| Abstract: |
| The Monge--Amp\`ere equation is a nonlinear second-order partial differential equation involving the determinant of the Hessian of an unknown function, with applications in areas such as Optimal Transport. Under appropriate conditions, the Monge--Amp\`ere equation has an equivalent Hamilton--Jacobi--Bellman (HJB) formulation in the classical and viscosity solution sense.
This talk studies a model Monge--Amp\`ere equation through its associated HJB formulation and the corresponding Stochastic Optimal Control problem. We establish the existence of strong controls for the control problem, and use a verification theorem to connect the value function to the HJB equation and hence to the original Monge--Amp\`ere equation.
Finally, we will showcase examples of using Reinforcement Learning to solve the problem numerically. |
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