Publications

2025

1. PinT-RK

   Convergence of ParaOpt for general Runge-Kutta time discretizations

   Felix Kwok, Julien Salomon, and Djahou N. Tognon

   Mar 2025

   Abs PDF

   ParaOpt is a time parallel method based on Parareal for solving optimality systems arising in optimal control problems. The method was presented in [M.J. Gander, F. Kwok and J. Salomon, SIAM J. Sci. Comput., 42 (2020), A2773–A2802] together with a convergence analysis in the case where implicit Euler is used to discretize the differential equations governing the system dynamics. However, its convergence behaviour for higher order time discretizations has not been considered. In this paper, we use an operator norm analysis to prove that the convergence rate of ParaOpt applied to a linear-quadratic optimal control problem has the same order as the Runge-Kutta time integration method used, provided that a few auxiliary order conditions are satisfied. We illustrate our theoretical results with numerical examples, before showing an additional test case not covered by our analysis, namely, a nonlinear optimal control problem involving a Schrödinger type system.

2. PinT-US

   ParaOpt for Unstable Systems

   Djahou N. Tognon

   Feb 2025

   Abs PDF

   ParaOpt is a two-level time-parallel method to solve the coupled forward/backward Euler-Lagrange system arising from partial differential equations (PDEs) constrained optimization. In this work, we present a convergence analysis of this algorithm in the case where the system under consideration is unstable. We complete this theoretical study with numerical experiments, where the properties of the algorithm are investigated on linear and nonlinear examples.

2024

1. PinT-OCP

   A Parallel in Time Algorithm Based on ParaExp for Optimal Control Problems

   Felix Kwok, and Djahou N. Tognon

   2024 IEEE 63rd Conference on Decision and Control (CDC), Milan, Italy, Dec 2024

   Abs PDF

   We propose a new parallel-in-time algorithm for solving optimal control problems constrained by discretized partial differential equations. Our approach, which is based on a deeper understanding of ParaExp, considers an overlapping time-domain decomposition in which we combine the solution of homogeneous problems using exponential propagation with the local solutions of inhomogeneous problems. The algorithm yields a linear system whose matrix-vector product can be fully performed in parallel. We then propose a preconditioner to speed up the convergence of GMRES in the special cases of the heat and wave equations. Numerical experiments are provided to illustrate the efficiency of our preconditioners.

2. NGS-FP

   A Dynamical Neural Galerkin Scheme for Filtering Problems

   Joubine Aghili, Joy Atokple Zialesi, Marie Billaud-Friess, and 3 more authors

   Jan 2024

   Abs arXiv

   This paper considers the filtering problem which consists in reconstructing the state of a dynamical system with partial observations coming from sensor measurements, and the knowledge that the dynamics are governed by a physical PDE model with unknown parameters. We present a filtering algorithm where the reconstruction of the dynamics is done with neural network approximations whose weights are dynamically updated using observational data. In addition to the estimate of the state, we also obtain time-dependent parameter estimations of the PDE parameters governing the observed evolution. We illustrate the behavior of the method in a one-dimensional KdV equation involving the transport of solutions with local support. Our numerical investigation reveals the importance of the location and number of the observations. In particular, it suggests to consider dynamical sensor placement.