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Register a new userLinear multistep methods for optimal control problems and applications to hyperbolic relaxation systems
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We are interested in high-order linear multistep schemes for time discretization of adjoint equations arising within optimal control problems. First we consider optimal control problems for ordinary differential equations and show loss of accuracy for Adams-Moulton and Adams-Bashford methods, whereas BDF methods preserve high--order accuracy. Subsequently we extend these results to semi--lagrangian discretizations of hyperbolic relaxation systems. Computational results illustrate theoretical findings.
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We consider the development of high order space and time numerical methods based on Implicit-Explicit (IMEX) multistep time integrators for hyperbolic systems with relaxation. More specifically, we consider hyperbolic balance laws in which the convection and the source term may have very different time and space scales. As a consequence the nature of the asymptotic limit changes completely, passing from a hyperbolic to a parabolic system. From the computational point of view, standard numerical methods designed for the fluid-dynamic scaling of hyperbolic systems with relaxation present several drawbacks and typically lose efficiency in describing the parabolic limit regime. In this work, in the context of Implicit-Explicit linear multistep methods we construct high order space-time discretizations which are able to handle all the different scales and to capture the correct asymptotic behavior, independently from its nature, without time step restrictions imposed by the fast scales. Several numerical examples confirm the theoretical analysis.
We consider the development of implicit-explicit time integration schemes for optimal control problems governed by the Goldstein-Taylor model. In the diffusive scaling this model is a hyperbolic approximation to the heat equation. We investigate the relation of time integration schemes and the formal Chapman-Enskog type limiting procedure. For the class of stiffly accurate implicit-explicit Runge-Kutta methods (IMEX) the discrete optimality system also provides a stable numerical method for optimal control problems governed by the heat equation. Numerical examples illustrate the expected behavior.
Time integration methods for solving initial value problems are an important component of many scientific and engineering simulations. Implicit time integrators are desirable for their stability properties, significantly relaxing restrictions on timestep size. However, implicit methods require solutions to one or more systems of nonlinear equations at each timestep, which for large simulations can be prohibitively expensive. This paper introduces a new family of linearly implicit multistep methods (LIMM), which only requires the solution of one linear system per timestep. Order conditions and stability theory for these methods are presented, as well as design and implementation considerations. Practical methods of order up to five are developed that have similar error coefficients, but improved stability regions, when compared to the widely used BDF methods. Numerical testing of a self-starting variable stepsize and variable order implementation of the new LIMM methods shows measurable performance improvement over a similar BDF implementation.
Identifying hidden dynamics from observed data is a significant and challenging task in a wide range of applications. Recently, the combination of linear multistep methods (LMMs) and deep learning has been successfully employed to discover dynamics, whereas a complete convergence analysis of this approach is still under development. In this work, we consider the deep network-based LMMs for the discovery of dynamics. We put forward error estimates for these methods using the approximation property of deep networks. It indicates, for certain families of LMMs, that the $ell^2$ grid error is bounded by the sum of $O(h^p)$ and the network approximation error, where $h$ is the time step size and $p$ is the local truncation error order. Numerical results of several physically relevant examples are provided to demonstrate our theory.
This work is concerned with the optimal control problems governed by a 1D wave equation with variable coefficients and the control spaces $mathcal M_T$ of either measure-valued functions $L_{w^*}^2(I,mathcal M(Omega))$ or vector measures $mathcal M(Omega,L^2(I))$. The cost functional involves the standard quadratic tracking terms and the regularization term $alpha|u|_{mathcal M_T}$ with $alpha>0$. We construct and study three-level in time bilinear finite element discretizations for this class of problems. The main focus lies on the derivation of error estimates for the optimal state variable and the error measured in the cost functional. The analysis is mainly based on some previous results of the authors. The numerical results are included.
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