scholarly journals Summability estimates on transport densities with Dirichlet regions on the boundary via symmetrization techniques

2018 ◽  
Vol 24 (3) ◽  
pp. 1167-1180 ◽  
Author(s):  
Samer Dweik ◽  
Filippo Santambrogio
Keyword(s):  
Shape Optimization ◽  
Bounded Domain ◽  
Transportation Problem ◽  
Mass Transportation ◽  
Surface Measure ◽  
Positive Mass ◽  
Counter Example ◽  
Diffuse Measure

In this paper we consider the mass transportation problem in a bounded domain Ω where a positive mass f+ in the interior is sent to the boundary ∂Ω. This problems appears, for instance in some shape optimization issues. We prove summability estimates on the associated transport density σ, which is the transport density from a diffuse measure to a measure on the boundary f− = P#f+ (P being the projection on the boundary), hence singular. Via a symmetrization trick, as soon as Ω is convex or satisfies a uniform exterior ball condition, we prove Lp estimates (if f+ ∈ Lp, then σ ∈ Lp). Finally, by a counter-example we prove that if f+ ∈ L∞ (Ω) and f− has bounded density w.r.t. the surface measure on ∂Ω, the transport density σ between f+ and f− is not necessarily in L∞ (Ω), which means that the fact that f− = P#f+ is crucial.

scholarly journals Design and fabrication of a freeform mirror generating a uniform illuminance distribution in a rectangular region

2020 ◽  
Vol 44 (4) ◽  
pp. 540-546
Author(s):  
E.S. Andreev ◽  
E.V. Byzov ◽  
D.A. Bykov ◽  
М.А. Moiseev ◽  
L.L. Doskolovich
Keyword(s):  
Numerical Simulations ◽  
Assignment Problem ◽  
Transportation Problem ◽  
Design Method ◽  
Experimental Measurements ◽  
Mass Transportation ◽  
Rectangular Region ◽  
Linear Assignment Problem ◽  
Linear Assignment ◽  
Good Agreement

The design of a freeform mirror generating a uniform illuminance distribution in a rectangular region with angular dimensions of 30°x15° is presented. The design method is based on the formulation of the problem of calculating the "ray-mapping" as a Monge-Kantorovich mass transportation problem and its subsequent reducing to a linear assignment problem. We describe a mirror fabrication process with the use of milling technology and present results of experimental measurements of the light distribution generated by the mirror. The experimental results are in good agreement with the results of numerical simulations and thus confirm the manufacturability of mirrors designed by the method proposed.

Optimal mass transportation problem in the design of freeform optical elements generating far-field irradiance distributions for plane incident beam

2019 ◽  
Vol 58 (33) ◽  
pp. 9131 ◽  
Author(s):  
Dmitry A. Bykov ◽  
Leonid L. Doskolovich ◽  
Albert A. Mingazov ◽  
Evgeni A. Bezus
Keyword(s):  
Transportation Problem ◽  
Incident Beam ◽  
Far Field ◽  
Mass Transportation ◽  
Optical Elements

scholarly journals Formulation of the inverse problem of calculating the optical surface for an illuminating beam with a plane wavefront as the Monge–Kantorovich problem

2019 ◽  
Vol 43 (5) ◽  
pp. 705-713 ◽  
Author(s):  
L.L. Doskolovich ◽  
A.A. Mingazov ◽  
D.A. Bykov ◽  
E.A. Bezus
Keyword(s):  
Cost Function ◽  
Transportation Problem ◽  
Specific Form ◽  
Mass Transportation ◽  
Optical Surface ◽  
Plane Wavefront ◽  
The Cost ◽  
Special Case ◽  
Illuminating Beam ◽  
Kantorovich Problem

A problem of calculating a refractive surface that forms a required irradiance distribution in the far field in the case of a plane illuminating beam is considered. We show that this problem can be formulated as a mass transportation problem. The specific form of the cost function for this problem is obtained. It is shown that with a certain choice of coordinates, the cost function becomes quadratic. The resulting mass transportation problem also describes a problem of calculating a mirror, which can be considered as a special case of the problem of calculating a refractive surface.

EXTREMAL DEPENDENCE FOR BILATERAL CREDIT VALUATION ADJUSTMENTS

2016 ◽  
Vol 19 (07) ◽  
pp. 1650042 ◽  
Author(s):  
MATTHIAS SCHERER ◽  
THORSTEN SCHULZ
Keyword(s):  
Default Risk ◽  
Transportation Problem ◽  
Parametric Model ◽  
Dependence Structure ◽  
Mass Transportation ◽  
Model Risk ◽  
Model Free ◽  
Valuation Process ◽  
Credit Valuation Adjustment ◽  
Model Free Approach

Recognizing counterparty default risk as integral part of the valuation process of financial derivatives has changed the classical view on option pricing. Calculating the bilateral credit valuation adjustment (BCVA) including wrong way risk (WWR) requires a sound model for the dependence structure between three quantities: the default times of the two contractual parties and the derivative/portfolio value at the first of the two default times. There exist various proposals, but no market consensus, on how this dependence structure should be modeled. Moreover, available mathematical tools depend strongly on the marginal models for the default times and the model for the underlying of the derivative. In practice, independence between all (or some) quantities is still a popular (over-)simplification, which completely misses the root of WWR. In any case, specifying the dependence structure imposes one to model risk and even within some parametric model one typically obtains a considerable interval of BCVA values when the parameters are taken to the extremes. In this work, we present a model-free approach to identify the dependence structure that implies the extremes of BCVA. This is achieved by solving a mass-transportation problem using tools from optimization.

scholarly journals An Explicit Solution for a Multimarginal Mass Transportation Problem

2020 ◽  
Vol 52 (4) ◽  
pp. 3666-3696
Author(s):  
Nikita A. Gladkov ◽  
Alexander P. Zimin
Keyword(s):  
Explicit Solution ◽  
Transportation Problem ◽  
Mass Transportation

scholarly journals On deterministic solutions for multi-marginal optimal transport with Coulomb cost

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ugo Bindini ◽  
Luigi De Pascale ◽  
Anna Kausamo
Keyword(s):  
Positive Result ◽  
Transportation Problem ◽  
Optimal Transport ◽  
Sufficient Condition ◽  
Mass Transportation ◽  
Necessary And Sufficient Condition ◽  
Infinite Class ◽  
Necessary And Sufficient

<p style='text-indent:20px;'>In this paper we study the three-marginal optimal mass transportation problem for the Coulomb cost on the plane <inline-formula><tex-math id="M1">\begin{document}$ \mathbb R^2 $\end{document}</tex-math></inline-formula>. The key question is the optimality of the so-called Seidl map, first disproved by Colombo and Stra. We generalize the partial positive result obtained by Colombo and Stra and give a necessary and sufficient condition for the radial Coulomb cost to coincide with a much simpler cost that corresponds to the situation where all three particles are aligned. Moreover, we produce an infinite class of regular counterexamples to the optimality of this family of maps.</p>

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