vector fields

Ghoussoub, Nassif; Moameni, Abbas

doi:10.1007/s00039-014-0287-2

Cited by 30 publications

(38 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the partition of the identity in (14) and in Theorem 2.5(iv) we conclude from (16) and Theorem 2.5(v) that any partial sum of the firmly nonexpansive mappings is also firmly nonexpansive. This is not the case for general partitions of the identity into sums of firmly nonexpansive mappings; indeed, an example where partial sums of a partition of the identity into firmly nonexpansive mappings fail to be firmly nonexpansive is provided in [4,Example 4.4].…”

Section: Remark 27mentioning

confidence: 72%

See 1 more Smart Citation

Multi-marginal maximal monotonicity and convex analysis

et al. 2019

View full text Add to dashboard Cite

Monotonicity and convex analysis arise naturally in the framework of multi-marginal optimal transport theory. However, a comprehensive multi-marginal monotonicity and convex analysis theory is still missing. To this end we study extensions of classical monotone operator theory and convex analysis into the multi-marginal setting. We characterize multi-marginal c-monotonicity in terms of classical monotonicity and firmly nonexpansive mappings. We provide Minty type, continuity and conjugacy criteria for multi-marginal maximal monotonicity. We extend the partition of the identity into a sum of firmly nonexpansive mappings and Moreau's decomposition of the quadratic function into envelopes and proximal mappings into the multi-marginal settings. We illustrate our discussion with examples and provide applications for the determination of multi-marginal maximal monotonicity and multi-marginal conjugacy. We also point out several open questions.2010 Mathematics Subject Classification: Primary 47H05, 26B25; Secondary 49N15, 49K30, 52A01, 91B68.

show abstract

Section: Remark 27mentioning

confidence: 72%

“…Finally, (14), (15) and (16) In order to extend our discussion of these formulas into the multi-marginal settings we will employ the following definitions and notations. We denote by ∆ the subset of X = X 1 × · · · × X N defined by…”

Section: A Characterization Of Multi-marginal C-monotonicity and Mintmentioning

confidence: 99%

Multi-marginal maximal monotonicity and convex analysis

et al. 2019

View full text Add to dashboard Cite

show abstract

“…. Interestingly, such multi-marginal Monge states generated by a single permutation have previously appeared in continuous optimal transport problems [GM13,CDD13]. Any Monge state not generated by a single permutation, i.e.…”

Section: Convex Geometry Of the Set Of Kantorovich Plansmentioning

confidence: 99%

A Simple Counterexample to the Monge Ansatz in Multimarginal Optimal Transport, Convex Geometry of the Set of Kantorovich Plans, and the Frenkel--Kontorova Model

Friesecke¹

2019

SIAM J. Math. Anal.

View full text Add to dashboard Cite

It is known from clever mathematical examples [Ca10] that the Monge ansatz may fail in continuous two-marginal optimal transport (alias optimal coupling alias optimal assignment) problems. Here we show that this effect already occurs for finite assignment problems with N = 3 marginals, = 3 'sites', and symmetric pairwise costs, with the values for N and both being optimal. Our counterexample is a transparent consequence of the convex geometry of the set of symmetric Kantorovich plans for N = = 3, which -as we show -possess 22 extreme points, only 7 of which are Monge. These extreme points have a simple physical meaning as irreducible molecular packings, and the example corresponds to finding the minimum energy packing for Frenkel-Kontorova interactions. Our finite example naturally gives rise, by superposition, to a continuous one, where failure of the Monge ansatz manifests itself as nonattainment and formation of 'microstructure'.

show abstract

“…The construction of V SCP int [n] for a given density n(r) is equivalent to an optimal transport (or mass transportation theory, a wellestablished field of mathematics and economics) problem with cost given by the interaction [41,42]. While several rigorous results have appeared recently in the mathematics literature [43][44][45][46][47][48][49][50], here we provide a simplified physical overview. The idea is that if we minimize the expectation of the interparticle interaction in a given density n(r), we must have a non-zero probability to find one particle wherever n(r) = 0.…”

mentioning

confidence: 99%

“…A few remarks are necessary on the kind of interactions v int (r) for which the KS-SCP DFT can be applied. Several rigorous results are available for convex repulsive long-ranged interactions depending on |r| only [34,40,[45][46][47][48][49][50]. In general, for the SCP formalism to be physically useful, the interaction v int (r) needs to be long-ranged, otherwise the SCP solution of Eq.…”

mentioning

confidence: 99%

Density-Functional Theory for Strongly Correlated Bosonic and Fermionic Ultracold Dipolar and Ionic Gases

et al. 2015

View full text Add to dashboard Cite

We introduce a density functional formalism to study the ground-state properties of stronglycorrelated dipolar and ionic ultracold bosonic and fermionic gases, based on the self-consistent combination of the weak and the strong coupling limits. Contrary to conventional density functional approaches, our formalism does not require a previous calculation of the interacting homogeneous gas, and it is thus very suitable to treat systems with tunable long-range interactions. Due to its asymptotic exactness in the regime of strong correlation, the formalism works for systems in which standard mean-field theories fail.Introduction -In contrast with its widespread use and success in areas as diverse as quantum chemistry [1], materials science [2] or semiconductor nanostructures [3], Density Functional Theory (DFT) has received relatively little attention in the very active field of ultracold atomic gases. It is well known that the Hohenberg-Kohn theorems, originally formulated in terms of the electron gas [4,5], hold for both fermionic and bosonic systems, as well as for interactions different than the Coulomb one. However, the lack of adequate density functionals has hindered the role of DFT in the study of ultracold atomic gases in favour of other well-established approaches, such as the widely used Gross-Pitaevskii (GP) method in the case of Bose gases. The latter is a mean-field approach and does not allow treating the effect of correlations, which play a crucial role in many different phenomena occurring in ultracold quantum gases [6]. One then often turns to configuration-interaction (CI), quantum Monte Carlo (QMC) or Green's-function methods (for recent reviews, see, e.g., Refs. 6-8).

show abstract

Symmetric Monge–Kantorovich problems and polar decompositions of vector fields

Cited by 30 publications

References 18 publications

Multi-marginal maximal monotonicity and convex analysis

Multi-marginal maximal monotonicity and convex analysis

A Simple Counterexample to the Monge Ansatz in Multimarginal Optimal Transport, Convex Geometry of the Set of Kantorovich Plans, and the Frenkel--Kontorova Model

Density-Functional Theory for Strongly Correlated Bosonic and Fermionic Ultracold Dipolar and Ionic Gases

Contact Info

Product

Resources

About