687 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]
Spatial Besov Regularity for Stochastic Partial Differential Equations on Lipschitz Domains
We use the scale of Besov spaces B^\alpha_{\tau,\tau}(O), \alpha>0,
1/\tau=\alpha/d+1/p, p fixed, to study the spatial regularity of the solutions
of linear parabolic stochastic partial differential equations on bounded
Lipschitz domains O\subset R^d. The Besov smoothness determines the order of
convergence that can be achieved by nonlinear approximation schemes. The proofs
are based on a combination of weighted Sobolev estimates and characterizations
of Besov spaces by wavelet expansions.Comment: 32 pages, 3 figure
Besov regularity of solutions to the p-Poisson equation
In this paper, we study the regularity of solutions to the -Poisson
equation for all . In particular, we are interested in smoothness
estimates in the adaptivity scale , , of Besov spaces. The regularity in this scale determines the
order of approximation that can be achieved by adaptive and other nonlinear
approximation methods. It turns out that, especially for solutions to
-Poisson equations with homogeneous Dirichlet boundary conditions on bounded
polygonal domains, the Besov regularity is significantly higher than the
Sobolev regularity which justifies the use of adaptive algorithms. This type of
results is obtained by combining local H\"older with global Sobolev estimates.
In particular, we prove that intersections of locally weighted H\"older spaces
and Sobolev spaces can be continuously embedded into the specific scale of
Besov spaces we are interested in. The proof of this embedding result is based
on wavelet characterizations of Besov spaces.Comment: 45 pages, 2 figure
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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
Adaptive Low-Rank Methods for Problems on Sobolev Spaces with Error Control in
Low-rank tensor methods for the approximate solution of second-order elliptic
partial differential equations in high dimensions have recently attracted
significant attention. A critical issue is to rigorously bound the error of
such approximations, not with respect to a fixed finite dimensional discrete
background problem, but with respect to the exact solution of the continuous
problem. While the energy norm offers a natural error measure corresponding to
the underlying operator considered as an isomorphism from the energy space onto
its dual, this norm requires a careful treatment in its interplay with the
tensor structure of the problem. In this paper we build on our previous work on
energy norm-convergent subspace-based tensor schemes contriving, however, a
modified formulation which now enforces convergence only in . In order to
still be able to exploit the mapping properties of elliptic operators, a
crucial ingredient of our approach is the development and analysis of a
suitable asymmetric preconditioning scheme. We provide estimates for the
computational complexity of the resulting method in terms of the solution error
and study the practical performance of the scheme in numerical experiments. In
both regards, we find that controlling solution errors in this weaker norm
leads to substantial simplifications and to a reduction of the actual numerical
work required for a certain error tolerance.Comment: 26 pages, 7 figure
An optimal adaptive wavelet method for First Order System Least Squares
In this paper, it is shown that any well-posed 2nd order PDE can be
reformulated as a well-posed first order least squares system. This system will
be solved by an adaptive wavelet solver in optimal computational complexity.
The applications that are considered are second order elliptic PDEs with
general inhomogeneous boundary conditions, and the stationary Navier-Stokes
equations.Comment: 40 page
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