A Parallel-in-Time Gradient-Type Method for Optimal Control Problems

 

Optimal control problems governed by partial differential equations (PDEs) arise
in many science and engineering applications. These problems are optimization
problems and can be solved using gradient based methods. However, the numerical
solution is computationally expensive because computation of the gradient of
the objective function and update of the control first requires the
expensive solutions of the forward-in-time the state equation (the governing PDE)
followed by the backward-in-time adjoint PDE.

To reduce this bottleneck, I introduce a new parallel-in-time gradient type method.
The time steps are split into N groups corresponding to time subintervals. At the
time subinterval boundaries state and adjoint information from the previous iteration
is used. On each time subinterval the forward-in-time state equation is solved, the
backward-in-time adjoint equation is solved, gradient-type information is generated,
and the control are updated. These computations can be performed in parallel across
time subintervals. State and adjoint information at time subinterval boundaries
is then exchanged with neighboring subintervals and the process is repeated.

Although this method does not compute gradient information, I prove that this
gradient-like information can be used in gradient-based methods. Numerical examples
on a 3D parabolic advection diffusion control problem and on a well rate optimization
problem for a two-phase immiscible reservoir shows good speed-up of the new method.