5 research outputs found
Space-Time Discontinuous Galerkin Solution of Convection Dominated Optimal Control Problems
In this paper, a space-time discontinuous Galerkin finite element method for
distributed optimal control problems governed by unsteady
diffusion-convection-reaction equations with control constraints is studied.
Time discretization is performed by discontinuous Galerkin method with
piecewise constant and linear polynomials, while symmetric interior penalty
Galerkin with upwinding is used for space discretization. The numerical results
presented confirm the theoretically observed convergence rates.Comment: 19 pages, 4 figures, contributed talk given at "3rd European
Conference on Computational Optimization 2013
Implicit Runge-Kutta schemes for optimal control problems with evolution equations
In this paper we discuss the use of implicit Runge-Kutta schemes for the time
discretization of optimal control problems with evolution equations. The
specialty of the considered discretizations is that the discretizations schemes
for the state and adjoint state are chosen such that discretization and
optimization commute. It is well known that for Runge-Kutta schemes with this
property additional order conditions are necessary. We give sufficient
conditions for which class of schemes these additional order condition are
automatically fulfilled. The focus is especially on implicit Runge-Kutta
schemes of Gauss, Radau IA, Radau IIA, Lobatto IIIA, Lobatto IIIB and Lobatto
IIIC collocation type up to order six. Furthermore we also use a SDIRK (singly
diagonally implicit Runge-Kutta) method to demonstrate, that for general
implicit Runge-Kutta methods the additional order conditions are not
automatically fulfilled.
Numerical examples illustrate the predicted convergence rates
A finite element method for Dirichlet boundary control problems governed by parabolic PDEs
Finite element approximations of Dirichlet boundary control problems governed
by parabolic PDEs on convex polygonal domains are studied in this paper. The
existence of a unique solution to optimal control problems is guaranteed based
on very weak solution of the state equation and as
control space. For the numerical discretization of the state equation we use
standard piecewise linear and continuous finite elements for the space
discretization of the state, while a dG(0) scheme is used for time
discretization. The Dirichlet boundary control is realized through a space-time
-projection. We consider both piecewise linear, continuous finite element
approximation and variational discretization for the controls and derive a
priori -error bounds for controls and states. We finally present numerical
examples to support our theoretical findings
Non-commutative Discretize-then-Optimize Algorithms for Elliptic PDE-Constrained Optimal Control Problems
In this paper, we analyze the convergence of several discretize-then-optimize
algorithms, based on either a second-order or a fourth-order finite difference
discretization, for solving elliptic PDE-constrained optimization or optimal
control problems. To ensure the convergence of a discretize-then-optimize
algorithm, one well-accepted criterion is to choose or redesign the
discretization scheme such that the resultant discretize-then-optimize
algorithm commutes with the corresponding optimize-then-discretize algorithm.
In other words, both types of algorithms would give rise to exactly the same
discrete optimality system. However, such an approach is not trivial. In this
work, by investigating a simple distributed elliptic optimal control problem,
we first show that enforcing such a stringent condition of commutative property
is only sufficient but not necessary for achieving the desired convergence. We
then propose to add some suitable semi-norm penalty/regularization terms
to recover the lost convergence due to the inconsistency caused by the loss of
commutativity. Numerical experiments are carried out to verify our theoretical
analysis and also validate the effectiveness of our proposed regularization
techniques.Comment: Revised on Aug 1, 2018. To appear in Journal of Computational and
Applied Mathematic
Fast Iterative Solver for the Optimal Control of Time-Dependent PDEs with Crank-Nicolson Discretization in Time
In this article, we derive a new, fast, and robust preconditioned iterative
solution strategy for the all-at-once solution of optimal control problems with
time-dependent PDEs as constraints, including the heat equation and the
non-steady convection--diffusion equation. After applying an
optimize-then-discretize approach, one is faced with continuous first-order
optimality conditions consisting of a coupled system of PDEs. As opposed to
most work in preconditioning the resulting discretized systems, where a
(first-order accurate) backward Euler method is used for the discretization of
the time derivative, we employ a (second-order accurate) Crank--Nicolson method
in time. We apply a carefully tailored invertible transformation for
symmetrizing the matrix, and then derive an optimal preconditioner for the
saddle-point system obtained. The key components of this preconditioner are an
accurate mass matrix approximation, a good approximation of the Schur
complement, and an appropriate multigrid process to apply this latter
approximation---these are constructed using our work in transforming the matrix
system. We prove the optimality of the approximation of the Schur complement
through bounds on the eigenvalues, and test our solver against a widely-used
preconditioner for the linear system arising from a backward Euler
discretization. These demonstrate the effectiveness and robustness of our
solver with respect to mesh-sizes, regularization parameter, and diffusion
coefficient