1,343 research outputs found
A New HDG Method for Dirichlet Boundary Control of Convection Diffusion PDEs II: Low Regularity
In the first part of this work, we analyzed a Dirichlet boundary control
problem for an elliptic convection diffusion PDE and proposed a new
hybridizable discontinuous Galerkin (HDG) method to approximate the solution.
For the case of a 2D polygonal domain, we also proved an optimal superlinear
convergence rate for the control under certain assumptions on the domain and on
the target state. In this work, we revisit the convergence analysis without
these assumptions; in this case, the solution can have low regularity and we
use a different analysis approach. We again prove an optimal convergence rate
for the control, and present numerical results to illustrate the convergence
theory
A Class of Embedded DG Methods for Dirichlet Boundary Control of Convection Diffusion PDEs
We investigated an hybridizable discontinuous Galerkin (HDG) method for a
convection diffusion Dirichlet boundary control problem in our earlier work
[SIAM J. Numer. Anal. 56 (2018) 2262-2287] and obtained an optimal convergence
rate for the control under some assumptions on the desired state and the
domain. In this work, we obtain the same convergence rate for the control using
a class of embedded DG methods proposed by Nguyen, Peraire and Cockburn [J.
Comput. Phys. vol. 302 (2015), pp. 674-692] for simulating fluid flows. Since
the global system for embedded DG methods uses continuous elements, the number
of degrees of freedom for the embedded DG methods are smaller than the HDG
method, which uses discontinuous elements for the global system. Moreover, we
introduce a new simpler numerical analysis technique to handle low regularity
solutions of the boundary control problem. We present some numerical
experiments to confirm our theoretical results
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]
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