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

## Juan Cáceres, Benjamín Barán, Christian Schaerer

DOI: http://dx.doi.org/10.15439/2014F340

Citation: Proceedings of the 2014 Federated Conference on Computer Science and Information Systems, M. Ganzha, L. Maciaszek, M. Paprzycki (eds). ACSIS, Vol. 2, pages 577–586 (2014)

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.