Optimal transport over nonlinear systems via infinitesimal generators on graphs

Figure 1.  Approximation of infinitesimal generator $A$. The entry $A_{ij}$ is proportional to flux across $B_i\cap B_j$ from $B_i$ to $B_j$, due to vector field $f$.

Figure 2.  (A) Some minimizing geodesics to the origin in the Grushin plane. (B) Analytically computed optimal transport solution between a uniform measure whose support is the disk $\Omega = \{(x, y)|x^2+(y-.8)^2<.15^2\}$, and a measure concentrated at the origin.

Figure 3.  (A)-(E) The optimal transport solution in the Grushin plane using graph based algorithm between a measure whose support is the disk $\Omega = \{(x, y)|x^2+(y-.8)^2<.15^2\}$, and delta measure at the origin. The parameters are $m = 10^4, k = 75$. (F) Convergence of optimal transport cost with number of time discretization steps $k$ and grid size $m$.

Figure 4.  Particle trajectories with feedback control computed using Eq. (47) from the optimal transport solution in the Grushin plane. Each box contained in the support of uniform initial measure $\mu_0$ is initially populated with 4 particles.

Figure 5.  nvariant manifolds and lobe-dynamics in the double-gyre system (reproduced from Ref. [39]).

Figure 6.  Optimal transport in the periodic double gyre system (Eqs.(57a-57b)) between measures at $t = 0$ and $t_f = 1$. The parameters are $m = 1800, \Delta t = \dfrac{1}{40}$.

Figure 7.  Optimal transport in the periodic double gyre system between measures at $t = 0$ and $t_f = 10, \Delta t = \dfrac{1}{40}$. The optimal transport solution shows a quantization phenomenon. Ten 'packets' are transported via lobe-dynamics from the left side to the right side of the domain. During $2<t<5$, the third packet is transported to the right side via the sequence $F^{-1}(A)\rightarrow A\rightarrow F(A) \rightarrow F^2(A)$, where set $A$ is defined in Fig. 5. The last packet gets transported to the right side over $9<t<10$. Animation available at: https://www.youtube.com/watch?v=Pu7sCkpm4RY

Figure 8.  The cost of optimal transport between two measures supported on two AIS for the double-gyre system, as a function of time-horizon of the problem.

Figure 9.  Initial and final measures shown on $(x, y)$ plane for two cases of optimal transport in the unicycle model. The green arrows indicate the third coordinate $\theta$. (A) $\mu_0$ is supported on $(0, 0.5, 0)$, $\mu_1$ is supported on $(1, 0, 0)$ and $(1, 1, 0)$. (B) $\mu_0$ is supported on $(0, 0.5, 0)$, $\mu_1$ is supported on $(1, 0, \frac{3\pi}{2})$ and $(1, 1, \frac{\pi}{2}).$

Figure 10.  The optimal transport solution of unicycle model shown in the $x-y$ plane for case 1. The grid size is $m = 25^3$, and $k = 20$.

Figure 11.  The optimal transport solution of unicycle model shown in the $x-y$ plane for case 2. The grid size is $m = 25^3$, and $k$ = 20.

Figure 12.  Illustration of the proof of Theorem 3.7. The existence of an trajectory $f$ approximating a curve $\gamma$ connecting $B_v$ to $B_w$, obtained by piecewise constant control, is guaranteed by the small time local controllability. By continuity, this leads to non-zero transition rates, and hence strong connectivity of the control graph $\mathcal{G}_c$.