Algorithm 1.

Solution of OMT:

û ← OMTsol(μ0, μ1);

Use μ0 and μ1 to compute a mass preserving u0

while true do

Solve (3.7) for δû

line search: set α = 1

while true do

${\widehat{u}}_{n+1}=\mathcal{P}\left({\widehat{u}}_n+\mathrm{\alpha }\mathrm{\delta }\widehat{u}\right)$

$\mathrm{if}{\Vert {\widehat{u}}_{n+1}-x\Vert }_{\mu 0}<{\Vert {\widehat{u}}_n-x\Vert }_{\mu 0}\text{and}Re\left(\mathrm{\lambda }\left({\nabla }_h{u}_{n+1}\right)\right)>0\mathrm{then}$

Break

end if

α ⇐ α/2

end while

end while