A new parallel-in-time gradient type method for the solution of time dependent optimal control problems is introduced. Each iteration of the classical gradient method requires the solution of the forward-in-time state equation followed by the solution of the backward-in-time adjoint equation to compute the gradient. To introduce parallelism, 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. Applied to a finite dimensional convex linear quadratic discrete time optimal control (DTOC) problem, this method can be interpreted as a so-called (2N-1)-part iteration scheme. Convergence of this new method applied to convex linear quadratic DTOC problems is proven for sufficiently small step sizes. Numerical examples on a 3D parabolic advection diffusion control problem and on a well rate optimization problem for a two-phase immiscible reservoir show good speed-up of the new method.

This is joint work with Xiaodi Deng.