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SUMMARY:Sparsity in classical and quantum optimal transport problems using
  moment constraints - Virginie Ehrlacher (ENPC - École des Ponts ParisTec
 h)
DTSTART:20240719T091500Z
DTEND:20240719T095500Z
UID:TALK218197@talks.cam.ac.uk
DESCRIPTION:Optimal Transport (OT) problems arise in a wide range of appli
 cations\, from physics to economics. Getting numerical approximate solutio
 n of these problems is a challenging issue of practical importance. In thi
 s work\, we investigate the relaxation of the OT problem when the marginal
  constraints are replaced by some moment constraints. Using Tchakaloff's t
 heorem\, we show that the Moment Constrained Optimal Transport problem (MC
 OT) is achieved by a finite discrete measure. Interestingly\, for multimar
 ginal OT problems\, the number of points weighted by this measure scales l
 inearly with the number of marginal laws\, which is encouraging to bypass 
 the curse of dimension. Interestingly\, the same type of sparsity results 
 also holds in for quantum optimal transport problems stemming from electro
 nic structure calculations. These sparsity results guided the design of ne
 w numerical schemes for the resolution of these problems which gave very i
 nteresting numerical results in high-dimensional contexts. The end of the 
 talk will be devoted to the remaining open problems related to the mathema
 tical analysis of these schemes.&nbsp\;
LOCATION:Seminar Room 1\, Newton Institute
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