The $L^1$ optimal transport density $\mu^*$ is the unique $L^\infty$ solution of the Monge-Kantorovich equations. It has been recently characterized also as the unique minimizer of the $L^1$-transport energy functional $\mathcal E$. In the present work we develop and we prove convergence of a numerical approximation scheme for $\mu^*.$ Our approach relies upon the combination of a FEM-inspired variational approximation of $\Ene$ with a minimization algorithm based on a gradient flow method.
We introduce the transport energy functional E (a variant of the Bouchitté–Buttazzo–Seppecher shape optimization functional) and prove that its unique minimizer is the optimal transport density μ∗, i.e., the solution of Monge–Kantorovich partial differential equations. We study the gradient flow of E and show that μ∗ is the unique global attractor of the flow. Next, we introduce a two parameter family {Eλ,δ}λ,δ>0 of strictly convex regularized functionals approximating E and prove the convergence of the minimizers μλ,δ∗ of Eλ,δ to μ∗ as we let δ→ 0 + and λ→ 0 +. We derive an evolution system of fully non-linear PDEs as the gradient flow of Eλ,δ in L2, showing existence and uniqueness of the solution for all times. We are able to prove that the trajectories of the flow converge in W01,p to the unique minimizer μλ,δ∗ of Eλ,δ. This allows us to characterize μλ,δ∗ by a non-linear system of PDEs that turns out to be a perturbation of the Monge–Kantorovich equations by a p-Laplacian.
$L^1$ Transport Energy
Facca E;
2022
Abstract
We introduce the transport energy functional E (a variant of the Bouchitté–Buttazzo–Seppecher shape optimization functional) and prove that its unique minimizer is the optimal transport density μ∗, i.e., the solution of Monge–Kantorovich partial differential equations. We study the gradient flow of E and show that μ∗ is the unique global attractor of the flow. Next, we introduce a two parameter family {Eλ,δ}λ,δ>0 of strictly convex regularized functionals approximating E and prove the convergence of the minimizers μλ,δ∗ of Eλ,δ to μ∗ as we let δ→ 0 + and λ→ 0 +. We derive an evolution system of fully non-linear PDEs as the gradient flow of Eλ,δ in L2, showing existence and uniqueness of the solution for all times. We are able to prove that the trajectories of the flow converge in W01,p to the unique minimizer μλ,δ∗ of Eλ,δ. This allows us to characterize μλ,δ∗ by a non-linear system of PDEs that turns out to be a perturbation of the Monge–Kantorovich equations by a p-Laplacian.I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
