An adjoint-based scalable algorithm for time-parallel integration

• This paper presents a new parallel in time discretization algorithm based on a nonlinear optimization approach.
• The objective cost function quantifies the mismatch of local solutions between adjacent subintervals.
• The optimization problem is solved iteratively using gradient-based methods.
• All the computational steps – forward solutions, gradients, and Hessian-vector products – involve only ideally parallel computations and therefore are highly scalable.

As parallel architectures evolve the number of available cores continues to increase. Applications need to display a high degree of concurrency in order to effectively utilize the available resources. Large scale partial differential equations mainly rely on a spatial domain decomposition approach, where the number of parallel tasks is limited by the size of the spatial domain. Time parallelism offers a promising approach to increase the degree of concurrency. ‘Parareal’ is an iterative parallel in time algorithm that uses both low and high accuracy numerical solvers. Though the high accuracy solvers are computed in parallel, the low accuracy ones are in serial.This paper revisits the parallel in time algorithm [11] using a nonlinear optimization approach. Like in the traditional ‘Parareal’ method, the time interval is partitioned into subintervals, and local time integrations are carried out in parallel. The objective cost function quantifies the mismatch of local solutions between adjacent subintervals. The optimization problem is solved iteratively using gradient-based methods. All the computational steps – forward solutions, gradients, and Hessian-vector products – involve only ideally parallel computations and therefore are highly scalable.The feasibility of the proposed algorithm is studied on three different model problems, namely, heat equation, Arenstorf's orbit, and the Lorenz model.