Implementation of a distributed parallel in time scheme using PETSc for a Parabolic Optimal Control Problem

Abstract

This work presents a parallel implementation of the Parareal method using Portable Extensible Toolkit for Scientific Computation (PETSc). An optimal control problem of a parabolic partial differential equation with known boundary conditions and initial state is solved, where the minimized cost function relates the controller v usage and the approximation of the solution y to an optimal known function y*, measured by ∥y∥ and ∥y*∥, respectively. The equations that model the process are discretized in space using Finite Elements and in time using Finite Differences. After the discretizations, the problem is transformed to a large linear system of algebraic equations, that is solved by the Conjugate Gradient method. A Parareal preconditioner is implemented to speed up the convergence of the Conjugate Gradient. The main advantage in using the Parareal approach is to speed up the resolution time, when comparing to implementations that use only the Conjugate Gradient or GMRES methods. The implementation developed in this work offers a parallelization relative efficiency for the strong scaling of approximately 70% each time the process count doubles. For weak scaling, 75% each time the process count doubles for a constant solution size per process and 96% each time the process count doubles for a constant data size per process.