Publications

2025

1. PinT-RK

   Convergence of ParaOpt for general Runge-Kutta time discretizations

   Felix Kwok, Julien Salomon, and Djahou N. Tognon

   Mar 2025

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   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

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   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.

3. NGS-FP

   A Dynamical Neural Galerkin Scheme for Filtering Problems

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

   ESAIM: Proceedings and Surveys, Jan 2025

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   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.

4. PhD-Thesis

   Time parallelization and machine learning for optimal control and inverse problems

   Djahou N. Tognon

   Jun 2025

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   Time-parallel algorithms (PinT) such as Parareal, ParaExp, etc., are well known in the literature for their ability to exploit the parallel architecture of today’s computers to solve initial value porblem. Recently, ParaOpt, an algorithm based on the principles of the Parareal algorithm, has made it possible to parallelize the solution of optimality systems. A first analysis of the convergence of this algorithm was presented in [M.J. Gander, F. Kwok and J. Salomon, SIAM J. Sci. Comput.,42 (2020), A2773-A2802] in the restricted case of the implicit Euler method for linear quadratic optimal control problems (LQOCP) involving dissipative systems. In this thesis, we first present a convergence analysis when the system under study is unstable. Secondly, we focus on the influence of the solver used for time resolution on convergence, considering the more general case of LQOCP discretization by Runge-Kutta methods. We show that the convergence rate of ParaOpt has the same order as the Runge-Kutta time integration method used, provided that the Runge-Kutta method satisfies some additional order conditions. We then consider a preconditioning problem, for which we introduce a new PinT algorithm, this time based on the ParaExp algorithm, for LQOCP solving. Our approach is based on an overlapping time-interval decomposition in which we combine the solution of homogeneous subproblems of optimality systems using exponential propagation with local solutions of inhomogeneous subproblems. The formulation leads to a linear system whose matrix-vector product can be fully computed in parallel. We then propose two preconditioners to accelerate the convergence of GMRES in the special cases of heat and wave equations. In a final independent chapter, we analyze a learning-based model correction method. The approach studied follows the principles of Aphynity [Yuan Yin et al J. Stat. Mech. (2021) 124012], a method consisting of introducing a corrective term in the form of a neural network into the PDE under consideration. Learning is then performed in the outer loop of the solver used, so that training takes place indirectly through a time scheme. We study the influence of the solver on the resulting network, and show in particular that the order of approximation of the corrected model is equal to the order of the scheme used. Finally, we show how a Richardson-type acceleration strategy can speed up convergence by considering two smaller, independently trained networks with different time steps.

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

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   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.