The regularized OMT (rOMT) problem60 considers the minimization of the energy functional
$$ {mathcal R} [mu ,v]=,{int }_{0}^{1}mathop{int }limits_{{mathfrak{X}}}frac{1}{2}mu (t,x)||v(t,x)|{|}^{2}dxdt$$
(1)
overall time-varying densities (mu =mu (t,x)ge 0) and sufficiently smooth velocity fields (v=v(t,x)in {{mathbb{R}}}^{n},) subject to the advection/diffusion (constraint) equation
$${partial }_{t}mu +nabla cdot (mu v)=nabla cdot (Dnabla mu ),$$
(2)
for all (xin {mathfrak{X}}), a connected Euclidean subdomain of ({{mathbb{R}}}^{n}), over the normalized time interval (tin [0,1]) with initial and final conditions
$$mu (0,x)=,{mu }_{init}(x),,mu (1,x)={mu }_{final}(x),,forall xin {mathfrak{X}}.$$
(3)
Here, ({mu }_{init}) and ({mu…
