13,996 research outputs found
Analytic Regularity and GPC Approximation for Control Problems Constrained by Linear Parametric Elliptic and Parabolic PDEs
This paper deals with linear-quadratic optimal control problems constrained by a parametric or stochastic elliptic or parabolic PDE. We address the (difficult) case that the state equation depends on a countable number of parameters i.e., on with , and that the PDE operator may depend non-affinely on the parameters. We consider tracking-type functionals and distributed as well as boundary controls. Building on recent results in [CDS1, CDS2], we show that the state and the control are analytic as functions depending on these parameters . We
establish sparsity of generalized polynomial chaos (gpc) expansions of both, state and control, in terms of the stochastic coordinate sequence of the random inputs, and prove convergence rates of best -term truncations of these expansions. Such truncations are the key for subsequent computations since they do {\em not} assume that the stochastic input data has a finite expansion. In the follow-up paper [KS2], we explain two methods how such best -term truncations can practically be computed, by greedy-type algorithms
as in [SG, Gi1], or by multilevel Monte-Carlo methods as in
[KSS]. The sparsity result allows in conjunction with adaptive wavelet Galerkin schemes for sparse, adaptive tensor discretizations of control problems constrained by linear elliptic and parabolic PDEs developed in [DK, GK, K], see [KS2]
Fully discrete finite element data assimilation method for the heat equation
We consider a finite element discretization for the reconstruction of the
final state of the heat equation, when the initial data is unknown, but
additional data is given in a sub domain in the space time. For the
discretization in space we consider standard continuous affine finite element
approximation, and the time derivative is discretized using a backward
differentiation. We regularize the discrete system by adding a penalty of the
-semi-norm of the initial data, scaled with the mesh-parameter. The
analysis of the method uses techniques developed in E. Burman and L. Oksanen,
Data assimilation for the heat equation using stabilized finite element
methods, arXiv, 2016, combining discrete stability of the numerical method with
sharp Carleman estimates for the physical problem, to derive optimal error
estimates for the approximate solution. For the natural space time energy norm,
away from , the convergence is the same as for the classical problem with
known initial data, but contrary to the classical case, we do not obtain faster
convergence for the -norm at the final time
Optimal Control of the Thermistor Problem in Three Spatial Dimensions
This paper is concerned with the state-constrained optimal control of the
three-dimensional thermistor problem, a fully quasilinear coupled system of a
parabolic and elliptic PDE with mixed boundary conditions. This system models
the heating of a conducting material by means of direct current. Local
existence, uniqueness and continuity for the state system are derived by
employing maximal parabolic regularity in the fundamental theorem of Pr\"uss.
Global solutions are addressed, which includes analysis of the linearized state
system via maximal parabolic regularity, and existence of optimal controls is
shown if the temperature gradient is under control. The adjoint system
involving measures is investigated using a duality argument. These results
allow to derive first-order necessary conditions for the optimal control
problem in form of a qualified optimality system. The theoretical findings are
illustrated by numerical results
Relaxation Methods for Mixed-Integer Optimal Control of Partial Differential Equations
We consider integer-restricted optimal control of systems governed by
abstract semilinear evolution equations. This includes the problem of optimal
control design for certain distributed parameter systems endowed with multiple
actuators, where the task is to minimize costs associated with the dynamics of
the system by choosing, for each instant in time, one of the actuators together
with ordinary controls. We consider relaxation techniques that are already used
successfully for mixed-integer optimal control of ordinary differential
equations. Our analysis yields sufficient conditions such that the optimal
value and the optimal state of the relaxed problem can be approximated with
arbitrary precision by a control satisfying the integer restrictions. The
results are obtained by semigroup theory methods. The approach is constructive
and gives rise to a numerical method. We supplement the analysis with numerical
experiments
Constrained Hyperbolic Divergence Cleaning for Smoothed Particle Magnetohydrodynamics
We present a constrained formulation of Dedner et al's hyperbolic/parabolic
divergence cleaning scheme for enforcing the \nabla\dot B = 0 constraint in
Smoothed Particle Magnetohydrodynamics (SPMHD) simulations. The constraint we
impose is that energy removed must either be conserved or dissipated, such that
the scheme is guaranteed to decrease the overall magnetic energy. This is shown
to require use of conjugate numerical operators for evaluating \nabla\dot B and
\nabla{\psi} in the SPMHD cleaning equations. The resulting scheme is shown to
be stable at density jumps and free boundaries, in contrast to an earlier
implementation by Price & Monaghan (2005). Optimal values of the damping
parameter are found to be {\sigma} = 0.2-0.3 in 2D and {\sigma} = 0.8-1.2 in
3D. With these parameters, our constrained Hamiltonian formulation is found to
provide an effective means of enforcing the divergence constraint in SPMHD,
typically maintaining average values of h |\nabla\dot B| / |B| to 0.1-1%, up to
an order of magnitude better than artificial resistivity without the associated
dissipation in the physical field. Furthermore, when applied to realistic, 3D
simulations we find an improvement of up to two orders of magnitude in momentum
conservation with a corresponding improvement in numerical stability at
essentially zero additional computational expense.Comment: 28 pages, 25 figures, accepted to J. Comput. Phys. Movies at
http://www.youtube.com/playlist?list=PL215D649FD0BDA466 v2: fixed inverted
figs 1,4,6, and several color bar
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