69 research outputs found
Parareal in time intermediate targets methods for optimal control problem
In this paper, we present a method that enables solving in parallel the
Euler-Lagrange system associated with the optimal control of a parabolic
equation. Our approach is based on an iterative update of a sequence of
intermediate targets that gives rise to independent sub-problems that can be
solved in parallel. This method can be coupled with the parareal in time
algorithm. Numerical experiments show the efficiency of our method.Comment: 14 page
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Mini-Workshop: Adaptive Methods for Control Problems Constrained by Time-Dependent PDEs
Optimization problems constrained by time-dependent PDEs (Partial Differential Equations) are challenging from a computational point of view: even in the simplest case, one needs to solve a system of PDEs coupled globally in time and space for the unknown solutions (the state, the costate and the control of the system). Typical and practically relevant examples are the control of nonlinear heat equations as they appear in laser hardening or the thermic control of flow problems (Boussinesq equations). Specifically for PDEs with a long time horizon, conventional time-stepping methods require an enormous storage of the respective other variables. In contrast, adaptive methods aim at distributing the available degrees of freedom in an a-posteriori-fashion to capture singularities and are, therefore, most promising
Diagonalization-based preconditioners and generalized convergence bounds for ParaOpt
The ParaOpt algorithm was recently introduced as a time-parallel solver for
optimal-control problems with a terminal-cost objective, and convergence
results have been presented for the linear diffusive case with implicit-Euler
time integrators. We reformulate ParaOpt for tracking problems and provide
generalized convergence analyses for both objectives. We focus on linear
diffusive equations and prove convergence bounds that are generic in the time
integrators used. For large problem dimensions, ParaOpt's performance depends
crucially on having a good preconditioner to solve the arising linear systems.
For the case where ParaOpt's cheap, coarse-grained propagator is linear, we
introduce diagonalization-based preconditioners, inspired by recent advances in
the ParaDiag family of methods. These preconditioners not only lead to a
weakly-scalable ParaOpt version, but are themselves invertible in parallel,
making maximal use of available concurrency. They have proven convergence
properties in the linear diffusive case that are generic in the time
discretization used, similarly to our ParaOpt results. Numerical results
confirm that the iteration count of the iterative solvers used for ParaOpt's
linear systems becomes constant in the limit of an increasing processor count.
The paper is accompanied by a sequential MATLAB implementation
Parareal in time 3D numerical solver for the LWR Benchmark neutron diffusion transient model
We present a parareal in time algorithm for the simulation of neutron
diffusion transient model. The method is made efficient by means of a coarse
solver defined with large time steps and steady control rods model. Using
finite element for the space discretization, our implementation provides a good
scalability of the algorithm. Numerical results show the efficiency of the
parareal method on large light water reactor transient model corresponding to
the Langenbuch-Maurer-Werner (LMW) benchmark [1]
Space-time block preconditioning for incompressible flow
Parallel-in-time methods have become increasingly popular in the simulation
of time-dependent numerical PDEs, allowing for the efficient use of additional
MPI processes when spatial parallelism saturates. Most methods treat the
solution and parallelism in space and time separately. In contrast, all-at-once
methods solve the full space-time system directly, largely treating time as
simply another spatial dimension. All-at-once methods offer a number of
benefits over separate treatment of space and time, most notably significantly
increased parallelism and faster time-to-solution (when applicable). However,
the development of fast, scalable all-at-once methods has largely been limited
to time-dependent (advection-)diffusion problems. This paper introduces the
concept of space-time block preconditioning for the all-at-once solution of
incompressible flow. By extending well-known concepts of spatial block
preconditioning to the space-time setting, we develop a block preconditioner
whose application requires the solution of a space-time (advection-)diffusion
equation in the velocity block, coupled with a pressure Schur complement
approximation consisting of independent spatial solves at each time-step, and a
space-time matrix-vector multiplication. The new method is tested on four
classical models in incompressible flow. Results indicate perfect scalability
in refinement of spatial and temporal mesh spacing, perfect scalability in
nonlinear Picard iterations count when applied to a nonlinear Navier-Stokes
problem, and minimal overhead in terms of number of preconditioner applications
compared with sequential time-stepping.Comment: 28 pages, 7 figures, 4 table
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